The ๐ฃ1-periodic part of the Adams spectral sequenceat an odd prime
by
Michael Joseph Andrews
MMath, University of Oxford (2009)
Submitted to the Department of Mathematicsin partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
c Massachusetts Institute of Technology 2015. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Mathematics
April 29, 2015
Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Haynes Miller
Professor of MathematicsThesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .William Minicozzi
Chairman, Department Committee on Graduate Students
2
The ๐ฃ1-periodic part of the Adams spectral sequence at an
odd prime
by
Michael Joseph Andrews
Submitted to the Department of Mathematicson April 29, 2015, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
Abstract
We tell the story of the stable homotopy groups of spheres for odd primes at chromaticheight 1 through the lens of the Adams spectral sequence. We find the โdancers to adiscordant system.โ
We calculate a Bockstein spectral sequence which converges to the 1-line of thechromatic spectral sequence for the odd primary Adams ๐ธ2-page. Furthermore, wecalculate the associated algebraic Novikov spectral sequence converging to the 1-lineof the ๐ต๐ chromatic spectral sequence. This result is also viewed as the calculationof a direct limit of localized modified Adams spectral sequences converging to thehomotopy of the ๐ฃ1-periodic sphere spectrum.
As a consequence of this work, we obtain a thorough understanding of a collectionof ๐0-towers on the Adams ๐ธ2-page and we obtain information about the differentialsbetween these towers. Moreover, above a line of slope 1/(๐2โ๐โ1) we can completelydescribe the ๐ธ2 and ๐ธ3-pages of the mod ๐ Adams spectral sequence, which accountsfor almost all the spectral sequence in this range.
Thesis Supervisor: Haynes MillerTitle: Professor of Mathematics
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4
Acknowledgments
Without the support of my mother and my advisor, Haynes, I have no doubt that
this thesis would have ceased to exist.
There are many things I would like to thank my mother for. Most relevant is the
time she dragged me to Oxford. I had decided, at sixteen years of age, that I was not
interested in going to Oxbridge for undergraduate study but she knew better. Upon
visiting Oxford, I experienced for the first time the wonder of being able to speak to
others who love maths as much as I do. My time there was mathematically fulfilling
and the friends I made, I hope, will be lifelong. Secondly, it was her who encouraged
me to apply to MIT for grad school. Thereโs no other way to put it, I was terrified
of moving abroad and away from the friends I had made. I would come to be the
happiest I could ever have been at MIT. Cambridge is a beautiful place to live and
the energy of the faculty and students at MIT is untouched by many institutes. Her
support during my first year away, during the struggle of qualifying exams, from over
3, 000 miles away, and throughout the rest of my life is never forgotten.
Haynes picked up the pieces many times during my first year at MIT. His emotional
support and kindness in those moments are the reasons I chose him to be my advisor.
He has always been a pleasure to talk with and I am particularly appreciative of
how he adapted to my requirements, always giving me the level of detail he knows
I need, while holding back enough so that our conversations remain exciting. His
mathematical influence is evident throughout this thesis. In particular, theorem 1.4.4
was his conjecture and the results of this thesis build on his work in [10] and [11].
It has been a pleasure to collaborate with him in subsequent work [2]. On the other
hand, it is wonderful to have an advisor that I consider a friend and who I can talk
to about things other than math. I will always remember him giving me strict orders
to go out and buy a guitar amp when he could tell I was suffering without. Thank
you, Haynes.
There are many friends to thank for their support during my time at MIT. I am
grateful to Michael, Dana, Jiayong, and Saul, particularly for their support during
5
my first year at MIT. I am grateful to Rosa, Stuart, Pat and Nate for putting up with
me as a roommate. Thank you, Nisa. I have been a better person since knowing you.
Thank you, Alex, for letting me talk your ear off about permanent cycles for months
and months, and for being the best friend one could hope for.
The weeks I spent collaborating with Will as he coded up spectral sequence charts
were some of my most enjoyable as a mathematician. Before Willโs work, no-one had
seen a trigraded spectral sequence plotted in 3D with rotation capability, or a 70 term
cocycle representative for ๐0. His programs made for a particularly memorable thesis
defense and will be useful for topologists for a long time, I am sure. Thank you, Will.
There are three courses I feel very lucky to have been a part of during my time at
MIT. They were taught by Haynes, Mark Behrens and Emily Riehl. Haynesโ course
on the Adams spectral sequence was the birthplace for this thesis.
Mark taught the best introductory algebraic topology course that you can imagine.
It was inspiring for my development as a topologist and a teacher. I wish to thank
him for the advice he gave me during the microteaching workshop, the conversations
we have had about topology and for the energy he brings to everything he is involved
in. I hope we will work together more in the future.
Emily made sure that I finally learned some categorical homotopy theory. Each
one of her classes was like watching Usain Bolt run the 100m over and over again for
an hour. They were incredible. I thank her for her rigour, her energy and for showing
me that abstract nonsense done right is beautiful. Although, the final version of this
thesis contains less categorical homotopy theory than the draft, her course gave me
the tools I needed to prove proposition 8.1.9.
Finally, I wish to thank Jessica Barton for her support, John Wilson who made it
possible for me to take my GRE exams, and Yan Zhang who helped me plot pictures
of my spectral sequences, which inspired the proof of proposition 5.4.5.
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Contents
1 Introduction 11
1.1 The stable homotopy groups of spheres . . . . . . . . . . . . . . . . . 11
1.2 Calculational tools in homotopy theory . . . . . . . . . . . . . . . . . 12
1.3 Some ๐ต๐*๐ต๐ -comodules and the corresponding ๐ -comodules . . . . 14
1.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Spectral sequence terminology 25
2.1 A correspondence approach . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Bockstein spectral sequences 33
3.1 The Hopf algebra ๐ and some ๐ -comodules . . . . . . . . . . . . . . 33
3.2 The ๐-Bockstein spectral sequence (๐-BSS) . . . . . . . . . . . . . . 35
3.3 The ๐โ0 -Bockstein spectral sequence (๐โ0 -BSS) . . . . . . . . . . . . . 38
3.4 The ๐-BSS and the ๐โ0 -BSS: a relationship . . . . . . . . . . . . . . . 39
3.5 The ๐โ11 -Bockstein spectral sequence (๐โ1
1 -BSS) . . . . . . . . . . . . 41
3.6 Multiplicativity of the BSSs . . . . . . . . . . . . . . . . . . . . . . . 43
4 Vanishing lines and localization 47
4.1 Vanishing lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 The localization map: the trigraded perspective . . . . . . . . . . . . 49
4.3 The localization map: the bigraded perspective . . . . . . . . . . . . 50
7
5 Calculating the 1-line of the ๐-CSS; its image in ๐ป*(๐ด) 53
5.1 The ๐ธ1-page of the ๐โ11 -BSS . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 The first family of differentials, principal towers . . . . . . . . . . . . 55
5.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.2 Quick proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.3 The proof of proposition 5.2.2.1 . . . . . . . . . . . . . . . . . 56
5.3 The second family of differentials, side towers . . . . . . . . . . . . . 64
5.3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.2 Quick proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3.3 A Kudo transgression theorem . . . . . . . . . . . . . . . . . . 65
5.3.4 Completing the proof of proposition 5.3.1.2 . . . . . . . . . . . 72
5.4 The ๐ธโ-page of the ๐โ11 -BSS . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 The localized algebraic Novikov spectral sequence 81
6.1 Algebraic Novikov spectral sequences . . . . . . . . . . . . . . . . . . 81
6.2 Evidence for the main result . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 The filtration spectral sequence (๐0-FILT) . . . . . . . . . . . . . . . 84
6.4 The ๐ธโ-page of the loc.alg.NSS . . . . . . . . . . . . . . . . . . . . . 88
7 Some permanent cycles in the ASS 91
7.1 Maps between stunted projective spaces . . . . . . . . . . . . . . . . 91
7.2 Homotopy and cohomotopy classes in stunted projective spaces . . . . 98
7.3 A permanent cycle in the ASS . . . . . . . . . . . . . . . . . . . . . . 102
8 Adams spectral sequences 105
8.1 Towers and their spectral sequences . . . . . . . . . . . . . . . . . . . 105
8.2 The modified Adams spectral sequence for ๐/๐๐ . . . . . . . . . . . . 112
8.3 The modified Adams spectral sequence for ๐/๐โ . . . . . . . . . . . . 115
8.4 A permanent cycle in the MASS-(๐+ 1) . . . . . . . . . . . . . . . . 116
8.5 The localized Adams spectral sequences . . . . . . . . . . . . . . . . . 117
8
8.6 Calculating the LASS-โ . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.7 The Adams spectral sequence . . . . . . . . . . . . . . . . . . . . . . 120
A Maps of spectral sequences 123
B Convergence of spectral sequences 129
9
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Chapter 1
Introduction
1.1 The stable homotopy groups of spheres
Algebraic topologists are interested in the class of spaces which can be built from
spheres. For this reason, when one tries to understand the continuous maps between
two spaces up to homotopy, it is natural to restrict attention to the maps between
spheres first. The groups of interest
๐๐+๐(๐๐) = homotopy classes of maps ๐๐+๐ โโ ๐๐
are called the homotopy groups of spheres.
Topologists soon realized that it is easier to work in a stable setting. Instead,
one asks about the stable homotopy groups of spheres or, equivalently, the homotopy
groups of the sphere spectrum
๐๐(๐0) = colim๐ ๐๐+๐(๐๐).
Calculating all of these groups is an impossible task but one can ask for partial
information. In particular, one can try to understand the global structure of these
groups by proving the existence of recurring patterns. These patterns are clearly
visible in spectral sequence charts for calculating ๐*(๐0) and this thesis came about
11
because of the authorโs desire to understand the mystery behind these powerful dots
and lines, which others in the field appeared so in awe of. It tells the story of the
stable homotopy groups of spheres for odd primes at chromatic height 1, through the
lens of the Adams spectral sequence.
1.2 Calculational tools in homotopy theory
The Adams spectral sequence (ASS) and the Adams-Novikov spectral sequence (ANSS)
are useful tools for homotopy theorists. Theoretically, they enable a calculation of the
stable homotopy groups but they have broader utility than this. Much of contempo-
rary homotopy theory has been inspired by analyzing the structure of these spectral
sequences.
The ASS has ๐ธ2-page given by the cohomology of the dual Steenrod algebra ๐ป*(๐ด)
and it converges ๐-adically to ๐*(๐0). The ANSS has as its ๐ธ2-page the cohomology
of the Hopf algebroid ๐ต๐*๐ต๐ given to us by the ๐-typical factor of complex cobordism
and it converges ๐-locally to ๐*(๐0).
The ANSS has the advantage that elements constructed using non-nilpotent self
maps occur in low filtration. This means that the classes they represent are less likely
to be hit by differentials in the spectral sequence and so proving such elements are
nontrivial in homotopy often comes down to an algebraic calculation of the ๐ธ2-page.
The ASS has the advantage that such elements have higher filtration and, therefore,
less indeterminacy in the spectral sequence. For this reason, among others, arguing
with both spectral sequences is fruitful.
๐ป*(๐ ;๐) CESS +3
alg.NSS
๐ป*(๐ด)
ASS
๐ป*(๐ต๐*๐ต๐ ) ANSS +3 ๐*(๐
0)
(1.2.1)
The relationship between the two spectral sequences is strengthened by the exis-
tence of an algebra ๐ป*(๐ ;๐), which serves as the ๐ธ2-page for two spectral sequences:
12
the Cartan-Eilenberg spectral sequence (CESS) which converges to ๐ป*(๐ด), and the
algebraic Novikov spectral sequence (alg.NSS) converging to ๐ป*(๐ต๐*๐ต๐ ). We will
say more about the algebra ๐ป*(๐ ;๐) shortly. For now it will be a black box and we
will give the relevant definitions in the next section.
Continuing our comparison of the two spectral sequences for calculating ๐*(๐0),
we note that the ASS has the advantage that its ๐ธ2-page can be calculated, in a
range, efficiently with the aid of a computer. The algebra required to calculate the
๐ธ2-page of the ANSS is more difficult. For this reason, the chromatic spectral sequence
(๐ฃ-CSS) was developed in [12] to calculate the 1 and 2-line.
โจ๐โฅ0๐ป
*(๐ ; ๐โ1๐ ๐/(๐โ0 , . . . , ๐
โ๐โ1))
๐-CSS +3
alg.NSS
๐ป*(๐ ;๐)
alg.NSS
โจ๐โฅ0๐ป
*(๐ต๐*๐ต๐ ; ๐ฃโ1๐ ๐ต๐*/(๐
โ, . . . , ๐ฃโ๐โ1))๐ฃ-CSS +3 ๐ป*(๐ต๐*๐ต๐ )
In [10, ยง5], Miller sets up a chromatic spectral sequence for computing ๐ป*(๐ ;๐).
To distinguish this spectral sequence from the more frequently used chromatic spectral
sequence of [12], we call it the ๐-CSS. At odd primes, Miller [10, ยง4] shows that the
๐ธ2-page of the ASS can be identified with ๐ป*(๐ ;๐) and so he compares the ๐-CSS
and the ๐ฃ-CSS to explain some differences between the Adams and Adams-Novikov
๐ธ2-terms. He also observes that it is almost trivial to calculate the 1-line in the ๐ต๐
case ([12, ยง4]), but notes that it is more difficult to calculate the 1-line of the ๐-CSS.
The main result of this thesis is a calculation of the 1-line of the ๐-CSS, that is, of
๐ป*(๐ ; ๐โ11 ๐/๐โ0 ).
The most interesting application of this work is a calculation of the ASS, at odd
primes, above a line of slope 1/(๐2 โ ๐ โ 1). We note that as the prime tends to
infinity, the fraction of the ASS described tends to 1. As a consequence of this work,
we are able to describe, for the first time, differentials of arbitrarily long length in the
ASS.
13
1.3 Some ๐ต๐*๐ต๐ -comodules and the corresponding
๐ -comodules
Our main result is the calculation of a Bockstein spectral sequence converging to
๐ป*(๐ ; ๐โ11 ๐/๐โ0 ), the 1-line of the chromatic spectral sequence for ๐ป*(๐ ;๐). First,
we recall how ๐ , ๐ and related ๐ -comodules are defined. They come from mimicking
constructions used in the chromatic spectral sequence for ๐ป*(๐ต๐*๐ต๐ ) and so we also
recall some relevant ๐ต๐*๐ต๐ -comodules. ๐ is an odd prime throughout this thesis.
Recall that the coefficient ring of the Brown-Peterson spectrum ๐ต๐ is a polynomial
algebra Z(๐)[๐ฃ1, ๐ฃ2, ๐ฃ3, . . .] on the Hazewinkel generators.
๐ โ ๐ต๐* and ๐ฃ๐๐โ1
1 โ ๐ต๐*/๐๐
are ๐ต๐*๐ต๐ -comodule primitives and so we have ๐ต๐*๐ต๐ -comodules ๐ฃโ11 ๐ต๐*/๐,
๐ต๐*/๐โ = colim(. . . โโ ๐ต๐*/๐
๐ ๐โโ ๐ต๐*/๐๐+1 โโ . . .), and
๐ฃโ11 ๐ต๐*/๐
โ = colim(. . . โโ (๐ฃ๐๐โ1
1 )โ1๐ต๐*/๐๐ ๐โโ (๐ฃ๐
๐
1 )โ1๐ต๐*/๐๐+1 โโ . . .).
By filtering the ๐ต๐ cobar construction by powers of the kernel of the augmentation
๐ต๐* โโ F๐ we obtain the algebraic Novikov spectral sequence
๐ป*(๐ ;๐) =โ ๐ป*(๐ต๐*๐ต๐ ).
๐ = F๐[๐1, ๐2, ๐3, . . .] is the polynomial sub Hopf algebra of the dual Steenrod algebra
๐ด and
๐ = gr*๐ต๐* = F๐[๐0, ๐1, ๐2, . . .]
is the associated graded of ๐ต๐*; ๐๐ denotes the class of ๐ฃ๐. Similarly to above, we
have ๐ -comodules ๐โ11 ๐/๐0, ๐/๐โ0 and ๐โ1
1 ๐/๐โ0 and there are appropriate algebraic
Novikov spectral sequences (the first three vertical spectral sequences in figure 1-2).
14
1.4 Main results
We have a Bockstein spectral sequence, the ๐โ11 -Bockstein spectral sequence (๐โ1
1 -BSS)
coming from ๐0-multiplication:
[๐ป*(๐ ; ๐โ1
1 ๐/๐0)[๐0
]]/๐โ0 =โ ๐ป*(๐ ; ๐โ1
1 ๐/๐โ0 ).
Our main theorem is the complete calculation of this spectral sequence, and this, as
we shall describe, tells us a lot about the Adams ๐ธ2-page.
The key input for the calculation is a result of Miller, which we recall presently.
Theorem 1.4.1 (Miller, [10, 3.6]).
๐ป*(๐ ; ๐โ11 ๐/๐0) = F๐[๐ยฑ1
1 ]โ ๐ธ[โ๐,0 : ๐ โฅ 1]โ F๐[๐๐,0 : ๐ โฅ 1].
Here โ๐,0 and ๐๐,0 are elements which can be written down explicitly, though their
formulae are not important for the current discussion. To state the main theorem in
a clear way we change these exterior and polynomial generators by units.
Notation 1.4.2. For ๐ โฅ 1, let ๐[๐] = ๐๐โ1๐โ1
, ๐๐ = ๐โ๐[๐]
1 โ๐,0, and ๐๐ = ๐1โ๐[๐+1]
1 ๐๐,0.
We have ๐ป*(๐ ; ๐โ11 ๐/๐0) = F๐[๐ยฑ1
1 ]โ ๐ธ[๐๐ : ๐ โฅ 1]โ F๐[๐๐ : ๐ โฅ 1].
We introduce some convenient notation for differentials in the ๐โ11 -BSS.
Notation 1.4.3. Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ; ๐โ11 ๐/๐0). We write ๐๐๐ฅ = ๐ฆ to mean that
for all ๐ฃ โ Z, ๐๐ฃ0๐ฅ and ๐๐ฃ+๐0 ๐ฆ survive until the ๐ธ๐-page and that ๐๐๐๐ฃ0๐ฅ = ๐๐ฃ+๐0 ๐ฆ. In this
case, notice that ๐๐ฃ0๐ฅ is a permanent cycle for ๐ฃ โฅ โ๐.
๐ป*(๐ ; ๐โ11 ๐/๐0) is an algebra and with the notation just introduced differentials
are derivations, i.e. from differentials ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ we deduce that ๐๐(๐ฅ๐ฅโฒ) =
๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ.
Using .= to denote equality up to multiplication by an element in Fร
๐ , we are now
ready to state the main theorem.
15
Theorem 1.4.4. In the ๐โ11 -BSS we have two families of differentials. For ๐ โฅ 1,
1. ๐๐[๐]๐๐๐๐โ1
1.
= ๐๐๐๐โ1
1 ๐๐, whenever ๐ โ Zโ ๐Z;
2. ๐๐๐โ1๐๐๐๐
1 ๐๐.
= ๐๐๐๐
1 ๐๐, whenever ๐ โ Z.
Together with the fact that ๐๐1 = 0 for ๐ โฅ 1, these two families of differentials
determine the ๐โ11 -BSS.
We describe the significance of this theorem in terms of the Adams spectral se-
quence ๐ธ2-page. To do so, we need to recall how the 1-line of the chromatic spectral se-
quence manifests itself in ๐ป*(๐ด). In the following zig-zag, ๐ฟ is the natural localization
map, ๐ is the boundary map coming from the short exact sequence of ๐ -comodules
0 โโ ๐ โโ ๐โ10 ๐ โโ ๐/๐โ0 โโ 0, and the isomorphism ๐ป*(๐ ;๐) โผ= ๐ป*(๐ด) is the
one given by Miller in [10, ยง4].
๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) ๐ป*(๐ ;๐/๐โ0 )๐ฟoo ๐ // ๐ป*(๐ ;๐) ๐ป*(๐ด)
โผ=oo (1.4.5)
If an element of ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) is a permanent cycle in the ๐-CSS, then we can
lift it under ๐ฟ and map via ๐ (and the isomorphism) to ๐ป*(๐ด). If there is no lift of
an element of ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) under ๐ฟ then it must support a nontrivial chromatic
differential.
We now turn to figure 1-1. Recall that ๐0 is the class detecting multiplication by ๐
in the ASS. Figure 1-1 displays selected โ๐0-towersโ in the ASS at the prime 3; most
of these are visible in the charts of Nassau [14]. In the range displayed, we see that
there are โprincipal towersโ in topological degrees which are one less than a multiple
of 2๐โ 2 and โside towersโ in topological degrees which are two less than a multiple
of ๐(2๐ โ 2). Under the zig-zag of (1.4.5) (lifting uniquely under ๐ and applying ๐ฟ)
we obtain ๐0-towers in ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ). The principal towers are sent to ๐0-towers
which correspond to differentials in the first family of 1.4.4. The side towers are sent
to ๐0-towers which correspond to differentials in the second family of 1.4.4. In the
ASS, in the range plotted, there are as many differentials as possible between each
16
180 185 190 195 200 205 210 21510
15
20
25
30
35
40
45
50
55
๐กโ ๐
๐
๐451 ๐2
๐451 ๐2
๐451
๐451 ๐3
๐461
๐471
๐481 ๐1
๐481 ๐1
๐481
๐481 ๐2
๐511
๐511 ๐2
๐541 ๐3
๐541 ๐3
๐541
๐541 ๐4
Figure 1-1: The relevant part of ๐ป๐ ,๐ก(๐ด) when ๐ = 3, in the range 175 < ๐กโ ๐ < 218,with a line of slope 1/(๐2โ ๐โ 1) = 1/5 drawn. Vertical black lines indicate multipli-cation by ๐0. The top and bottom of selected ๐0-towers are labelled by the source andtarget, respectively, of the corresponding Bockstein differential. Red arrows indicateAdams differentials up to higher Cartan-Eilenberg filtration.
17
principal tower and its side towers. Some permanent cycles are left at the top of each
principal tower. They detect ๐ฃ1-periodic elements in the given dimension.
Almost all of what we have described about figure 1-1 is true in general.
In each positive dimension ๐ท which is one less than a multiple of 2๐โ 2 there is a
โprincipal tower.โ As long as ๐ = (๐ท+ 1)/(2๐โ 2) is not a power of ๐, the principal
tower maps under the zig-zag (1.4.5) to the ๐0-tower corresponding to the Bockstein
differential on ๐๐1 . If ๐ = (๐ท+1)/(2๐โ2) is a power of ๐, so that ๐ท = ๐๐(2๐โ2)โ1
where ๐ โฅ 0, the principal tower has length ๐๐ and it starts on the 1-line at โ1,๐.
This is a statement about the existence of chromatic differentials: for ๐ โฅ 1, there
are chromatic differentials on the ๐0-tower corresponding to the Bockstein differential
on ๐๐๐
1 .
In each positive dimension ๐ท which is two less than a multiple of ๐(2๐โ 2) there
are โside towers.โ If ๐๐ is the highest power of ๐ dividing ๐ = (๐ท+ 2)/(2๐โ 2), then
there are ๐ side towers. In most cases, the ๐th side tower (we order from higher Adams
filtration to lower Adams filtration) maps under the zig-zag (1.4.5) to the ๐0-tower
corresponding to the Bockstein differential on ๐๐1 ๐๐. However, if ๐ = (๐ท+2)/(2๐โ2)
is a power of ๐ so that ๐ท = ๐๐(2๐โ 2)โ 2 where ๐ โฅ 1, the ๐th side tower has length
๐๐โ๐[๐] and it starts on the 2-line at ๐1,๐โ1; for ๐ โฅ 2, there are chromatic differentials
on the ๐0-tower corresponding to the Bockstein differential on ๐๐๐
1 ๐๐.
To make the assertions above we have to calculate some differentials in a Bockstein
spectral sequence for ๐ป*(๐ ;๐). We omit stating the relevant result here.
We have not described all the elements in ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ). The remaining ele-
ments line up in a convenient way but to be more precise we must talk about the
localized algebraic Novikov spectral sequence (loc.alg.NSS)
๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) =โ ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐โ).
This is also important if we are to address the Adams differentials between principal
towers and their side towers.
Theorem 1.4.4 allows us to understand the associated graded of the ๐ธ2-page of the
18
loc.alg.NSS with respect to the Bockstein filtration. Since the Bockstein filtration is
respected by ๐loc.alg.NSS2 : ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐โ0 ]๐ก) โโ ๐ป๐ +1,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก+1) we have a
filtration spectral sequence (๐0-FILT)
๐ธ0(๐0-FILT) = ๐ธโ(๐โ11 -BSS) =โ ๐ธ3(loc.alg.NSS).
Theorem 1.4.4 enables us to write down some obvious permanent cycles in the ๐โ11 -
BSS. The next theorem tells us that they are the only elements which appear on the
๐ธ1-page of the ๐0-FILT.
Theorem 1.4.6. ๐ธ1(๐0-FILT) has an F๐-basis given by the following elements.
๐๐ฃ0 : ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โฅ 1, ๐ โ Zโ ๐Z, โ๐[๐] โค ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐
1 ๐๐ : ๐ โฅ 1, ๐ โ Z, 1โ ๐๐ โค ๐ฃ < 0
This theorem tells us that the ๐2 differentials in the loc.alg.NSS which do not
increase Bockstein filtration kill all the ๐0-towers except those corresponding to the
differentials of theorem 1.4.4. This is precisely what we meant when we said that โthe
remaining elements line up in a convenient way.โ Once theorem 1.4.6 is proved, the
calculation of the remainder of the loc.alg.NSS is straightforward because one knows
๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) by [12, ยง4].
We now turn to the Adams differentials between principal towers and their side
towers, which is the motivation for drawing figure 1-2. In [11], Miller uses the square
analogous to (1.2.1) for the mod ๐ Moore spectrum to deduce Adams differentials
(up to higher Cartan-Eilenberg filtration) from algebraic Novikov differentials. The
algebraic Novikov spectral sequence he calculates is precisely the one labelled as the
๐ฃ1-alg.NSS in figure 1-2 and this is the key input to proving theorem 1.4.6. We can use
the same techniques to deduce Adams differentials for the sphere from differentials in
the alg.NSS. We make this statement precise (see also, [2, S8]).
19
๐ป*(๐ ; ๐โ11 ๐/๐0)
๐ฃ1-alg.NSS
// ๐ป*(๐ ; ๐โ11 ๐/๐โ0 )
loc.alg.NSS
๐ป*(๐ ;๐/๐โ0 )
๐ฟoo ๐ // ๐ป*(๐ ;๐)
alg.NSS
CESS +3 ๐ป*(๐ด)
ASS
โผ=xx
๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐) // ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐โ) ๐ป*(๐ต๐*๐ต๐ ;๐ต๐*/๐
โ)๐ฟoo ๐ // ๐ป*(๐ต๐*๐ต๐ ) ANSS +3 ๐*(๐0)
Figure 1-2: Obtaining information about the Adams spectral sequence from the Millerโs ๐ฃ1-algebraic Novikov spectral sequence.Having calculated the ๐โ1
1 -BSS, Millerโs calculation of the ๐ฃ1-alg.NSS allows us to calculate the loc.alg.NSS. Above a line ofslope 1/(๐2 โ ๐ โ 1) the ๐ธ2-page of the loc.alg.NSS is isomorphic to the ๐ธ2-page of the alg.NSS. Thus, our localized algebraicNovikov differentials allow us to deduce unlocalized ones, which can, in turn, be used to deduce Adams ๐2 differentials up tohigher Cartan-Eilenberg filtration.
20
Theorem 1.4.7 (Miller, [11, 6.1]). Suppose ๐ฅ โ ๐ป๐ ,๐ข(๐ ;๐๐ก). Use the identification
๐ป*(๐ด) = ๐ป*(๐ ;๐) to view ๐ฅ as lying in ๐ป๐ +๐ก,๐ข+๐ก(๐ด). Then we have
๐ASS2 ๐ฅ โ
๐ก+1โจ๐โฅ0
๐ป๐ +๐+1,๐ข+๐(๐ ;๐๐กโ๐+1) โ ๐ป๐ +๐ก+2,๐ข+๐ก+1(๐ด),
where the zero-th coordinate is ๐alg.NSS2 ๐ฅ โ ๐ป๐ +1,๐ข(๐ ;๐๐ก+1).
Moreover, the map ๐ : ๐ป*(๐ ;๐/๐โ0 ) โโ ๐ป*(๐ ;๐) is an isomorphism away from
low topological degrees, since ๐ป*(๐ ; ๐โ10 ๐) = F๐[๐ยฑ1
0 ] and we have the following result
concerning the localization map L.
Proposition 1.4.8. The localization map
๐ฟ : ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก)
is an isomorphism if (๐ข+ ๐ก) < ๐(๐โ 1)(๐ + ๐ก)โ 2. In particular, the localization map
is an isomorphism above a line of slope 1/(๐2 โ ๐โ 1) when we plot elements in the
(๐ขโ ๐ , ๐ + ๐ก)-plane, the plane that corresponds to the usual way of drawing the Adams
spectral sequence.
The upshot of all of this is that as long as we are above a particular line of
slope 1/(๐2 โ ๐ โ 1), the ๐2 differentials in the loc.alg.NSS can be transferred to ๐2
differentials in the unlocalized spectral sequence (the alg.NSS), and using theorem
1.4.7 we obtain ๐2 differentials in the Adams spectral sequence. In fact, we can do
even better. Proposition 1.4.8 states the isomorphism range which one proves when
one chooses to use the bigrading (๐, ๐) = (๐ + ๐ก, ๐ข + ๐ก). We can also prove a version
which makes full use of the trigrading (๐ , ๐ก, ๐ข) and this allows one to obtain more
information. In particular, it allows one to show that the bottom of a principal tower
in the Adams spectral sequence always supports ๐2 differentials which map to the last
side tower.
To complete the story we discuss the higher Adams differentials between principal
towers and their side towers. Looking at figure 1-1 one would hope to prove that if a
21
principal tower has ๐ side towers, then the ๐th side tower is the target for nontrivial
๐๐โ๐+2 differentials. We have just addressed the case when ๐ = ๐ and one finds that in
the loc.alg.NSS everything goes as expected. The issue is that theorem 1.4.7 does not
exist for higher differentials. For instance, ๐alg.NSS2 ๐ฅ = 0, simply says that ๐ASS
2 ๐ฅ has
higher Cartan-Eilenberg filtration. In this case ๐alg.NSS3 ๐ฅ lives in the wrong trigrading
to give any more information about ๐ASS2 ๐ฅ. Instead, we set up and calculate a spectral
sequence which converges to the homotopy of the ๐ฃ1-periodic sphere spectrum
๐ฃโ11 ๐/๐โ = hocolim(. . . โโ (๐ฃ๐
๐โ1
1 )โ1๐/๐๐๐โโ (๐ฃ๐
๐
1 )โ1๐/๐๐+1 โโ . . .).
This is the localized Adams spectral sequence for the ๐ฃ1-periodic sphere (LASS-โ)
๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) =โ ๐*(๐ฃ
โ11 ๐/๐โ).
This spectral sequence behaves as one would like with respect to differentials between
principal towers and their side towers (i.e. in the same way as the loc.alg.NSS) and
moreover, the zig-zag of (1.4.5) consists of maps of spectral sequences, which enables
a comparison with the Adams spectral sequence. It is this calculation that allows
us to describe differentials of arbitrarily long length in the ASS. They come from
differentials between primary towers and side towers. We find such differentials in
the LASS-โ, sufficiently far above the line of slope 1/(๐2โ ๐โ 1), and transfer them
across to the ASS.
In order to set up the LASS-โ we prove an odd primary analog of a result of
Davis and Mahowald, which appears in [6]. This is of interest in its own right and we
state it below.
In [1] Adams shows that there is a CW spectrum ๐ต with one cell in each positive
dimension congruent to 0 or โ1 modulo ๐ = 2๐ โ 2 such that ๐ต โ (ฮฃโ๐ตฮฃ๐)(๐).
Denote the skeletal filtration by a superscript in square brackets. We use the following
notation.
Notation 1.4.9. For 1 โค ๐ โค ๐ let ๐ต๐๐ = ๐ต[๐๐]/๐ต[(๐โ1)๐].
22
The following theorem allows a very particular construction of a ๐ฃ1 self-map for
๐/๐๐+1.
Theorem 1.4.10. The element ๐๐๐โ๐โ1
0 โ1,๐ โ ๐ป๐๐โ๐,๐๐(๐+1)โ๐โ1(๐ด) is a permanent
cycle in the Adams spectral sequence represented by a map
๐ผ : ๐๐๐๐โ1 ๐ // ๐ต๐๐
๐๐โ๐๐ // ๐ต๐๐โ1
๐๐โ๐โ1// . . . // ๐ต๐+2
2
๐ // ๐ต๐+11
๐ก // ๐0.
Here, ๐ comes from the fact that the top cell of ๐ต[๐๐๐โ1]/๐ต[(๐๐โ๐โ1)๐โ1] splits off, ๐ก
is obtained from the transfer map ๐ตโ1 โโ ๐0, and each ๐ is got by factoring a
multiplication-by-๐ map.
Moreover, there is an element โ ๐๐๐๐(๐/๐๐+1) whose image in ๐ต๐๐๐๐(๐/๐) is
๐ฃ๐๐
1 , and whose desuspension maps to ๐ผ under
๐๐๐๐โ1(ฮฃโ1๐/๐๐+1) โโ ๐๐๐๐โ1(๐
0).
1.5 Outline of thesis
Chapter 2 is an expository chapter on spectral sequences. A correspondence approach
is presented, terminology is defined, and we say what it means for a spectral sequence
to converge. In chapter 3 we introduce all the Bockstein spectral sequences that we
use and prove their important properties, namely, that differentials in the ๐-BSS and
the ๐โ0 -BSS coincide, and that the differentials in the ๐โ11 -BSS are derivations.
Chapter 4 contains our first important result. After finding some vanishing lines
we examine the range in which the localization map๐ป*(๐ ;๐/๐โ0 )โ ๐ป*(๐ ; ๐โ11 ๐/๐โ0 )
is an isomorphism. We do this from a trigraded and a bigraded perspective.
Chapter 5 contains our main results. We calculate the ๐โ11 -BSS and find some
differentials in the ๐-BSS. We address the family of differentials corresponding to the
principal towers using an explicit argument with cocycles. The family of differentials
corresponding to the side towers is obtained using a Kudo transgression theorem. A
combinatorial argument gives the ๐ธโ-page of the ๐โ11 -BSS.
23
Chapter 6 contains the calculation of the localized algebraic Novikov spectral se-
quence. The key ingredients for the calculation are the combinatorics used to describe
the ๐ธโ-page of the ๐โ11 -BSS and Millerโs calculation of the ๐ฃ1-algebraic Novikov spec-
tral sequence.
In chapter 7 we construct representatives for some permanent cycles in the Adams
spectral sequence using the geometry of stunted projective spaces and the transfer
map.
In chapter 8 we set up the localized Adams spectral sequence for the ๐ฃ1-periodic
sphere (LASS-โ), calculate it, and demonstrate the consequences the calculation
has for the Adams spectral sequence for the sphere. Along the way we construct
a modified Adams spectral sequence for the mod ๐๐ Moore spectrum and the Prรผfer
sphere. We lift the permanent cycles of the previous chapter to permanent cycles in
these spectral sequences and we complete the proof of the last theorem stated in the
introduction.
In the appendices we construct various maps of spectral sequences and check the
convergence of our spectral sequences.
24
Chapter 2
Spectral sequence terminology
Spectral sequences are used in abundance throughout this thesis. Graduate students
in topology often live in fear of spectral sequences and so we take this opportunity to
give a presentation of spectral sequences, which, we hope, shows that they are not all
that bad. We also fix the terminology which is used throughout the rest of the thesis.
All of this chapter is expository. Everything we say is surely documented in [3].
2.1 A correspondence approach
The reader is probably familiar with the notion of an exact couple which is one of the
most common ways in which a spectral sequence arises.
Definition 2.1.1. An exact couple consists of abelian groups ๐ด and ๐ธ together with
homomorphisms ๐, ๐ and ๐ such that the following triangle is exact.
๐ด
๐
๐ด๐oo
๐ธ
๐
;;
Given an exact couple, one can form the associated derived exact couple. Iterating
this process gives rise to a spectral sequence. Experience has led the author to
conclude that, although this inductive definition is slick, it disguises some of the
25
important features that spectral sequences have and which are familiar to those who
work with them on a daily basis. Various properties become buried in the induction
and the author feels that first time users should not have to struggle for long periods
of time to discover these properties however rewarding that process might be.
An alternative approach exploits correspondences. A correspondence ๐ : ๐บ1 โโ
๐บ2 is a subgroup ๐ โ ๐บ1 ร ๐บ2. The images of ๐ under the projection maps are the
domain dom(๐) and the image im(๐) of the correspondence. We can also define the
kernel of a correspondence ker (๐) โ dom(๐).
We will find that the picture becomes clearer, especially once gradings are intro-
duced, when we spread out the exact couple:
. . . ๐ดoo
๐ดoo . . .๐oo ๐ด๐oo
๐
. . .oo
๐ธ
๐
;;
๐ธ
Let ๐ : ๐ธร๐ดร๐ดร๐ธ โโ ๐ธร๐ธ be the projection map. Then we make the following
definitions.
Definition 2.1.2. For each ๐ โฅ 1 let
๐๐ = (๐ฅ, , ๐ฆ, ๐ฆ) โ ๐ธ ร ๐ดร ๐ดร ๐ธ : ๐๐ฅ = = ๐๐โ1๐ฆ and ๐๐ฆ = ๐ฆ
and ๐๐ = ๐( ๐๐). Let ๐0 = ๐ธ ร 0 โ ๐ธ ร ๐ธ.
. . .๐oo ๐ฆ๐oo_
๐
๐ฅ
8๐
;;
๐ฆ
Since ๐, ๐, ๐ and ๐ are homomorphisms of abelian groups ๐๐ and ๐๐ are subgroups of
๐ธร๐ดร๐ดร๐ธ and ๐ธร๐ธ, respectively. In particular, ๐๐ : ๐ธ โโ ๐ธ is a correspondence.
We note that ๐0 is the zero homomorphism and that ๐1 = ๐๐.
26
Notation 2.1.3. We write ๐๐๐ฅ = ๐ฆ if (๐ฅ, ๐ฆ) โ ๐๐.
We have the following useful observations.
Lemma 2.1.4.
1. For ๐ โฅ 1, ๐๐๐ฅ is defined if and only if ๐๐โ1๐ฅ = 0, i.e.
(๐ฅ, 0) โ ๐๐โ1 โโ โ๐ฆ : (๐ฅ, ๐ฆ) โ ๐๐.
2. For ๐ โฅ 1, ๐๐0 = ๐ฆ if and only if there exists an ๐ฅ with ๐๐โ1๐ฅ = ๐ฆ, i.e.
(0, ๐ฆ) โ ๐๐ โโ โ๐ฅ : (๐ฅ, ๐ฆ) โ ๐๐โ1.
We note that the first part of the lemma says that dom(๐๐) = ker (๐๐โ1) for ๐ โฅ 1.
The second part of the lemma has the following corollary.
Corollary 2.1.5. For ๐ โฅ 1, the following conditions are equivalent:
1. ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅ = ๐ฆโฒ;
2. ๐๐๐ฅ = ๐ฆ and there exists an ๐ฅโฒ with ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ.
It is also immediate from the definitions that the following lemma holds.
Lemma 2.1.6. Suppose ๐ โฅ 1 and that ๐๐๐ฅ = ๐ฆ. Then ๐๐ ๐ฆ = 0 for any ๐ โฅ 1.
Spectral sequences consist of pages.
Definition 2.1.7. For ๐ โฅ 1, let ๐ธ๐ = ker ๐๐โ1/ im ๐๐โ1. This is the ๐th page of the
spectral sequence.
One is often taught that a spectral sequence begins with an ๐ธ1 or ๐ธ2-page and
that one obtains successive pages by calculating differentials and taking homology.
We relate our correspondence approach to this one presently.
27
We have a surjection ๐ธ๐ โโ ker ๐๐โ1/โ๐ im ๐๐ , an injection
โ๐ ker ๐๐ / im ๐๐โ1 โโ
๐ธ๐, and the preceding lemmas show that ๐๐ defines a homomorphism allowing us to
form the following composite which, for now, we call ๐ฟ๐.
๐ธ๐ โโ ker ๐๐โ1/โ๐
im ๐๐ โโโ๐
ker ๐๐ / im ๐๐โ1 โโ ๐ธ๐.
We have an identification of the ๐ธ๐+1-page as the homology of the ๐ธ๐-page with
respect to the differential ๐ฟ๐. We will blur the distinction between the correspondence
๐๐ and the differential ๐ฟ๐, calling them both ๐๐.
We note that the ๐ธ1 page is ๐ธ. Our Bockstein spectral sequences have convenient
descriptions from the ๐ธ1-page and so we use the correspondence approach. Conse-
quently, all our differentials will be written in terms of elements on the ๐ธ1-page. Our
topological spectral sequences have better descriptions from the ๐ธ2-page. The corre-
spondence approach also allows us to write all our formulae in terms of elements of
the ๐ธ2-pages.
Here is some terminology that we will use freely throughout this thesis.
Definition 2.1.8. Suppose ๐๐๐ฅ = ๐ฆ. Then ๐ฅ is said to survive to the ๐ธ๐-page and
support a ๐๐ differential. ๐ฆ is said to be the target of a ๐๐ differential, to be hit by a
๐๐ differential, and to be a boundary. If, in addition, ๐ฆ /โ im ๐๐โ1, then the differential
is said to be nontrivial and ๐ฅ is said to support a nontrivial differential.
Definition 2.1.9. Elements ofโ๐ ker ๐๐ are called permanent cycles.
We write ๐ธโ forโ๐ ker ๐๐ /
โ๐ im ๐๐ , permanent cycles modulo boundaries, the
๐ธโ-page of the spectral sequence.
Note that lemma 2.1.6 says that targets of differentials are permanent cycles or,
said another way, elements that are hit by a differential survive to all pages of the
spectral sequence. In particular, note that we use the word hit, not kill.
28
2.2 Convergence
The purpose of a spectral sequence is to give a procedure to calculate an abelian
group of interest ๐ . This procedure can be viewed as having three steps, which we
outline below, but first, we give some terminology.
Definition 2.2.1. A filtration of an abelian group ๐ is a sequence of subgroups
๐ โ . . . โ ๐น ๐ โ1๐ โ ๐น ๐ ๐ โ ๐น ๐ +1๐ โ . . . โ 0, ๐ โ Z.
The associated graded abelian group corresponding to this filtration is the graded
abelian groupโจ
๐ โZ ๐น๐ ๐/๐น ๐ +1๐ .
The ๐ธโ-page of a spectral sequence should tell us about the associated graded of
an abelian group ๐ we are trying to calculate. In particular, the ๐ธโ-page should be
Z-graded, so we consider the story described in the previous section, with the added
assumption that ๐ด and ๐ธ have a Z-grading ๐ , that ๐ : ๐ด๐ +1 โโ ๐ด๐ , ๐ : ๐ด๐ โโ ๐ธ๐
and ๐ : ๐ธ๐ โโ ๐ด๐ +1. We can redraw the exact couple as follows.
. . . ๐ด๐ oo
๐ด๐ +1oo . . .
๐oo ๐ด๐ +๐๐oo
๐
. . .oo
๐ธ๐
๐
::
๐ธ๐ +๐
We see that ๐๐ has degree ๐ and so ๐ธโ becomes Z-graded. In all the cases we consider
in the thesis the abelian group ๐ we are trying to calculate will be either the limit
or colimit of the directed system ๐ด๐ ๐ โZ.
We now describe the way in which a spectral sequence can be used to calculate
an abelian group ๐ .
1. Define a filtration of ๐ and an identification ๐ธ๐ โ = ๐น ๐ ๐/๐น ๐ +1๐ between the
๐ธโ-page of the spectral sequence and the associated graded of ๐ .
2. Resolve extension problems. Depending on circumstances this will give us either
๐น ๐ ๐ for each ๐ or ๐/๐น ๐ ๐ for each ๐ .
29
3. Recover ๐ . Depending on circumstances this will either be via an isomorphism
๐ โโ lim๐ ๐/๐น ๐ ๐ or an isomorphism colim๐ ๐น๐ ๐ โโ๐ .
In all the cases we consider, ๐ will be graded and the filtration will respect this
grading. Thus, the associated graded will be bigraded. Correspondingly, the exact
couple will be bigraded. There are three cases which arise for us. We highlight how
each affects the procedure above.
1. Each case is determined by the way in which the filtration behaves.
(a) ๐น 0๐ = ๐ andโ๐น ๐ ๐ = 0.
(b) ๐น 0๐ = 0 andโ๐น ๐ ๐ = ๐ .
(c)โ๐น ๐ ๐ = ๐ and keeping track of the additional gradings the identifica-
tion in the first part of the procedure becomes ๐ธ๐ ,๐กโ = ๐น ๐ ๐๐กโ๐ /๐น
๐ +1๐๐กโ๐ ;
moreover, for each ๐ข there exists an ๐ such that ๐น ๐ ๐๐ข = 0.
2. The way in which we would go about resolving extension problems varies ac-
cording to which case we are in.
(a) ๐/๐น 0๐ = 0 so suppose that we know ๐/๐น ๐ ๐ where ๐ โฅ 0. The first
part of the procedure gives us ๐น ๐ ๐/๐น ๐ +1๐ and so resolving an extension
problem gives ๐/๐น ๐ +1๐ . By induction, we know ๐/๐น ๐ ๐ for all ๐ .
(b) ๐น 0๐ = 0 so suppose that we know ๐น ๐ +1๐ where ๐ < 0. The first part of
the procedure gives us ๐น ๐ ๐/๐น ๐ +1๐ and so resolving an extension problem
gives ๐น ๐ ๐ . By induction, we know ๐น ๐ ๐ for all ๐ .
(c) This is similar to (2b). Fixing ๐ข, there exists an ๐ 0 with ๐น ๐ 0๐๐ข = 0. Sup-
pose that we know ๐น ๐ +1๐๐ข where ๐ < ๐ 0. The first part of the procedure
gives us ๐น ๐ ๐๐ข/๐น๐ +1๐๐ข and so resolving an extension problem gives ๐น ๐ ๐๐ข.
By induction, we know ๐น ๐ ๐๐ข for all ๐ . We can now vary ๐ข.
3. In case (๐) we need an isomorphism ๐ โโ lim๐ ๐/๐น ๐ ๐ . In cases (๐) and (๐)
we have an isomorphism colim๐ ๐น๐ ๐ โโ๐ .
30
When we say that our spectral sequences converge we ignore whether or not we
can resolve the extension problems. This is paralleled by the fact that, when making
such a statement, we ignore whether or not it is possible to calculate the differentials
in the spectral sequence. The point is, that theoretically, both of these issues can be
overcome even if it is extremely difficult to do so in practice. We conclude that the
important statements for convergence are given in stages (1) and (3) of our procedure
and we make the requisite definition.
Definition 2.2.2. Suppose given a graded abelian group ๐ and a spectral sequence
๐ธ** . Suppose that ๐ is filtered, that we have an identification ๐ธ๐
โ = ๐น ๐ ๐/๐น ๐ +1๐
and that one of the following conditions holds.
1. ๐น 0๐ = ๐ ,โ๐น ๐ ๐ = 0 and the natural map ๐ โ lim๐ ๐/๐น ๐ ๐ is an isomor-
phism.
2. ๐น 0๐ = 0 andโ๐น ๐ ๐ = ๐ .
3.โ๐น ๐ ๐ = ๐ , if we keep track of the additional gradings then we have ๐ธ๐ ,๐ก
โ =
๐น ๐ ๐๐กโ๐ /๐น๐ +1๐๐กโ๐ , and for each ๐ข there exists an ๐ such that ๐น ๐ ๐๐ข = 0.
Then the spectral sequence is said to converge and we write ๐ธ๐ 1
๐ =โ ๐ or ๐ธ๐
2๐
=โ ๐
depending on which page of the spectral sequence has the more concise description.
It would appear that the notation ๐ธ๐ 1
๐ =โ ๐ is over the top since ๐ appears twice,
but once other gradings are recorded it is the ๐ above the โ =โ โ that indicates the
filtration degree.
Suppose that ๐ธ๐ 1
๐ =โ ๐ (or equivalently ๐ธ๐
2๐
=โ ๐). We have some terminol-
ogy to describe the relationship between permanent cycles and elements of ๐ .
Definition 2.2.3. Suppose that ๐ฅ is a permanent cycle defined in ๐ธ๐ ๐ (usually ๐ = 1
or ๐ = 2) and ๐ง โ ๐น ๐ ๐ . Then we say that ๐ฅ detects ๐ง or that ๐ง represents ๐ฅ, to mean
that the image of ๐ฅ in ๐ธ๐ โ and the image of ๐ง in ๐น ๐ ๐/๐น ๐ +1๐ correspond under the
given identification ๐ธ๐ โ = ๐น ๐ ๐/๐น ๐ +1๐ .
Suppose that ๐ฅ โ ๐ธ๐ ๐ detects ๐ง โ ๐น ๐ ๐ . Notice that ๐ฅ is a boundary if and only
if ๐ง โ ๐น ๐ +1๐ .
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32
Chapter 3
Bockstein spectral sequences
In this chapter, we set up all the Bockstein spectral sequences used in this thesis and
prove the properties that we require of them.
3.1 The Hopf algebra ๐ and some ๐ -comodules
Throughout this thesis ๐ is an odd prime.
Definition 3.1.1. Let ๐ denote the polynomial algebra over F๐ on the Milnor gen-
erators ๐๐ : ๐ โฅ 1 where |๐๐| = 2๐๐ โ 2. ๐ is a Hopf algebra when equipped with
the Milnor diagonal
๐ โโ ๐ โ ๐, ๐๐ โฆโโ๐โ๐=0
๐๐๐
๐โ๐ โ ๐๐, (๐0 = 1).
Definition 3.1.2. Let ๐ denote the polynomial algebra over F๐ on the generators
๐๐ : ๐ โฅ 0 where |๐๐| = 2๐๐ โ 2. ๐ is an algebra in ๐ -comodules when equipped
with the coaction map
๐ โโ ๐ โ๐, ๐๐ โฆโโ๐โ๐=0
๐๐๐
๐โ๐ โ ๐๐.
We write ๐๐ก for the sub-๐ -comodule consisting of monomials of length ๐ก.
Note that the multiplication on ๐ is commutative, which is the same as graded
33
commutative since everything lives in even degrees. We shall see later that ๐ก is the
Novikov weight. Miller [10] also refers to ๐ก as the Cartan degree.
๐0 is a ๐ -comodule primitive and so ๐/๐0 and ๐โ10 ๐ are ๐ -comodules.
Definition 3.1.3. Define๐/๐โ0 by the following short exact sequence of ๐ -comodules.
0 // ๐ // ๐โ10 ๐ // ๐/๐โ0 // 0
๐/๐โ0 is a ๐-module in ๐ -comodules.
We find that ๐1 โ ๐/๐0 is a comodule primitive so we may define ๐โ11 ๐/๐0 which
is an algebra in ๐ -comodules. We may also define ๐โ11 ๐/๐โ0 , a ๐-module in ๐ -
comodules, but this requires a more sophisticated construction, which we now outline.
Definition 3.1.4. For ๐ โฅ 1, ๐๐ is the sub-๐ -comodule of ๐/๐โ0 defined by the
following short exact sequence of ๐ -comodules. ๐๐ is a ๐-module in ๐ -comodules.
0 // ๐ // ๐โจ๐โ๐0 โฉ //๐๐// 0.
Lemma 3.1.5. ๐๐๐โ1
1 : ๐๐ โโ๐๐ is a homomorphism of ๐-modules in ๐ -comodules.
Definition 3.1.6. For each ๐ โฅ 0 let ๐๐(๐) = ๐๐. ๐โ11 ๐๐ is defined to be the
colimit of the following diagram.
๐๐(0)๐๐
๐โ1
1 //๐๐(1)๐๐
๐โ1
1 //๐๐(2)๐๐
๐โ1
1 //๐๐(3)๐๐
๐โ1
1 // . . .
Definition 3.1.7. We have homomorphisms ๐โ11 ๐๐ โโ ๐โ1
1 ๐๐+1 induced by the
inclusions ๐๐ โโ๐๐+1. ๐โ11 ๐/๐โ0 is defined to be the colimit of following diagram.
๐โ11 ๐1
// ๐โ11 ๐2
// ๐โ11 ๐3
// ๐โ11 ๐4
// . . .
Notation 3.1.8. If Q is a ๐ -comodule then we write ฮฉ*(๐ ;Q) for the cobar con-
struction on ๐ with coefficients in Q. In particular, we have
ฮฉ๐ (๐ ;Q) = ๐โ๐ โQ
34
where ๐ = F๐ โ ๐ as F๐-modules. We write [๐1| . . . |๐๐ ]๐ for ๐1 โ . . .โ ๐๐ โ ๐. We set
ฮฉ*๐ = ฮฉ*(๐ ;F๐).
We recall (see [10, pg. 75]) that the differentials are given by an alternating sum
making use of the diagonal and coaction maps. We also recall that if Q is an algebra in
๐ -comodules then ฮฉ*(๐ ;Q) is a DG-F๐-algebra; if Qโฒ is a Q-module in ๐ -comodules
then ฮฉ*(๐ ;Qโฒ) is a DG-ฮฉ*(๐ ;Q)-module.
Definition 3.1.9. If Q is a ๐ -comodule then๐ป*(๐ ;Q) is the cohomology of ฮฉ*(๐ ;Q).
We remark that in our setting ๐ป*(๐ ;Q) will always have three gradings. There
is the cohomological grading ๐ . ๐ and its comodules are graded and so we have an
internal degree ๐ข. The Novikov weight ๐ก on ๐ persists to ๐/๐0, ๐โ10 ๐, ๐/๐โ0 , ๐โ1
1 ๐/๐0,
and ๐โ11 ๐/๐โ0 .
Later on, we will use an algebraic Novikov spectral sequence. From this point of
view, right ๐ -comodules are more natural (see [2], for instance). However, Millerโs
paper [10] is such a strong source of guidance for this work that we choose to use left
๐ -comodules as he does there.
3.2 The ๐-Bockstein spectral sequence (๐-BSS)
Applying ๐ป*(๐ ;โ) to the short exact sequence of ๐ -comodules
0 // ๐๐0 // ๐ // ๐/๐0 // 0
gives a long exact sequence. We also have a trivial long exact sequence consisting of
the zero group every three terms and ๐ป*(๐ ;๐) elsewhere. Intertwining these long
exact sequences gives an exact couple, the nontrivial part, of which, looks as follows
35
(๐ฃ โฅ 0).
๐ป๐ ,๐ข(๐ ;๐๐กโ๐ฃ)
๐ป๐ ,๐ข(๐ ;๐๐กโ๐ฃโ1)oo . . .๐0oo ๐ป๐ ,๐ข(๐ ;๐๐กโ๐ฃโ๐)
๐0oo
๐ป๐ ,๐ข(๐ ; [๐/๐0]
๐กโ๐ฃ)โจ๐๐ฃ0โฉ
๐
66
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐กโ๐ฃโ๐)โจ๐๐ฃ+๐0 โฉ
Here ๐ raises the degree of ๐ by one relative to what is indicated and the powers
of ๐0 are used to distinguish copies of ๐ป*(๐ ;๐/๐0) from one another.
Definition 3.2.1. The spectral sequence arising from this exact couple is called the
๐-Bockstein spectral sequence (๐-BSS). It has ๐ธ1-page given by
๐ธ๐ ,๐ก,๐ข,๐ฃ1 (๐-BSS) =
โงโชโจโชโฉ๐ป๐ ,๐ข(๐ ; [๐/๐0]
๐กโ๐ฃ)โจ๐๐ฃ0โฉ if ๐ฃ โฅ 0
0 if ๐ฃ < 0
and ๐๐ has degree (1, 0, 0, ๐).
The spectral sequence converges to ๐ป*(๐ ;๐) and the filtration degree is given by
๐ฃ. In particular, we have an identification
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐-BSS) = ๐น ๐ฃ๐ป๐ ,๐ข(๐ ;๐๐ก)/๐น ๐ฃ+1๐ป๐ ,๐ข(๐ ;๐๐ก)
where ๐น ๐ฃ๐ป*(๐ ;๐) = im(๐๐ฃ0 : ๐ป*(๐ ;๐) โโ ๐ป*(๐ ;๐)) for ๐ฃ โฅ 0. The identification
is given by lifting an element of ๐น ๐ฃ๐ป*(๐ ;๐) to the ๐ฃth copy of ๐ป*(๐ ;๐) and mapping
this lift down to ๐ป*(๐ ;๐/๐0)โจ๐๐ฃ0โฉ to give a permanent cycle.
Remark 3.2.2. One can describe the ๐ธ1-page of the ๐-BSS more concisely as the
algebra ๐ป*(๐ ;๐/๐0)[๐0]. The first three gradings (๐ , ๐ก, ๐ข) are obtained from the grad-
ings on the elements of ๐ป*(๐ ;๐/๐0) and ๐0; the adjoined polynomial generator ๐0 has
๐ฃ-grading 1, whereas elements of ๐ป*(๐ ;๐/๐0) have ๐ฃ-grading 0.
Notation 3.2.3. Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ;๐/๐0). We write ๐๐๐ฅ = ๐ฆ to mean that for
every ๐ฃ โฅ 0, ๐๐ฃ0๐ฅ survives to the ๐ธ๐-page, ๐๐ฃ0๐ฆ is a permanent cycle, and ๐๐๐๐ฃ0๐ฅ =
36
๐๐ฃ+๐0 ๐ฆ. In this case, ๐ฅ is said to support a ๐๐ differential. If one of the differentials
๐๐๐๐ฃ0๐ฅ = ๐๐ฃ+๐0 ๐ฆ is nontrivial, then ๐ฅ is said to support a nontrivial differential.
Lemma 3.2.4. Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ;๐/๐0). Then ๐๐๐ฅ = ๐ฆ in the ๐-BSS if and only
if there exist ๐ and ๐ in ฮฉ*(๐ ;๐) with ๐๐ = ๐๐0๐ such that their images in ฮฉ*(๐ ;๐/๐0)
are cocycles representing ๐ฅ and ๐ฆ, respectively.
Proof. Suppose that ๐๐๐ฅ = ๐ฆ in the ๐-BSS. By definition 2.1.2 there exist and ๐ฆ
fitting into the following diagram.
๐ป๐ +1,๐ข(๐ ;๐๐กโ1) . . .๐0oo ๐ป๐ +1,๐ข(๐ ;๐๐กโ๐)
๐0oo
๐ป๐ ,๐ข(๐ ; [๐/๐0]
๐ก))
๐
77
๐ป๐ +1,๐ข(๐ ; [๐/๐0]๐กโ๐)
. . .๐0oo ๐ฆ๐0oo_
๐ฅ
.
๐
66
๐ฆ
Let ๐ด โ ฮฉ*(๐ ;๐/๐0) be a representative for ๐ฅ and ๐ต โ ฮฉ*(๐ ;๐) be a representative
for ๐ฆ. There exists an ๐ด โ ฮฉ*(๐ ;๐) representing , and an ๐โฒ and ๐ผโฒ fitting into the
following diagram.
ฮฉ*(๐ ;๐)๐0 //
ฮฉ*(๐ ;๐) //
๐
ฮฉ*(๐ ;๐/๐0)
ฮฉ*(๐ ;๐)
๐0 // ฮฉ*(๐ ;๐) // ฮฉ*(๐ ;๐/๐0)
๐โฒ //_
๐ด_
๐ด // ๐ผโฒ // 0
Moreover, there exists ๐ถ โ ฮฉ*(๐ ;๐) such that ๐ด = ๐๐โ10๐ต + ๐ ๐ถ. Let ๐ = ๐โฒ โ ๐0 ๐ถ.
37
We see that ๐, like ๐โฒ, gives a lift of ๐ด, and that
๐๐ = ๐ผโฒ โ ๐0๐ ๐ถ = ๐0( ๐ดโ ๐ ๐ถ) = ๐0(๐๐โ10๐ต) = ๐๐0 ๐ต.
Taking ๐ = ๐ต completes the โonly ifโ direction.
The โifโ direction is clear.
3.3 The ๐โ0 -Bockstein spectral sequence (๐โ0 -BSS)
Applying ๐ป*(๐ ;โ) to the short exact sequence of ๐ -comodules
0 // ๐/๐0 // ๐/๐โ0๐0 // ๐/๐โ0 // 0
gives a long exact sequence. We also have a trivial long exact sequence consisting
of the zero group every three terms and ๐ป*(๐ ;๐/๐โ0 ) elsewhere. Intertwining these
long exact sequences gives an exact couple, the nontrivial part, of which, looks as
follows (๐ฃ < 0).
๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐กโ๐ฃ+๐)
๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐กโ๐ฃ+๐โ1)oo . . .๐0oo ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐กโ๐ฃ)
๐0oo
๐
๐ป๐ ,๐ข(๐ ; [๐/๐0]
๐กโ๐ฃ+๐)โจ๐๐ฃโ๐0 โฉ
66
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐กโ๐ฃ)โจ๐๐ฃ0โฉ
Here ๐ raises the degree of ๐ by one relative to what is indicated and the powers of
๐0 are used to distinguish copies of ๐ป*(๐ ;๐/๐0) from one another.
Definition 3.3.1. The spectral sequence arising from this exact couple is called the
๐โ0 -Bockstein spectral sequence (๐โ0 -BSS). It has ๐ธ1-page given by
๐ธ๐ ,๐ก,๐ข,๐ฃ1 (๐โ0 -BSS) =
โงโชโจโชโฉ๐ป๐ ,๐ข(๐ ; [๐/๐0]
๐กโ๐ฃ)โจ๐๐ฃ0โฉ if ๐ฃ < 0
0 if ๐ฃ โฅ 0
and ๐๐ has degree (1, 0, 0, ๐). The spectral sequence converges to ๐ป*(๐ ;๐/๐โ0 ) and
38
the filtration degree is given by ๐ฃ. In particular, we have an identification
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ0 -BSS) = ๐น ๐ฃ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก)/๐น ๐ฃ+1๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก)
where ๐น ๐ฃ๐ป*(๐ ;๐/๐โ0 ) = ker (๐โ๐ฃ0 : ๐ป*(๐ ;๐/๐โ0 ) โโ ๐ป*(๐ ;๐/๐โ0 )) for ๐ฃ โค 0. The
identification is given by taking a permanent cycle in ๐ป*(๐ ;๐/๐0)โจ๐๐ฃ0โฉ, mapping it
up to ๐ป*(๐ ;๐/๐โ0 ) and pulling this element back to the 0th copy of ๐ป*(๐ ;๐/๐โ0 ).
Remark 3.3.2. One can describe the ๐ธ1-page of the ๐โ0 -BSS more concisely as the
๐ป*(๐ ;๐/๐0)[๐0]-module [๐ป*(๐ ;๐/๐0)
[๐0
]]/๐โ0 .
Notation 3.3.3. Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ;๐/๐0). We write ๐๐๐ฅ = ๐ฆ to mean that for
all ๐ฃ โ Z, ๐๐ฃ0๐ฅ and ๐๐ฃ0๐ฆ survive until the ๐ธ๐-page and that ๐๐๐๐ฃ0๐ฅ = ๐๐ฃ+๐0 ๐ฆ. In this case,
notice that ๐๐ฃ0๐ฅ is a permanent cycle for ๐ฃ โฅ โ๐ and that ๐๐ฃ0๐ฆ is a permanent cycle
for all ๐ฃ โ Z.
Again, ๐ฅ is said to support a ๐๐ differential. If one of the differentials ๐๐๐๐ฃ0๐ฅ = ๐๐ฃ+๐0 ๐ฆ
is nontrivial, then ๐ฅ is said to support a nontrivial differential.
3.4 The ๐-BSS and the ๐โ0 -BSS: a relationship
Suppose that ๐ฅ โ ๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก), ๐ฆ โ ๐ป๐ +1,๐ข(๐ ; [๐/๐0]
๐กโ๐). 3.2.3 and 3.3.3 give
meanings to the equation ๐๐๐ฅ = ๐ฆ in the ๐-BSS and the ๐โ0 -BSS, respectively. It
appears, a priori, that the truth of the equation ๐๐๐ฅ = ๐ฆ depends on which spectral
sequence we are working in. The following lemma shows that this is not the case.
Lemma 3.4.1. Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ;๐/๐0). Then ๐๐๐ฅ = ๐ฆ in the ๐-BSS if and only
if ๐๐๐ฅ = ๐ฆ in the ๐โ0 -BSS.
Proof. Suppose that ๐๐๐ฅ = ๐ฆ in the ๐-BSS. By lemma 3.2.4, we find that there
exist ๐ and ๐ in ฮฉ*(๐ ;๐) with ๐๐ = ๐๐0๐ such that their images in ฮฉ*(๐ ;๐/๐0) are
cocycles representing ๐ฅ and ๐ฆ, respectively. Let ๐ and ๐ be the images of ๐ and ๐ in
ฮฉ*(๐ ;๐/๐0), respectively.
39
Then we have
ฮฉ*(๐ ;๐/๐0) //
ฮฉ*(๐ ;๐/๐โ0 )๐0 //
๐
ฮฉ*(๐ ;๐/๐โ0 )
ฮฉ*(๐ ;๐/๐0) // ฮฉ*(๐ ;๐/๐โ0 )
๐0 // ฮฉ*(๐ ;๐/๐โ0 )
๐/๐๐+10
//_
๐/๐๐0_
๐ // ๐/๐0
// 0
and so
๐ป*(๐ ;๐/๐โ0 ) . . .๐0oo ๐ป*(๐ ;๐/๐โ0 )
๐0oo
๐
๐ป*(๐ ;๐/๐0)
55
๐ป*(๐ ;๐/๐0)
๐/๐0 . . .๐0oo ๐/๐๐0
๐0oo
๐
๐ฅ = ๐
55
๐ฆ = ๐
giving ๐๐๐ฅ = ๐ฆ in the ๐โ0 -BSS.
We prove the converse using induction on ๐. The result is clear for ๐ = 0 since,
by convention, ๐0 is zero for both spectral sequences. For ๐ โฅ 1 we have
๐๐๐ฅ = ๐ฆ in the ๐โ0 -BSS
=โ ๐๐โ1๐ฅ = 0 in the ๐โ0 -BSS (Lemma 2.1.4)
=โ ๐๐โ1๐ฅ = 0 in the ๐-BSS (Induction)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐-BSS for some ๐ฆโฒ (Lemma 2.1.4)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐โ0 -BSS (1st half of proof)
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐โ0 -BSS for some ๐ฅโฒ (Corollary 2.1.5)
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐-BSS (Induction)
=โ ๐๐๐ฅ = ๐ฆ in the ๐-BSS (Corollary 2.1.5)
40
which completes the proof.
3.5 The ๐โ11 -Bockstein spectral sequence (๐โ1
1 -BSS)
We can mimic the construction of the ๐โ0 -BSS using the following short exact sequence
of ๐ -comodules.
0 // ๐โ11 ๐/๐0 // ๐โ1
1 ๐/๐โ0๐0 // ๐โ1
1 ๐/๐โ0 // 0 (3.5.1)
Definition 3.5.2. The spectral sequence arising from this exact couple is called the
๐โ11 -Bockstein spectral sequence (๐โ1
1 -BSS). It has ๐ธ1-page given by
๐ธ1(๐โ11 -BSS) =
[๐ป*(๐ ; ๐โ1
1 ๐/๐0)[๐0
]]/๐โ0
and ๐๐ has degree (1, 0, 0, ๐). The spectral sequence converges to ๐ป*(๐ ; ๐โ11 ๐/๐โ0 )
and the filtration degree is given by ๐ฃ. In particular, we have an identification
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ1
1 -BSS) = ๐น ๐ฃ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก)/๐น ๐ฃ+1๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐โ0 ]๐ก)
where, as in the ๐โ0 -BSS, ๐น ๐ฃ = ker ๐โ๐ฃ0 for ๐ฃ โค 0. The identification is given by
taking a permanent cycle in ๐ป*(๐ ; ๐โ11 ๐/๐0)โจ๐๐ฃ0โฉ, mapping it up to ๐ป*(๐ ; ๐โ1
1 ๐/๐โ0 )
and pulling this element back to the 0th copy of ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ).
We follow the notational conventions in 3.3.3.
We notice, that as a consequence of lemma 3.4.1, a ๐๐-differential in the ๐โ0 -BSS
can be validated using only elements in ฮฉ*(๐ ;๐๐+1) (see definition 3.1.4). The same
can be said of the ๐โ11 -BSS and the proof is similar. The following lemma statement
makes use of the connecting homomorphism in the long exact sequence coming from
the short exact sequence of ๐ -comodules
0 // ๐โ11 ๐/๐0 // ๐โ1
1 ๐๐+1๐0 // ๐โ1
1 ๐๐// 0.
41
Lemma 3.5.3. Suppose that ๐ฅ, ๐ฆ โ ๐ป*(๐โ11 ๐/๐0) and that ๐๐๐ฅ = ๐ฆ in the ๐โ1
1 -BSS.
Then there exist โ ๐ป*(๐ ; ๐โ11 ๐1) and ๐ฆ โ ๐ป*(๐ ; ๐โ1
1 ๐๐) with the properties that
= ๐๐โ10 ๐ฆ, and under the maps
๐ป*(๐ ; ๐โ11 ๐/๐0)
โผ=โโ ๐ป*(๐ ; ๐โ11 ๐1), ๐ : ๐ป*(๐ ; ๐โ1
1 ๐๐) โโ ๐ป*(๐โ11 ๐/๐0),
๐ฅ is mapped to , and ๐ฆ is mapped to ๐ฆ, respectively. We summarize this situation by
saying that ๐๐๐ฅ = ๐ฆ in the ๐๐+1-zig-zag.
Proof. The result is clear for ๐ = 1 and so we proceed by induction on ๐. For ๐ > 1
we have
๐๐๐ฅ = ๐ฆ in the ๐โ11 -BSS
=โ ๐๐โ1๐ฅ = 0 in the ๐โ11 -BSS (Lemma 2.1.4)
=โ ๐๐โ1๐ฅ = 0 in the ๐๐-zig-zag (Induction)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐๐+1-zig-zag for some ๐ฆโฒ (Lemma 2.1.4)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐โ11 -BSS
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐โ1
1 -BSS for some ๐ฅโฒ (Corollary 2.1.5)
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐๐-zig-zag (Induction)
=โ ๐๐๐ฅ = ๐ฆ in the ๐๐+1-zig-zag (Corollary 2.1.5)
which completes the proof.
We note the following simple result.
Lemma 3.5.4. In the ๐โ11 -BSS we have ๐๐๐โ1๐
ยฑ๐๐1 = 0.
Proof. One sees that ๐ยฑ๐๐
1 /๐๐๐
0 โ ฮฉ*(๐ ; ๐โ11 ๐/๐โ0 ) is a cocycle.
We have an evident map of spectral sequences
๐ธ*,*,*,** (๐โ0 -BSS) โโ ๐ธ*,*,*,*
* (๐โ11 -BSS).
42
3.6 Multiplicativity of the BSSs
The ๐-BSS is multiplicative because ฮฉ*(๐ ;๐) โโ ฮฉ*(๐ ;๐/๐0) is a map of DG
algebras.
Lemma 3.6.1. Suppose ๐ฅ, ๐ฅโฒ, ๐ฆ, ๐ฆโฒ โ ๐ป*(๐ ;๐/๐0) and that ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ in
the ๐-BSS. Then
๐๐(๐ฅ๐ฅโฒ) = ๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ.
Here |๐ฅ| and |๐ฆ| denote the cohomological gradings of ๐ฅ and ๐ฆ, respectively, since
every element of ๐ , ๐ and ๐/๐0 has even ๐ข grading.
Proof. Suppose ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ.
Lemma 3.2.4 tells us that there exist ๐, ๐โฒ, ๐, ๐โฒ โ ฮฉ*(๐ ;๐) such that their images
in ฮฉ*(๐ ;๐/๐0) represent ๐ฅ, ๐ฅโฒ, ๐ฆ, ๐ฆโฒ, respectively, and such that ๐๐ = ๐๐0๐, ๐๐โฒ = ๐๐0๐โฒ.
The image of ๐๐โฒ โ ฮฉ*(๐ ;๐) in ฮฉ*(๐ ;๐/๐0) represents ๐ฅ๐ฅโฒ and the image of
๐๐โฒ + (โ1)|๐|๐๐โฒ โ ฮฉ*(๐ ;๐)
in ฮฉ*(๐ ;๐/๐0) represents ๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ. Since ๐(๐๐โฒ) = ๐๐0(๐๐โฒ + (โ1)|๐|๐๐โฒ), lemma
3.2.4 completes the proof.
Corollary 3.6.2. We have a multiplication
๐ธ๐ ,๐ก,๐ข,๐ฃ1 (๐-BSS)โ ๐ธ๐ โฒ,๐กโฒ,๐ขโฒ,๐ฃโฒ
1 (๐-BSS) โโ ๐ธ๐ +๐ โฒ,๐ก+๐กโฒ,๐ข+๐ขโฒ,๐ฃ+๐ฃโฒ
1 (๐-BSS)
restricting to the following maps.
ker ๐๐ โ im ๐๐ //
im ๐๐
โ๐ ker ๐๐ โ
โ๐ im ๐๐ //
โ๐ im ๐๐
ker ๐๐ โ ker ๐๐ // ker ๐๐
โ๐ ker ๐๐ โ
โ๐ ker ๐๐ //
โ๐ ker ๐๐
im ๐๐ โ ker ๐๐ //
OO
im ๐๐
OO
โ๐ im ๐๐ โ
โ๐ ker ๐๐ //
OO
โ๐ im ๐๐
OO
43
Thus we have induced maps
๐ธ๐ ,๐ก,๐ข,๐ฃ๐ (๐-BSS)โ ๐ธ๐ โฒ,๐กโฒ,๐ขโฒ,๐ฃโฒ
๐ (๐-BSS) โโ ๐ธ๐ +๐ โฒ,๐ก+๐กโฒ,๐ข+๐ขโฒ,๐ฃ+๐ฃโฒ
๐ (๐-BSS)
for 1 โค ๐ โค โ. Moreover,
๐ธ๐ ,๐ก,๐ข,*โ (๐-BSS)โ ๐ธ๐ โฒ,๐กโฒ,๐ขโฒ,*
โ (๐-BSS) โโ ๐ธ๐ +๐ โฒ,๐ก+๐กโฒ,๐ข+๐ขโฒ,*โ (๐-BSS)
is the associated graded of the map
๐ป๐ ,๐ข(๐ ;๐๐ก)โ๐ป๐ โฒ,๐ขโฒ(๐ ;๐๐กโฒ) โโ ๐ป๐ +๐ โฒ,๐ข+๐ขโฒ(๐ ;๐๐ก+๐กโฒ).
Lemma 3.4.1 means that we have the following corollary to the previous lemma.
Corollary 3.6.3. Suppose ๐ฅ, ๐ฅโฒ, ๐ฆ, ๐ฆโฒ โ ๐ป*(๐ ;๐/๐0) and that ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ
in the ๐โ0 -BSS. Then
๐๐(๐ฅ๐ฅโฒ) = ๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ.
The ๐โ0 -BSS is not multiplicative in the sense that we do not have a strict analogue
of corollary 3.6.2. This is unsurprising because๐ป*(๐ ;๐/๐โ0 ) does not have an obvious
algebra structure. However, we do have a pairing between the ๐-BSS and the ๐โ0 -BSS
converging to the ๐ป*(๐ ;๐)-module structure map of ๐ป*(๐ ;๐/๐โ0 ).
An identical result to lemma 3.6.1 holds for the ๐โ11 -BSS.
Lemma 3.6.4. Suppose ๐ฅ, ๐ฅโฒ, ๐ฆ, ๐ฆโฒ โ ๐ป*(๐ ; ๐โ11 ๐/๐0) and that ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ
in the ๐โ11 -BSS. Then
๐๐(๐ฅ๐ฅโฒ) = ๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ.
Proof. Suppose that ๐๐๐ฅ = ๐ฆ in the ๐โ11 -BSS. We claim that for large enough ๐ the
elements ๐๐๐๐
1 ๐ฅ and ๐๐๐๐
1 ๐ฆ lift to elements ๐ and ๐ in ๐ป*(๐ ;๐/๐0) with the property
that ๐๐๐ = ๐ in the ๐โ0 -BSS.
By lemma 3.5.3, we have and ๐ฆ demonstrating that ๐๐๐ฅ = ๐ฆ in the ๐๐+1-zig-zag.
Using definition 3.1.6 and the fact that filtered colimits commute with tensor products
and homology, we can find a ๐ such that ๐๐๐๐
1 ๐ฅ and ๐๐๐๐
1 ๐ฆ lift to ๐ โ ๐ป*(๐ ;๐/๐0) and
44
๐ โ ๐ป*(๐ ;๐๐), respectively, and such that their images in ๐ป*(๐ ;๐1) coincide. Let
๐ โ ๐ป*(๐ ;๐/๐0) be the image of ๐ . Then ๐ lifts ๐๐๐๐
1 ๐ฆ and ๐๐๐ = ๐ in the ๐โ0 -BSS,
proving the claim.
Suppose that ๐๐๐ฅ = ๐ฆ and ๐๐๐ฅโฒ = ๐ฆโฒ in the ๐โ11 -BSS. For large enough ๐, we obtain
elements ๐, ๐ โฒ, ๐ and ๐ โฒ lifting ๐๐๐๐
1 ๐ฅ, ๐๐๐๐
1 ๐ฅโฒ, ๐๐๐๐
1 ๐ฆ and ๐๐๐๐
1 ๐ฆโฒ, respectively, and
differentials ๐๐๐ = ๐ and ๐๐๐ โฒ = ๐ โฒ in the ๐โ0 -BSS. The previous corollary gives
๐๐(๐๐โฒ) = ๐ ๐ โฒ + (โ1)|๐|๐๐ โฒ.
Mapping into the ๐โ11 -BSS and using lemma 3.5.3 we obtain
๐๐(๐2๐๐๐
1 (๐ฅ๐ฅโฒ)) = ๐2๐๐๐
1 (๐ฆ๐ฅโฒ + (โ1)|๐ฅ|๐ฅ๐ฆโฒ)
in the ๐๐+1-zig-zag. Dividing through by ๐2๐๐๐
1 completes the proof.
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46
Chapter 4
Vanishing lines and localization
In this chapter we prove some vanishing lines for ๐ป*(๐ ;Q) with various choices of Q.
We also analyze the localization map ๐ป*(๐ ;๐/๐โ0 ) โโ ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ).
4.1 Vanishing lines
We make note of vanishing lines for ๐ป*(๐ ;Q) in the cases (see section 3.1 for defini-
tions)
Q = ๐/๐0, ๐โ11 ๐/๐0, ๐๐, ๐
โ11 ๐๐, ๐/๐
โ0 , ๐
โ11 ๐/๐โ0 .
Notation 4.1.1. We write ๐ for |๐1| = 2๐โ 2.
Definition 4.1.2. For ๐ โ Zโฅ0 let ๐(2๐ ) = ๐๐๐ and ๐(2๐ + 1) = ๐๐๐ + ๐ and write
๐(โ1) =โ.
In [10] Miller uses the following result.
Lemma 4.1.3. ๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) = 0 when ๐ข < ๐(๐ ) + ๐๐ก.
Since ๐1 has (๐ก, ๐ข) bigrading (1, ๐) we obtain the following corollary.
Corollary 4.1.4. ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก) = 0 when ๐ข < ๐(๐ ) + ๐๐ก.
Lemma 4.1.5. For each ๐ โฅ 1, ๐ป๐ ,๐ข(๐ ; [๐๐]๐ก) = 0 whenever ๐ข < ๐(๐ ) + ๐(๐ก+ 1).
47
Proof. We proceed by induction on ๐.
The previous corollary together with the isomorphism ๐ป*(๐ ;๐/๐0) โผ= ๐ป*(๐ ;๐1)
gives the base case.
The long exact sequence associated to the short exact sequence of ๐ -comodules
0 โโ ๐1 โโ ๐๐+1๐0โโ ๐๐ โโ 0 shows ๐ป๐ ,๐ข(๐ ; [๐๐+1]
๐ก) is zero provided that
๐ป๐ ,๐ข(๐ ; [๐1]๐ก) and ๐ป๐ ,๐ข([๐๐]๐ก+1) are zero. Since ๐ข < ๐(๐ ) + ๐(๐ก + 1) implies that
๐ข < ๐(๐ ) + ๐((๐ก+ 1) + 1) the inductive step is complete.
Corollary 4.1.6. For Q = ๐๐, ๐โ11 ๐๐, ๐/๐
โ0 , or ๐โ1
1 ๐/๐โ0 we have
๐ป๐ ,๐ข(๐ ;Q๐ก) = 0 whenever ๐ข < ๐(๐ ) + ๐(๐ก+ 1).
Notation 4.1.7. We write (๐, ๐) for (๐ + ๐ก, ๐ข+ ๐ก).
Since (๐ + 1)๐ โ 1 โค ๐(๐ ) we have the following corollaries.
Corollary 4.1.8. For Q = ๐/๐0 or ๐โ11 ๐/๐0 we have
๐ป๐ ,๐ข(๐ ;Q๐ก) = 0 whenever ๐โ ๐ < ๐๐ โ 1.
Corollary 4.1.9. For Q = ๐๐, ๐/๐โ0 , ๐
โ11 ๐๐, or ๐โ1
1 ๐/๐โ0 we have
๐ป๐ ,๐ข(๐ ;Q๐ก) = 0 whenever ๐โ ๐ < ๐(๐ + 1)โ 1.
Lemma 4.1.10. For ๐ โฅ 1, ๐ป๐ ,๐ข(๐ ; [๐๐]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) is
1. surjective when ๐โ ๐ = ๐๐โ1๐ and ๐ โฅ ๐๐โ1 โ ๐.
2. injective when ๐โ ๐ = ๐๐โ1๐ โ 1 and ๐ โฅ ๐๐โ1 โ ๐+ 1;
Proof. The previous corollary tells us that ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) = 0 when ๐โ ๐ = ๐๐โ1๐
and ๐ โฅ ๐๐โ1. The following exact sequence completes the proof.
๐ป๐ โ1,๐ข(๐ ; [๐/๐โ0 ]๐ก+๐) // ๐ป๐ ,๐ข(๐ ; [๐๐]๐ก) // ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) // ๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก+๐)
48
4.2 The localization map: the trigraded perspective
In this section we analyze the map ๐ป*(๐ ;๐/๐โ0 ) โโ ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ). In particular,
we find a range in which it is an isomorphism. The result which allows us to do this
follows. Throughout this section ๐ โฅ 0. Recall definition 4.1.2.
Proposition 4.2.1 ([10, pg. 81]). The localization map
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐0]๐ก)
1. is injective if ๐ข < ๐(๐ โ 1) + (2๐2 โ 2)(๐ก+ 1)โ ๐;
2. is surjective if ๐ข < ๐(๐ ) + (2๐2 โ 2)(๐ก+ 1)โ ๐.
This allows us to prove the following lemma which explains how we can transfer
differentials between the ๐โ0 -BSS and the ๐โ11 -BSS.
Lemma 4.2.2. Suppose ๐ข < ๐(๐ )+(2๐2โ2)(๐ก+2)โ๐ so that proposition 4.2.1 gives a
surjection ๐ธ๐ ,๐ก,๐ข,*1 (๐โ0 -BSS)โ ๐ธ๐ ,๐ก,๐ข,*
1 (๐โ11 -BSS) and an injection ๐ธ๐ +1,๐ก,๐ข,*
1 (๐โ0 -BSS)โ
๐ธ๐ +1,๐ก,๐ข,*1 (๐โ1
1 -BSS).
Suppose ๐ฅ โ ๐ธ๐ ,๐ก,๐ข,*1 (๐โ0 -BSS) maps to ๐ฅ โ ๐ธ๐ ,๐ก,๐ข,*
1 (๐โ11 -BSS) and that ๐๐๐ฅ = ๐ฆ in
the ๐โ11 -BSS. Then, in fact, ๐ฆ lies in ๐ธ๐ +1,๐ก,๐ข,*
1 (๐โ0 -BSS) and ๐๐๐ฅ = ๐ฆ in the ๐โ0 -BSS.
Proof. We proceed by induction on ๐. The result is true in the case ๐ = 0 where
๐0 = 0 and the case ๐ = 1 where ๐๐ is a function. Suppose ๐ > 1. Then
๐๐๐ฅ = ๐ฆ in the ๐โ11 -BSS
=โ ๐๐โ1๐ฅ = 0 in the ๐โ11 -BSS (Lemma 2.1.4)
=โ ๐๐โ1๐ฅ = 0 in the ๐โ0 -BSS (Induction)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐โ0 -BSS for some ๐ฆโฒ (Lemma 2.1.4)
=โ ๐๐๐ฅ = ๐ฆโฒ in the ๐โ11 -BSS (Map of SSs)
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐โ1
1 -BSS for some ๐ฅโฒ (Corollary 2.1.5)
=โ ๐๐โ1๐ฅโฒ = ๐ฆโฒ โ ๐ฆ in the ๐โ0 -BSS (Induction)
=โ ๐๐๐ฅ = ๐ฆ in the ๐โ0 -BSS (Corollary 2.1.5)
49
We remark that the statement about ๐ฆ lying in ๐ธ๐ +1,๐ก,๐ข,*1 (๐โ0 ) is actually trivial: the
map ๐ธ๐ +1,๐ก,๐ข,*1 (๐โ0 -BSS) โโ ๐ธ๐ +1,๐ก,๐ข,*
1 (๐โ11 -BSS) is an isomorphism since ๐ โฅ 0 implies
๐(๐ ) < ๐(๐ + 1).
Corollary 4.2.3. ๐ธ๐ ,๐ก,๐ข,*โ (๐โ0 -BSS) โโ ๐ธ๐ ,๐ก,๐ข,*
โ (๐โ11 -BSS) is
1. injective if ๐ข < ๐(๐ โ 1) + (2๐2 โ 2)(๐ก+ 2)โ ๐;
2. surjective if ๐ข < ๐(๐ ) + (2๐2 โ 2)(๐ก+ 2)โ ๐.
Proof. Suppose ๐ข < ๐(๐ ) + (2๐2โ 2)(๐ก+ 2)โ ๐ and that ๐ฆ โ ๐ธ๐ +1,๐ก,๐ข,*โ (๐โ0 -BSS) maps
to zero in ๐ธ๐ +1,๐ก,๐ข,*โ (๐โ1
1 -BSS). This says that ๐๐๐ฅ = ๐ฆ for some ๐ฅ in ๐ธ๐ ,๐ก,๐ข,*1 (๐โ0 -BSS).
By the previous lemma ๐๐๐ฅ = ๐ฆ, which says that ๐ฆ is zero in ๐ธ๐ +1,๐ก,๐ข,*โ (๐โ0 -BSS). This
proves the first statement when ๐ > 0. For ๐ = 0, the result is clear since it holds at
the ๐ธ1-page and the only boundary is zero.
Suppose ๐ข < ๐(๐ )+(2๐2โ2)(๐ก+2)โ๐ and we have an element of ๐ธ๐ ,๐ก,๐ข,*โ (๐โ1
1 -BSS).
We can write this element as ๐ฅ for ๐ฅ โ ๐ธ๐ ,๐ก,๐ข,*1 (๐โ0 -BSS). Moreover, since ๐๐๐ฅ = 0 for
each ๐ the previous lemma tells us that each ๐๐๐ฅ = 0 for each ๐, i.e. ๐ฅ is a permanent
cycle, as is required to prove the second statement.
Proposition 4.2.4. The localization map
๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก)
1. is injective if ๐ข < ๐(๐ โ 1) + (2๐2 โ 2)(๐ก+ 2)โ ๐;
2. is surjective if ๐ข < ๐(๐ ) + (2๐2 โ 2)(๐ก+ 2)โ ๐.
Proof. We have ๐ป*(๐ ;Q) =โ๐ฃ ๐น
๐ฃ๐ป*(๐ ;Q) and ๐น 0๐ป*(๐ ;Q) = 0 when Q = ๐/๐โ0
or ๐โ11 ๐/๐โ0 and so the result follows from the previous corollary.
4.3 The localization map: the bigraded perspective
Recall the bigrading (๐, ๐) of definition 4.1.7. We prove the analogues of the results
of the last section with respect to this bigrading.
50
Proposition 4.3.1 ([10, 4.7(๐)]). The localization map
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐0]๐ก)
1. is a surjection if ๐ โฅ 0 and ๐ < ๐(๐ + 1)โ ๐ โ 1;
2. is an isomorphism if ๐ โฅ 0 and ๐ < ๐(๐)โ ๐ โ 1.
Corollary 4.3.2. The localization map
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐0]๐ก)
1. is a surjection if ๐ < ๐(๐โ 1)๐ โ 1, i.e. ๐โ ๐ < (๐2 โ ๐โ 1)๐ โ 1;
2. is an isomorphism if ๐ < ๐(๐โ 1)(๐ โ 1)โ 1,
i.e. ๐โ ๐ < (๐2 โ ๐โ 1)(๐ โ 1)โ 2.
Proof. Consider ๐(๐) = ๐(๐โ 1)๐ โ ๐(๐) for ๐ โฅ 0. We have ๐(1) = ๐(๐โ 3) + 2 โฅ
0 = ๐(0) and ๐(๐ + 2) = ๐(๐). Thus ๐(๐โ 1)๐ โ ๐(๐) โค ๐(๐โ 3) + 2 and so
๐(๐โ 1)(๐ โ 1)โ 1 โค[๐(๐) + ๐(๐โ 3) + 2
]โ ๐(๐โ 1)โ 1 = ๐(๐)โ ๐ โ 1.
Together with the previous proposition, this proves the claim for ๐ โฅ 0.
When ๐ < 0, ๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) = 0 and so the localization map is injective. We
just need to prove that ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก) = 0 whenever ๐โ ๐ < (๐2 โ ๐โ 1)๐ โ 1
and ๐ < 0. We can only have [(๐โ ๐) + 1]/(๐2 โ ๐โ 1) < ๐ < 0 if (๐โ ๐) + 1 < 0.
But then [(๐ โ ๐) + 1]/๐ < ๐ < 0 and the vanishing line of corollary 4.1.8 gives the
result.
This allows us to prove bigraded versions of all the results of the previous subsec-
tion. In particular, we have the following proposition.
51
Proposition 4.3.3. The localization map
๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก) โโ ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก)
1. is a surjection if ๐ < ๐(๐โ 1)(๐ + 1)โ 2, i.e. ๐โ ๐ < (๐2 โ ๐โ 1)(๐ + 1)โ 1;
2. is an isomorphism if ๐ < ๐(๐โ 1)๐ โ 2, i.e. ๐โ ๐ < (๐2 โ ๐โ 1)๐ โ 2.
52
Chapter 5
Calculating the 1-line of the ๐-CSS;
its image in ๐ป*(๐ด)
This chapter contains our main result. We calculate the ๐โ11 -BSS
[๐ป*(๐ ; ๐โ1
1 ๐/๐0)[๐0
]]/๐โ0
๐ฃ=โ ๐ป*(๐ ; ๐โ1
1 ๐/๐โ0 ).
In the introduction we discussed โprincipal towersโ and their โside towers.โ Our
presentation of the results is divided up in this way, too.
5.1 The ๐ธ1-page of the ๐โ11 -BSS
Our starting place for the calculation of the ๐โ11 -BSS is a result of Miller in [10] which
gives a description of ๐ธ1(๐โ11 -BSS).
Definition 5.1.1. Denote by ๐ โฒ the Hopf algebra obtained from ๐ by quotienting
out the ideal generated by the image of the ๐-th power map ๐ โโ ๐ , ๐ โฆโโ ๐๐.
We can make F๐[๐1] into an algebra in ๐ โฒ-comodules by defining ๐1 to be a comod-
ule primitive. The map ๐/๐0 โโ ๐/(๐0, ๐2, ๐3, . . .) = F๐[๐1] is an algebra map over
the Hopf algebra map ๐ โโ ๐ โฒ. Thus, we have the following induced map.
ฮฉ*(๐ ; ๐โ11 ๐/๐0) โโ ฮฉ*(๐ โฒ;F๐[๐ยฑ1
1 ]) (5.1.2)
53
Theorem 5.1.3 (Miller, [10, 4.4]). The map ๐ป*(๐ ; ๐โ11 ๐/๐0) โโ ๐ป*(๐ โฒ;F๐[๐ยฑ1
1 ]) is
an isomorphism.
[๐๐] andโ๐โ1
๐=1(โ1)๐โ1
๐[๐๐๐|๐๐โ๐๐ ] are cocycles in ฮฉ*(๐ โฒ) and so they define elements
โ๐,0 and ๐๐,0 in ๐ป*(๐ โฒ;F๐). The cohomology of a primitively generated Hopf algebra
is well understood and the following lemma is a consequence.
Lemma 5.1.4. ๐ป*(๐ โฒ;F๐) = ๐ธ[โ๐,0 : ๐ โฅ 1]โ F๐[๐๐,0 : ๐ โฅ 1].
Corollary 5.1.5. ๐ป*(๐ ; ๐โ11 ๐/๐0) = F๐[๐ยฑ1
1 ]โ๐ธ[โ๐,0 : ๐ โฅ 1]โ F๐[๐๐,0 : ๐ โฅ 1]. The
(๐ , ๐ก, ๐ข) trigradings are as follows.
|๐1| = (0, 1, 2๐โ 2), |โ๐,0| = (1, 0, 2๐๐ โ 2), |๐๐,0| = (2, 0, ๐(2๐๐ โ 2)).
For our work it is convenient to change these exterior and polynomial generators
by units.
Notation 5.1.6. For ๐ โฅ 1, let ๐[๐] = ๐๐โ1๐โ1
, ๐๐ = ๐โ๐[๐]
1 โ๐,0, and ๐๐ = ๐โ๐ยท๐[๐]
1 ๐๐,0.
Let ๐[0] = 0 and note that we have ๐[๐+1] = ๐๐ + ๐[๐] = ๐ ยท ๐[๐] + 1 for ๐ โฅ 0.
Corollary 5.1.7. ๐ป*(๐ ; ๐โ11 ๐/๐0) = F๐[๐ยฑ1
1 ] โ ๐ธ[๐๐ : ๐ โฅ 1] โ F๐[๐๐ : ๐ โฅ 1]. The
(๐ , ๐ก, ๐ข) trigradings are as follows.
|๐1| = (0, 1, 2๐โ 2), |๐๐| = (1,โ๐[๐], 0), |๐๐| = (2, 1โ ๐[๐+1], 0).
We make note of some elements that lift uniquely to ๐ป*(๐ ;๐/๐0).
Lemma 5.1.8. The elements
1, ๐2๐๐โ1
1 ๐๐, ๐1,0 = ๐๐1๐1, ๐2๐๐
1 ๐๐ โ ๐ป*(๐ ; ๐โ11 ๐/๐0)
have unique lifts to ๐ป*(๐ ;๐/๐0). The same is true after multiplying by ๐๐1 as long as
๐ โฅ 0.
54
Proof. We use proposition 4.2.1. The (๐ , ๐ก, ๐ข) trigradings of the elements in the lemma
are
(0, 0, 0), (1, 2๐๐โ1 โ ๐[๐], 2๐๐๐โ1), (2, 0, ๐๐), (2, 2๐๐ โ ๐[๐+1] + 1, 2๐๐๐),
respectively. In each case (๐ , ๐ก, ๐ข) satisfies ๐ข < ๐(๐ โ 1) + (2๐2 โ 2)(๐ก + 1) โ ๐ and
๐ข < ๐(๐ ) + (2๐2 โ 2)(๐ก+ 1)โ ๐; the key inequalities one needs are ๐ < 2๐2 โ 2 and
2๐๐๐โ1 < (2๐2 โ 2)(2๐๐โ1 โ ๐[๐] + 1)โ ๐. (5.1.9)
The latter inequality is equivalent to (๐ + 1)๐[๐] < 2๐๐ + ๐, which holds because
๐ โฅ 3. Since ๐ < 2๐2 โ 2 multiplication by a positive power of ๐1 only makes things
better.
5.2 The first family of differentials, principal towers
5.2.1 Main results
Notation 5.2.1.1. We write .= to denote equality up to multiplication by an element
in Fร๐ .
The main results of this section are as follows. The first concerns the ๐โ11 -BSS
and the second gives the corresponding result in the ๐-BSS.
Proposition 5.2.1.2. For ๐ โฅ 1 and ๐ โ Z โ ๐Z we have the following differential
in the ๐โ11 -BSS.
๐๐[๐]๐๐๐๐โ1
1.
= ๐๐๐๐โ1
1 ๐๐
Proposition 5.2.1.3. Let ๐ โฅ 1. We have the following differential in the ๐-BSS.
๐๐๐โ1๐๐๐โ1
1.
= โ1,๐โ1
Moreover, if ๐ โ Zโ ๐Z and ๐ > 1, ๐๐[๐]๐๐๐๐โ1
1 is defined in the ๐-BSS.
55
5.2.2 Quick proofs
The differentials in the ๐โ11 -BSS are derivations (lemma 3.6.4) and ๐๐[๐]๐
โ๐๐1 = 0
(lemma 3.5.4). This means that proposition 5.2.1.2 follows quickly from the following
sub-proposition.
Proposition 5.2.2.1. For ๐ โฅ 1 we have the following differential in the ๐โ11 -BSS.
๐๐[๐]๐๐๐โ1
1.
= ๐๐๐โ1
1 ๐๐
This is the consuming calculation of the section. Supposing this result for now,
we prove proposition 5.2.1.3.
Proof of proposition 5.2.1.3. The formula ๐(
[ ]๐๐๐โ1
1
)= [๐๐
๐โ1
1 ]๐๐๐โ1
0 in ฮฉ*(๐ ;๐), to-
gether with lemma 3.2.4 proves the first statement.
By lemma 3.4.1, we can verify the second statement in the ๐โ0 -BSS. We have
๐๐ฃ0๐๐๐๐โ1
1 โ ๐ธ0,๐๐๐โ1+๐ฃ,๐๐๐๐โ1,๐ฃ1 (๐โ0 -BSS)
and we will show that ๐โ๐[๐]โ1
0 ๐๐๐๐โ1
1 survives to the ๐ธ๐[๐]-page. Proposition 5.2.1.2 and
lemma 4.2.2 say that it is enough to verify that (๐ , ๐ก, ๐ข) = (0, ๐๐๐โ1โ ๐[๐]โ 1, ๐๐๐๐โ1)
satisfies ๐ข < ๐(๐ ) + (2๐2 โ 2)(๐ก + 2) โ ๐. Since ๐ < 2๐2 โ 2 the worst case is when
๐ = 2 where the inequality is (5.1.9).
5.2.3 The proof of proposition 5.2.2.1
We prove proposition 5.2.2.1 via the following cocycle version of the statement.
Proposition 5.2.3.1. For each ๐ โฅ 1, there exist cocycles
๐ฅ๐ โ ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ), ๐ฆ๐ โ ฮฉ1(๐ ; ๐โ1
1 ๐/๐0)
such that
1. ๐๐[๐]โ1
0 ๐ฅ๐ = ๐โ10 ๐๐
๐โ1
1 ,
56
2. ๐ฆ๐ = ๐0๐(๐โ10 ๐ฅ๐),
3. the image of ๐ฆ๐ in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) is (โ1)๐โ1[๐๐]๐โ๐
[๐โ1]
1 .
In the expression ๐0๐(๐โ10 ๐ฅ๐), ๐โ1
0 ๐ฅ๐ denotes the element of ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ) with
the following two properties:
1. multiplying by ๐0 gives ๐ฅ๐;
2. the denominators of the terms in ๐โ10 ๐ฅ๐ have ๐0 raised to a power greater than
or equal to 2.
Thus, ๐0๐(๐โ10 ๐ฅ๐) gives a particular representative for the image of the class of ๐ฅ๐
under the boundary map ๐ : ๐ป0(๐ ; ๐โ11 ๐/๐โ0 ) โโ ๐ป1(๐ ; ๐โ1
1 ๐/๐0) coming from the
short exact sequence (3.5.1).
To illuminate the statement of the proposition we draw the relevant diagrams.
ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ) ฮฉ0(๐ ; ๐โ1
1 ๐/๐โ0 )๐๐
[๐]โ10oo
๐0๐(๐โ10 (โ))
ฮฉ0(๐ ; ๐โ1
1 ๐/๐0)
55
ฮฉ1(๐ ; ๐โ11 ๐/๐0)
ฮฉ0(๐ โฒ;F๐[๐ยฑ1
1 ]) ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ])
๐โ10 ๐๐
๐โ1
1 ๐ฅ๐๐๐
[๐]โ10oo
_
๐0๐(๐โ10 (โ))
๐๐
๐โ1
1_
,
66
๐ฆ๐_
๐๐๐โ1
1 (โ1)๐โ1[๐๐]๐โ๐[๐โ1]
1
Passing to cohomology and using theorem 5.1.3, we see that the proposition implies
that ๐๐[๐]๐๐๐โ1
1 = (โ1)๐โ1๐โ๐[๐โ1]
1 โ๐,0.
= ๐๐๐โ1
1 ๐๐, as required.
57
We note that for the ๐ = 1 and ๐ = 2 cases of the proposition we can take
๐ฅ1 = ๐โ10 ๐1, ๐ฆ1 = [๐1], ๐ฅ2 = ๐โ๐โ1
0 ๐๐1 โ ๐โ10 ๐โ1
1 ๐2, ๐ฆ2 = [๐2]๐โ11 + [๐1]๐
โ21 ๐2.
Sketch proof of proposition 5.2.3.1. We proceed by induction on ๐. So suppose that
we have cocycles ๐ฅ๐ and ๐ฆ๐ satisfying the statements in the proposition. Write ๐ 0๐ฅ๐
and ๐ 0๐ฆ๐ for the cochains in which we have raised every symbol to the ๐th power. We
claim that:
1. ๐ 0๐ฅ๐ and ๐ 0๐ฆ๐ are cocyles;
2. ๐๐[๐+1]โ2
0 ๐ 0๐ฅ๐ = ๐โ10 ๐๐
๐
1 ;
3. ๐ 0๐ฆ๐ = ๐0๐(๐โ10 ๐ 0๐ฅ๐).
Since ๐ฆ๐ maps to (โ1)๐โ1[๐๐]๐โ๐[๐โ1]
1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) and ๐๐๐ is zero in ๐ โฒ, ๐ 0๐ฆ๐
maps to 0. By theorem 5.1.3, we deduce that there exists a ๐ค๐ โ ฮฉ0(๐ ; ๐โ11 ๐/๐0)
with ๐๐ค๐ = ๐ 0๐ฆ๐. We summarize some of this information in the following diagram.
ฮฉ*(๐ ; ๐โ11 ๐/๐0) // ฮฉ*(๐ ; ๐โ1
1 ๐/๐โ0 ) // ฮฉ*(๐ ; ๐โ11 ๐/๐โ0 )
๐ค๐_
๐
๐โ10 ๐ 0๐ฅ๐
//_
๐
๐ 0๐ฅ๐
๐ 0๐ฆ๐ // ๐โ1
0 ๐ 0๐ฆ๐
Let ๐ฅ๐+1 = ๐โ10 ๐ 0๐ฅ๐โ ๐โ1
0 ๐ค๐, a cocycle in ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ) and ๐ฆ๐+1 = ๐0๐(๐โ1
0 ๐ฅ๐+1),
a cocycle in ฮฉ1(๐ ; ๐โ11 ๐/๐0). We claim that:
1. ๐๐[๐+1]โ1
0 ๐ฅ๐+1 = ๐โ10 ๐๐
๐
1 ;
2. ๐ฆ๐+1 = ๐0๐(๐โ10 ๐ฅ๐+1);
3. the image of ๐ฆ๐+1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) is (โ1)๐[๐๐+1]๐
โ๐[๐]
1 .
58
The first claim follows from the claim above that ๐๐[๐+1]โ2
0 ๐ 0๐ฅ๐ = ๐โ10 ๐๐
๐
1 , since then
๐๐[๐+1]โ1
0 ๐ฅ๐+1 = ๐๐[๐+1]โ1
0 [๐โ10 ๐ 0๐ฅ๐ โ ๐โ1
0 ๐ค๐] = ๐๐[๐+1]โ2
0 ๐ 0๐ฅ๐ = ๐โ10 ๐๐
๐
1 . The second
claim holds by definition of ๐ฆ๐+1.
In order to convert the sketch proof into a proof we must prove the first three
claims and the final claim. The next lemma takes care of the first two claims.
Lemma 5.2.3.2. Suppose that ๐ฅ โ ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ) and ๐ฆ โ ฮฉ1(๐ ; ๐โ1
1 ๐/๐0) are
cocycles. Then ๐ 0๐ฅ and ๐ 0๐ฆ are cocyles, too. Moreover, ๐๐[๐]โ1
0 ๐ฅ = ๐โ10 ๐๐
๐โ1
1 implies
๐๐[๐+1]โ2
0 ๐ 0๐ฅ = ๐โ10 ๐๐
๐
1 .
Proof. The result is clear for ๐ 0๐ฆ since Fr : ๐ โโ ๐ , ๐ โฆโโ ๐๐ is a Hopf algebra map
and ๐โ11 ๐/๐0 โโ ๐โ1
1 ๐/๐0, q โฆโโ q๐ is an algebra map over Fr.
Suppose that ๐ฅ and ๐ 0๐ฅ involve negative powers of ๐0 at worst ๐โ๐0 and that ๐ฅ
involves negative powers of ๐1 at worst ๐โ๐1 . Then we have the following sequence of
injections (recall definitions 3.1.4 through 3.1.7).
ฮฉ*(๐ ;๐๐(๐)) // ฮฉ*(๐ ;๐๐(๐๐)) // ฮฉ*(๐ ; ๐โ11 ๐๐) // ฮฉ*(๐ ; ๐โ1
1 ๐/๐โ0 )
๐๐๐๐โ1
1 ๐ฅ // ๐ฅ // ๐ฅ
๐๐๐๐
1 ๐ 0๐ฅ // ๐ 0๐ฅ // ๐ 0๐ฅ
Since ๐ฅ is a cocycle in ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ), ๐๐๐
๐โ1
1 ๐ฅ is a cocycle in ฮฉ0(๐ ;๐๐(๐)). Thus,
๐๐๐๐
1 ๐ 0๐ฅ is a cocycle in ฮฉ0(๐ ;๐๐(๐๐)) and ๐ 0๐ฅ is a cocycle in ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ). Also,
๐๐[๐]โ1
0 ๐ฅ = ๐โ10 ๐๐
๐โ1
1 =โ ๐๐ยท๐[๐]โ๐
0 ๐ 0๐ฅ = ๐โ๐0 ๐๐๐
1 =โ ๐๐ยท๐[๐]โ1
0 ๐ 0๐ฅ = ๐โ10 ๐๐
๐
1 .
The proof is completed by noting that ๐ ยท ๐[๐] โ 1 = ๐[๐+1] โ 2.
The next lemma takes care of the third claim.
Lemma 5.2.3.3. Suppose ๐ฅ โ ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ) is a cocycle and that
๐0๐(๐โ10 ๐ฅ) = ๐ฆ โ ฮฉ1(๐ ; ๐โ1
1 ๐/๐0).
Then ๐0๐(๐โ10 ๐ 0๐ฅ) = ๐ 0๐ฆ โ ฮฉ1(๐ ; ๐โ1
1 ๐/๐0).
59
Proof. Suppose that ๐โ10 ๐ฅ and ๐โ๐0 ๐ 0๐ฅ involve negative powers of ๐0 at worst ๐โ๐0 and
that ๐โ10 ๐ฅ involves negative powers of ๐1 at worst ๐โ๐1 . Then we have the following
sequence of injections (recall definitions 3.1.4 through 3.1.7).
ฮฉ*(๐ ;๐๐(๐)) // ฮฉ*(๐ ;๐๐(๐๐)) // ฮฉ*(๐ ; ๐โ11 ๐๐) // ฮฉ*(๐ ; ๐โ1
1 ๐/๐โ0 )
๐๐๐๐โ1
1 ๐โ10 ๐ฅ // ๐โ1
0 ๐ฅ // ๐โ10 ๐ฅ
๐๐๐๐
1 ๐โ๐0 ๐ 0๐ฅ // ๐โ๐0 ๐ 0๐ฅ // ๐โ๐0 ๐ 0๐ฅ
We have
๐(๐๐๐๐
1 ๐โ๐0 ๐ 0๐ฅ) = ๐ 0๐(๐๐๐๐โ1
1 ๐โ10 ๐ฅ) โ ฮฉ1(๐ ;๐๐(๐๐))
and so
๐(๐โ๐0 ๐ 0๐ฅ) = ๐โ๐๐๐
1 ๐(๐๐๐๐
1 ๐โ๐0 ๐ 0๐ฅ) = ๐ 0
[๐โ๐๐
๐โ1
1 ๐(๐๐๐๐โ1
1 ๐โ10 ๐ฅ)
]= ๐ 0๐(๐โ1
0 ๐ฅ)
(5.2.3.4)
in ฮฉ1(๐ ; ๐โ11 ๐/๐โ0 ). We obtain
๐0๐(๐โ10 ๐ 0๐ฅ) = ๐0๐(๐๐โ1
0 (๐โ๐0 ๐ 0๐ฅ)) = ๐๐0๐(๐โ๐0 ๐ 0๐ฅ) = ๐ 0(๐0๐(๐โ10 ๐ฅ)) = ๐ 0๐ฆ
where the penultimate equality comes from the preceding observation.
Proof of proposition 5.2.3.1. We are just left with the final claim, that the image of
๐ฆ๐+1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) is (โ1)๐[๐๐+1]๐
โ๐[๐]
1 .
Recall that ๐ฆ๐+1 is defined to be ๐0๐(๐โ10 ๐ฅ๐+1) and that ๐ฅ๐+1 is ๐โ1
0 ๐ 0๐ฅ๐ โ ๐โ10 ๐ค๐.
We summarize this in the following diagram.
ฮฉ*(๐ ; ๐โ11 ๐/๐0) // ฮฉ*(๐ ; ๐โ1
1 ๐/๐โ0 ) // ฮฉ*(๐ ; ๐โ11 ๐/๐โ0 )
๐โ10 ๐ฅ๐+1 = ๐โ2
0 ๐ 0๐ฅ๐ โ ๐โ20 ๐ค๐
//_
๐
๐ฅ๐+1
๐ฆ๐+1 // ๐โ1
0 ๐ฆ๐+1
60
When considering the image of ๐ฆ๐+1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) we can ignore contributions
arising from ๐โ20 ๐ 0๐ฅ๐ since (5.2.3.4) gives
๐(๐โ20 ๐ 0๐ฅ๐) = ๐๐โ2
0 ๐ 0๐(๐โ10 ๐ฅ๐)
and so all terms involve a ๐๐ raised to a ๐-th power. Let
๐คโฒ๐ = ๐ค๐ + (โ1)๐๐โ๐
[๐]
1 ๐๐+1 โ ฮฉ0(๐ ; ๐โ11 ๐/๐0)
so that
โ๐โ20 ๐ค๐ = (โ1)๐๐โ2
0 ๐โ๐[๐]
1 ๐๐+1 โ ๐โ20 ๐คโฒ
๐ โ ฮฉ0(๐ ; ๐โ11 ๐/๐โ0 ).
As an example, we recall that
๐ฅ1 = ๐โ10 ๐1, ๐ฆ1 = [๐1], ๐ฅ2 = ๐โ๐โ1
0 ๐๐1 โ ๐โ10 ๐โ1
1 ๐2, ๐ฆ2 = [๐2]๐โ11 + [๐1]๐
โ21 ๐2;
we have
๐ค1 = ๐โ11 ๐2, ๐ค
โฒ1 = 0, ๐ค2 = ๐โ2๐โ1
1 ๐๐+12 โ ๐โ๐โ1
1 ๐3, ๐คโฒ2 = ๐โ2๐โ1
1 ๐๐+12 .
We consider the contributions from the two terms in the expression for โ๐โ20 ๐ค๐
separately.
Lemma 5.2.3.5. The only term of ๐(๐โ20 ๐โ๐
[๐]
1 ๐๐+1), which is relevant to the image
of ๐ฆ๐+1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]), is [๐๐+1]๐
โ10 ๐โ๐
[๐]
1 .
Proof. Recall definitions 3.1.4 and 3.1.6. We have a ๐ -comodule map
๐2(๐[๐โ1] + 1) โโ ๐โ1
1 ๐2 โ ๐โ11 ๐/๐โ0 , ๐โ2
0 ๐๐โ11 ๐๐+1 โฆโโ ๐โ2
0 ๐โ๐[๐]
1 ๐๐+1.
Under the coaction map ๐ โโ ๐ โ๐, we have
๐๐โ11 โฆโโ
โ๐+๐=๐โ1
(โ1)๐๐๐1 โ ๐๐0๐๐1 and ๐๐+1 โฆโโ
โ๐+๐ =๐+1
๐๐๐
๐ โ ๐๐ .
61
Under the coaction map ๐โ10 ๐ โโ ๐ โ ๐โ1
0 ๐, we have
๐โ20 ๐๐โ1
1 ๐๐+1 โฆโโโ
๐+๐=๐โ1
โ๐+๐ =๐+1
(โ1)๐๐๐1๐๐๐
๐ โ ๐๐โ20 ๐๐1๐๐
so that under the coaction map ๐โ11 ๐/๐โ0 โโ ๐ โ ๐โ1
1 ๐/๐โ0 , we have
๐โ20 ๐โ๐
[๐]
1 ๐๐+1 โฆโโโ
๐+๐=๐โ1
๐=0,1
โ๐+๐ =๐+1
(โ1)๐๐๐1๐๐๐
๐ โ ๐๐โ20 ๐
๐โ๐(๐[๐โ1]+1)1 ๐๐ .
We know that terms involving ๐โ20 must eventually cancel in some way so we ignore
these. Because we are concerned with an image in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]) we ignore terms
involving ๐๐โs raised to a power greater than or equal to ๐ and terms involving ๐๐โs
other than ๐1 and ๐0. Since ๐ โฅ 1, we are left with the term corresponding to ๐ = 0,
๐ = ๐+ 1, ๐ = 0 and ๐ = ๐โ 1: it is ๐๐+1 โ ๐โ10 ๐โ๐
[๐]
1 .
The proof of proposition 5.2.3.1 is almost complete. We just need to show that
๐(๐โ20 ๐คโฒ
๐) contributes nothing to the image of ๐ฆ๐+1 in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]). Recall that
๐คโฒ๐ = ๐ค๐ + (โ1)๐๐โ๐
[๐]
1 ๐๐+1 โ ฮฉ0(๐ ; ๐โ11 ๐/๐0)
and that ๐๐ค๐ = ๐ 0๐ฆ๐.
Denote by ๐ โฒโฒ the Hopf algebra obtained from ๐ by quotienting out the ideal
generated by the image of the map ๐ โโ ๐ , ๐ โฆโโ ๐๐2 .
Lemma 5.2.3.6.
๐๐คโฒ๐ = ๐ 0๐ฆ๐ + (โ1)๐
โ๐+๐=๐+1
๐,๐โฅ1
[๐๐๐
๐ ]๐โ๐[๐]
1 ๐๐ โ ฮฉ1(๐ ; ๐โ11 ๐/๐0)
is in the kernel of the map ๐ โ ๐โ11 ๐/๐0 โโ ๐ โฒโฒ โ F๐[๐ยฑ1
1 ].
Proof. By the inductive hypothesis ๐ฆ๐ โก (โ1)๐โ1[๐๐]๐โ๐[๐โ1]
1 modulo the kernel of
๐ โ ๐โ11 ๐/๐0 โโ ๐ โฒ โ F๐[๐ยฑ1
1 ]. So ๐ 0๐ฆ๐ โก (โ1)๐โ1[๐๐๐]๐โ๐[๐]+1
1 modulo the kernel of
62
๐ โ ๐โ11 ๐/๐0 โโ ๐ โฒโฒโF๐[๐ยฑ1
1 ]. (โ1)๐โ1[๐๐๐]๐โ๐[๐]+1
1 cancels with the ๐ = 1 term of the
summation in the lemma statement.
Corollary 5.2.3.7. For each monomial ๐ of ๐คโฒ๐ not equal to a power of ๐1, there
exists a ๐ > 1 such that ๐๐๐ divides ๐ .
Proof. The map ๐โ11 ๐/๐0 โโ ๐ โ ๐โ1
1 ๐/๐0 โโ ๐ โ F๐[๐ยฑ11 ] takes
๐๐1๐1ยท ยท ยท ๐๐๐๐๐
โฆโโ ๐๐1๐๐1โ1 ยท ยท ยท ๐๐๐๐๐๐โ1 โ ๐
โ๐๐
1
and so it is injective with image F๐[๐๐1 , ๐๐2 , ๐
๐3 , . . .] โ F๐[๐ยฑ1
1 ]. One sees that elements
๐๐1๐1ยท ยท ยท ๐๐๐๐๐
with ๐ โฅ 2, 1 = ๐1 < . . . < ๐๐, ๐1 โ Z, ๐2, . . . , ๐๐ โ 1, 2, . . . , ๐โ 1 are not
sent to ker (๐ โโ ๐ โฒโฒ) โ F๐[๐ยฑ11 ]. By the previous lemma, each monomial of ๐คโฒ
๐ not
equal to a power of ๐1 must contain some ๐๐ (๐ > 1) raised to a power greater than
or equal to ๐.
Since powers of ๐1 โ ฮฉ0(๐ ; ๐โ11 ๐/๐0) are cocycles we can assume that ๐ค๐ and ๐คโฒ
๐
do not contain powers of ๐1 as monomials.
Suppose no power of ๐1 worse than ๐โ๐๐1 appears in ๐คโฒ๐. Making use of the map
(see definitions 3.1.4 and 3.1.6)
ฮฉ*(๐ ;๐2(๐)) โโ ฮฉ*(๐ ; ๐โ11 ๐2) โ ฮฉ*(๐ ; ๐โ1
1 ๐/๐โ0 ), ๐โ20 ๐๐๐1 ๐ค
โฒ๐ โฆโโ ๐โ2
0 ๐คโฒ๐
we see that it is sufficient to analyze ๐(๐โ20 ๐๐๐1 ๐ค
โฒ๐). Viewing ๐๐๐1 ๐คโฒ
๐ as lying in ฮฉ0(๐ ;๐),
we care about terms of ๐(๐๐๐1 ๐คโฒ๐) involving a single power of ๐0. From the previous
corollary we see that the boundary of every monomial in ๐คโฒ๐ will involve terms which
consist of either a ๐๐ raised to a power greater than or equal to ๐ or a ๐๐ with ๐ > 1.
We conclude that the contribution from ๐(๐โ20 ๐คโฒ
๐) is zero in ฮฉ1(๐ โฒ;F๐[๐ยฑ11 ]).
Proving the part of proposition 5.3.1.2 which is left to section 5.3.4 relies heavily
on the ideas used in the previous proof. One may like to look ahead to that proof
while the ideas are still fresh.
63
5.3 The second family of differentials, side towers
5.3.1 Main results
The main results of this section are as follows. The first concerns the ๐โ11 -BSS and
the second gives the corresponding result in the ๐-BSS.
Proposition 5.3.1.1. For ๐ โฅ 1 and ๐ โ Z we have the following differential in the
๐โ11 -BSS.
๐๐๐โ1๐๐๐๐
1 ๐๐.
= ๐๐๐๐
1 ๐๐
Proposition 5.3.1.2. Let ๐ โฅ 1. Then ๐๐๐
1 ๐๐ โ ๐ป*(๐ ; ๐โ11 ๐/๐0) lifts to an element
๐ป*(๐ ;๐/๐0) which we also denote by ๐๐๐
1 ๐๐. We have the following differential in the
๐-BSS.
๐๐๐โ๐[๐]๐๐๐
1 ๐๐.
= ๐1,๐โ1
Moreover, if ๐ โ Z and ๐ > 1, ๐๐๐โ1๐๐๐๐
1 ๐๐ is defined in the ๐-BSS.
5.3.2 Quick proofs
The differentials in the ๐โ11 -BSS are derivations (lemma 3.6.4) and ๐๐๐โ1๐
ยฑ๐๐1 = 0
(lemma 3.5.4). This means that proposition 5.3.1.1 follows quickly from the following
sub-proposition.
Proposition 5.3.2.1. For ๐ โฅ 1 we have the following differential in the ๐โ11 -BSS.
๐๐๐โ1๐๐๐(๐+1)1 ๐๐
.= ๐
๐๐(๐+1)1 ๐๐
In this subsection, we prove this proposition assuming the following Kudo trans-
gression theorem.
Recall lemma 5.1.8, which says that ๐๐๐โ1(๐+1)
1 ๐๐ and ๐๐๐(๐+1)1 ๐๐ have unique lifts
to ๐ป*(๐ ;๐/๐0). We denote the lifts by the same name.
Proposition 5.3.2.2 (Kudo transgression). Suppose ๐ฅ, ๐ฆ โ ๐ป*(๐ ;๐/๐0), ๐ฅ has co-
homological degree 0, ๐ฆ has cohomological degree 1, and that ๐๐๐ฅ = ๐ฆ in the ๐-BSS.
64
Then we have ๐(๐โ1)๐๐ฅ๐โ1๐ฆ
.= โจ๐ฆโฉ๐, where โจ๐ฆโฉ๐ will be defined in the course of the
proof.
Moreover, โจ๐๐๐โ1(๐+1)
1 ๐๐โฉ๐ = ๐๐๐(๐+1)1 ๐๐ in ๐ป*(๐ ;๐/๐0).
The Kudo transgression theorem is the consuming result of this section. Supposing
it for now, we prove proposition 5.3.2.1 and proposition 5.3.1.2, save for the claim
about ๐๐๐โ๐[๐]๐๐๐
1 ๐๐.
Proof of proposition 5.3.2.1. Proposition 5.2.1.2, proposition 5.2.1.3 and lemma 5.1.8
tells us that
๐๐[๐]๐๐๐โ1(๐+1)1
.= ๐
๐๐โ1(๐+1)1 ๐๐
in the ๐-BSS. By the Kudo transgression theorem we have
๐๐๐โ1๐๐๐(๐+1)1 ๐๐
.= ๐
๐๐(๐+1)1 ๐๐
in the ๐-BSS and hence (lemma 3.4.1), the ๐โ11 -BSS.
Proof of part of proposition 5.3.1.2. By lemma 3.4.1, we can verify the last statement
in the ๐โ0 -BSS. We have
๐๐ฃ0๐๐๐๐
1 ๐๐ โ ๐ธ1,๐๐๐โ๐[๐]+๐ฃ,๐๐๐๐,๐ฃ1 (๐โ1 -BSS).
Consider the case ๐ฃ = โ๐๐. By proposition 5.3.1.1 and lemma 4.2.2, it is enough to
show that (๐ , ๐ก, ๐ข) = (1, ๐๐๐โ ๐[๐]โ ๐๐, ๐๐๐๐) satisfies ๐ข < ๐(๐ ) + (2๐2โ 2)(๐ก+ 2)โ ๐.
The worst case is when ๐ = 2 where the inequality is implied by (5.1.9).
5.3.3 A Kudo transgression theorem
Suppose given a connected commutative Hopf algebra P, a commutative algebra Q in
P-comodules, and suppose that all nontrivial elements of P and Q have even degree.
In order to prove proposition 5.3.2.2, we mimic theorem 3.1 of [9] to define natural
operations
๐ฝ๐ 0 : ฮฉ0(P;Q) โโ ฮฉ1(P;Q).
65
Once these operations have been defined and we have observed their basic properties
the proof of the Kudo transgression proposition follows quickly.
The reader should refer to [10, pg. 75-76] for notation regarding twisting mor-
phisms and twisted tensor products. We write ๐ for the universal twisting morphism
instead of [ ].
The first step towards proving the existence of the operation ๐ฝ๐ 0 is to describe a
map
ฮฆ : ๐ โ ฮฉ*(P;Q)โ๐ โโ ฮฉ*(P;Q),
which acts as the ๐ appearing in [9, theorem 3.1]. This can be obtained by dualizing
the construction in [9, lemma 11.3]. Conveniently, this has already been documented
in [5, lemma 2.3].
0 // Q //
๐Q ๐0
0 //
๐1
0 //
๐2
. . .
0 // PโQ๐ //
๐โ1 ๐0
PโPโQ๐ //
๐1
PโPโPโQ๐ //
๐2
. . .
0 // Q // 0 // 0 // . . .
Consider the diagram above. The top and bottom row are equal to the chain complex
consisting of Q concentrated in cohomological degree zero and the middle row is the
chain complex P โ๐ ฮฉ*(P;Q). We have the counit ๐ : P โโ F๐ and the coaction
๐Q : Q โโ P โQ. The definition of a P-comodule gives 1 โ ๐๐ = 0. We also have
1โ ๐๐ = ๐๐ + ๐๐ where ๐ is the contraction defined by
๐(๐0[๐1| . . . |๐๐ ]๐) = ๐(๐0)๐1[๐2| . . . |๐๐ ]๐.
[Note that just for this section ๐ no longer means 2๐โ 2.]
Let ๐ถ๐ denote the cyclic group of order ๐ and let ๐ be the standard F๐[๐ถ๐]-free
resolution of F๐ (see [5, definition 2.2]). We are careful to note that the boundary map
66
in ๐ decreases degree. Following Brunerโs account in [5, lemma 2.3], we can extend
the multiplication map displayed at the top of the following diagram and construct
ฮฆ, a ๐ถ๐-equivariant map of DG P-comodules (with ฮฆ(๐๐โ [Pโ๐ ฮฉ*(P;Q))โ๐]๐) = 0
if ๐๐ > (๐โ 1)๐).
Qโ๐ //
๐0โ๐โ๐
Q
๐
๐ โ (Pโ๐ ฮฉ*(P;Q))โ๐ ฮฆ // Pโ๐ ฮฉ*(P;Q)
Precisely, we make the following definition.
Definition 5.3.3.1.
ฮฆ : ๐ โ (Pโ๐ ฮฉ*(P;Q))โ๐ โโ Pโ๐ ฮฉ*(P;Q)
is the map obtained by applying [5, lemma 2.3] to the following set up:
1. ๐ = ๐, ๐ = โจ(1 2 ยท ยท ยท ๐)โฉ = ๐ถ๐ and ๐ฑ = ๐ ;
2. (๐ ,๐ด) = (F๐,P), ๐ = ๐ = Q and ๐พ = ๐ฟ = Pโ๐ ฮฉ*(P;Q);
3. ๐ : ๐โ๐ โโ ๐ is the iterated multiplication Qโ๐ โโ Q.
Letโs recall the construction. Bruner defines
ฮฆ๐,๐ : ๐๐ โ [Pโ๐ ฮฉ*(P;Q)โ๐]๐ โโ Pโ๐ ฮฉ๐โ๐(P;Q)
inductively. The gradings here are all (co)homological gradings.
As documented in [16, pg. 325, A1.2.15] there is a natural associative multiplica-
tion
(Pโ๐ ฮฉ*(P;Q))โฮ (Pโ๐ ฮฉ*(P;Q)) โโ Pโ๐ ฮฉ*(P;Q)
๐[๐1| ยท ยท ยท |๐๐ ]๐ ยท ๐โฒ[๐โฒ1| ยท ยท ยท ๐โฒ๐ก]๐โฒ =โ
๐๐โฒ(0)[๐1๐โฒ(1)| ยท ยท ยท |๐๐ ๐โฒ(๐ )|๐(1)๐โฒ1| ยท ยท ยท |๐(๐ก)๐โฒ๐ก]๐(๐ก+1)๐
โฒ.
(5.3.3.2)
67
Here,โ๐โฒ(0) โ ยท ยท ยท โ ๐โฒ(๐ ) โ Pโ(๐ +1) is the ๐ -fold diagonal of ๐โฒ โ P and
โ๐(1) โ ยท ยท ยท โ
๐(๐ก+1) โ Pโ๐กโQ is the ๐ก-fold diagonal of ๐ โ Q. Also, โฮ denotes the internal tensor
product in the category of P-comodules as in [10, pg. 74]; one checks directly that
the multiplication above is a P-comodule map.
Iterating this multiplication gives a map
(Pโ๐ ฮฉ*(P;Q))โ๐ โโ Pโ๐ ฮฉ*(P;Q)
which determines ฮฆ0,*.
Suppose we have defined ฮฆ๐โฒ,๐ for ๐โฒ < ๐. Since ฮฆ๐,๐ = 0 for ๐ < ๐ we may suppose
that we have defined ฮฆ๐,๐โฒ for ๐โฒ < ๐. We define ฮฆ๐,๐ using ๐ถ๐-equivariance, the
adjunction
P-comodulesforget // F๐-modules๐โ(โ)
oo ๐ // ๐
and the contracting homotopy
๐ =
๐โ๐=1
(๐๐)๐โ1 โ ๐ โ 1๐โ๐.
In particular, we define ฮฆ๐,๐ on ๐๐ โ ๐ฅ by
ฮฆ๐,๐ = ([๐ฮฆ๐,๐โ1]โผ โ [ฮฆ๐โ1,๐โ1(๐โ 1)]โผ)(1โ ๐ ).
Our choice of ฮฆ is natural in P and Q because we specified the multiplication
determining ฮฆ0,* and the contracting homotopy ๐ in a natural way.
ฮฆ restricts to a natural ๐ถ๐-equivariant DG homomorphism
ฮฆ : ๐ โ ฮฉ*(P;Q)โ๐ โโ ฮฉ*(P;Q).
In the proof of proposition 5.3.2.2 we need the fact that ฮฆ interacts nicely with
P-comodule primitives.
68
Definition 5.3.3.3. Suppose that ๐ฅ โ Pโ๐ฮฉ*(P;Q) and that ๐ โ Q is a P-comodule
primitive. We write ๐๐ฅ for ๐ฅ ยท 1[]๐.
Lemma 5.3.3.4. Suppose that ๐ โ Q is P-comodule primitive. Then
ฮฆ(๐๐ โ ๐๐1๐ฅ1 โ ยท ยท ยท โ ๐๐๐๐ฅ๐) = ๐โ
๐ ๐๐ฮฆ(๐๐ โ ๐ฅ1 โ ยท ยท ยท โ ๐ฅ๐).
Proof. A special case of formula (5.3.3.2) gives
๐โฒ[๐โฒ1| ยท ยท ยท |๐โฒ๐ ]๐โฒ ยท 1[]๐ = ๐โฒ[๐โฒ1| ยท ยท ยท |๐โฒ๐ ]๐โฒ๐.
Since ๐ โ Q is a P-comodule primitive we also obtain
1[]๐ ยท ๐โฒ[๐โฒ1| ยท ยท ยท ๐โฒ๐ก]๐โฒ = ๐โฒ[๐โฒ1| ยท ยท ยท |๐โฒ๐ก]๐โฒ๐;
left and right multiplication by 1[]๐ agree. This observation proves the ๐ = 0 case of
the result since ฮฆ0,*(๐0 โ โ โ . . . โ โ) is equal to the map (P โ๐ ฮฉ*(P;Q))โ๐ โโ
Pโ๐ ฮฉ*(P;Q). We can now make use of the inductive formula
ฮฆ๐,๐ = ([๐ฮฆ๐,๐โ1]โผ โ [ฮฆ๐โ1,๐โ1(๐โ 1)]โผ)(1โ ๐ ).
๐Q, ๐ โ 1, and ๐ commute with multiplication by ๐ and so 1 โ ๐ commutes with
multiplication by 1โ๐๐1โ . . .โ๐๐๐ . By an inductive hypothesis we can suppose ฮฆ๐,๐โ1
and ฮฆ๐โ1,๐โ1 have the required property. It follows that ๐ฮฆ๐,๐โ1 and ฮฆ๐โ1,๐โ1(๐ โ 1)
have the required property. The same is true of their adjoints and so the result holds
for the adjoint of ฮฆ๐,๐ and thus for ฮฆ๐,๐ itself.
We finally define ๐ฝ๐ 0 : ฮฉ0(P;Q) โโ ฮฉ1(P;Q) and note a couple of its properties.
One should read the proof of [9, theorem 3.1]; this definition mimics that of ๐ฝ๐0 :
๐พ0 โ ๐พโ1. In particular, we take ๐ = ๐ = 0 and the reader will note that we omit a
๐(โ1) in our definition.
Definition 5.3.3.5. Let ๐ โ ฮฉ0(P;Q). We define ๐ฝ๐ 0๐ โ ฮฉ1(P;Q) as follows.
69
1. Let ๐ = ๐๐ โ ฮฉ1(P;Q).
2. We define ๐ก๐ โ ฮฉ*(P;Q)โ๐ for 0 < ๐ < ๐.
In the following two formulae juxtaposition denotes tensor product.
Write ๐ = 2๐+ 1 and define for 0 < ๐ โค ๐
๐ก2๐ = (๐ โ 1)!โ๐ผ
๐๐1๐2๐๐2๐2 ยท ยท ยท ๐๐๐๐2
summed over all ๐-tuples ๐ผ = (๐1, . . . , ๐๐) such thatโ
๐ ๐๐ = ๐โ 2๐.
Define for 0 โค ๐ < ๐
๐ก2๐+1 = ๐!โ๐ผ
๐๐1๐2 ยท ยท ยท ๐๐๐๐2๐๐๐+1๐
summed over all (๐ + 1)-tuples ๐ผ = (๐1, . . . , ๐๐+1) such thatโ
๐ ๐๐ = ๐โ 2๐ โ 1.
3. Define ๐ โ ๐ โ ฮฉ*(P;Q)โ๐ by
๐ =๐โ๐=1
(โ1)๐ [๐๐โ2๐โ1 โ ๐ก2๐ โ ๐๐โ2๐ โ ๐ก2๐โ1] ,
so ๐๐ = โ๐๐โ2 โ ๐๐ [9, 3.1(8)].
4. ๐ฝ๐ 0๐ is defined to be ฮฆ๐.
Naturality of ๐ฝ๐ 0 follows from the naturality of ฮฆ. Using the observation made
in part (3) of the definition we immediately obtain the following lemma.
Lemma 5.3.3.6. Let ๐ โ ฮฉ0(P;Q). Then ๐(๐ฝ๐ 0๐) = โฮฆ(๐๐โ2 โ (๐๐)๐).
Moreover, we make the following definition.
Definition 5.3.3.7. Given ๐ โ ฮฉ1(P;Q), we define โจ๐โฉ๐ to be the element
ฮฆ(๐๐โ2 โ ๐๐) โ ฮฉ2(P;Q).
70
If ๐ฆ โ ๐ป1(P;Q) is represented by ๐, then โจ๐ฆโฉ๐ โ ๐ป2(P;Q) is defined to be the class
of โจ๐โฉ๐.
The fact that โจ๐ฆโฉ๐ is well-defined is used in [9, definition 2.2].
We are now ready to prove the first statement in the proposition.
Proof of the first part of proposition 5.3.2.2. By lemma 3.2.4 there exists ๐ and ๐ in
ฮฉ*(๐ ;๐) with ๐๐ = ๐๐0๐ such that their images ๐ and ๐ in ฮฉ*(๐ ;๐/๐0) are cocycles
representing ๐ฅ and ๐ฆ, respectively.
Consider ๐ฝ๐ 0๐. To get a grasp on what this element looks like we need to go back
to definition 5.3.3.5. Since ๐๐ = ๐๐0๐ we should stare at the definition but replace ๐ by
๐๐0๐. We note that the sum defining ๐ involves ๐ก1, . . . , ๐ก2๐. ๐ก2๐ is given by
(๐โ 1)!๐โ1โ๐=0
๐2๐(๐๐0๐)๐2๐โ2๐.
There are only single (๐๐0๐)โs in each term, whereas the terms in the sums defining
๐ก1, . . . , ๐ก2๐โ1 all involve at least two (๐๐0๐)โs. By lemma 5.3.3.4, ๐ฝ๐ 0๐ is divisible by
๐๐0 and the image of ๐ด = (๐ฝ๐ 0๐)/๐๐0 in ฮฉ1(๐ ;๐/๐0) is a unit multiple of the image of
ฮฆ(๐0 โ ๐ก2๐)/๐๐0 in ฮฉ1(๐ ;๐/๐0). This latter image is equal to
๐ด = (๐โ 1)!๐โ1โ๐=0
๐2๐ ๐ ๐2๐โ2๐,
where juxtaposition now denotes multiplication.
On the other hand, lemma 5.3.3.6, lemma 5.3.3.4 and definition 5.3.3.7 give
๐(๐ฝ๐ 0๐).
= ฮฆ(๐๐โ2 โ (๐๐0๐)๐) = ๐๐๐0 ฮฆ(๐๐โ2 โ ๐๐) = ๐๐๐0 โจ๐โฉ๐.
Letting ๐ต = ๐(๐ฝ๐ 0๐)/๐๐๐0 gives ๐๐ด = ๐(๐โ1)๐0 ๐ต, and the image ๐ต of ๐ต in ฮฉ2(๐ ;๐/๐0)
is a unit multiple of โจ๐โฉ๐, which represents โจ๐ฆโฉ๐.
The formula for ๐ด above, shows that it represents a unit multiple of ๐ฅ๐โ1๐ฆ and so
we deduce from lemma 3.2.4 that ๐(๐โ1)๐๐ฅ๐โ1๐ฆ
.= โจ๐ฆโฉ๐.
71
To complete the proof of proposition 5.3.2.2 we need the following lemma.
Lemma 5.3.3.8. Let P be the primitively generated Hopf algebra F๐[๐]/(๐๐) where
the degree of ๐ is even. Let โ and ๐ be classes in ๐ป*(P;F๐) which are represented in
ฮฉ*P by [๐] and๐โ1โ๐=1
(โ1)๐โ1
๐[๐๐|๐๐โ๐],
respectively. Then โจโโฉ๐ .= ๐.
Proof. This follows from remarks 6.9 and 11.11 of [9]. Beware of the different use
of notation: our โจ๐ฆโฉ๐ is Mayโs ๐ฝ๐ 0๐ฆ and May defines โจ๐ฆโฉ๐ using the โช1-product
associated to ฮฉ*P.
Finishing the proof of proposition 5.3.2.2. The previous lemma gives โจโ๐,0โฉ๐.
= ๐๐,0 in
๐ป*(F๐[๐๐]/(๐๐๐);F๐). Since ๐1 is primitive, definition 5.3.3.7 and lemma 5.3.3.4 show
that
โจ๐๐๐โ1(๐+1)
1 ๐๐โฉ๐ = โจ๐๐๐โ๐[๐โ1]
1 โ๐,0โฉ๐.
= ๐๐๐+1โ๐ยท๐[๐โ1]
1 ๐๐,0 = ๐๐๐(๐+1)1 ๐๐
in ๐ป*(F๐[๐๐]/(๐๐๐);F๐[๐ยฑ11 ]). We use naturality to transfer the required identity from
๐ป*(F๐[๐๐]/(๐๐๐);F๐[๐ยฑ11 ]) to ๐ป*(๐ ;๐/๐0). We have homomorphisms
๐ป*(F๐[๐๐]/(๐๐๐);F๐[๐ยฑ11 ]) // ๐ป*(๐ โฒ;F๐[๐ยฑ1
1 ]) ๐ป*(๐ ; ๐โ11 ๐/๐0)oo ๐ป*(๐ ;๐/๐0).oo
The first is induced by the inclusion F๐[๐๐]/(๐๐๐) โโ ๐ โฒ. Theorem 5.1.3 tells us that
the second is an isomorphism. Lemma 5.1.8 says that ๐๐๐โ1(๐+1)
1 ๐๐ and ๐๐๐(๐+1)
1 ๐๐ have
unique lifts to ๐ป*(๐ ;๐/๐0). This completes the proof.
5.3.4 Completing the proof of proposition 5.3.1.2
We are left to show that ๐๐๐โ๐[๐]๐๐๐
1 ๐๐.
= ๐1,๐โ1 for ๐ โฅ 1. The ๐ = 1 case
๐๐โ1๐๐โ11 โ1,0
.= ๐1,0
72
is given by proposition 5.3.1.1 and lemma 5.1.8 or by noting the following formula in
ฮฉ*(๐ ;๐) and using lemma 3.2.4.
๐
[๐โ1โ๐=1
(โ1)๐
๐[๐๐1]๐
๐โ10 ๐๐โ๐1
]=
๐โ1โ๐=1
(โ1)๐โ1
๐[๐๐1|๐
๐โ๐1 ]๐๐โ1
0
Suppose that for some ๐ โฅ 1 we have ๐๐ โ ฮฉ1(๐ ;๐) and ๐๐ โ ฮฉ2(๐ ;๐), such that
1. ๐๐ maps to (โ1)๐[๐๐]๐๐๐โ๐[๐]
1 in ฮฉ1(๐ โฒ;F๐[๐1]);
2. ๐๐ maps toโ๐โ1
๐=1(โ1)๐โ1
๐[๐๐๐
๐โ1
1 |๐(๐โ๐)๐๐โ1
1 ] in ฮฉ2(๐ ;๐/๐0);
3. ๐๐๐ = ๐๐๐โ๐[๐]
0 ๐๐.
๐ 0๐๐ lies in the injectivity range of proposition 4.2.1 and so using theorem 5.1.3
together with the diagram below we see that ฮฉ*(๐ ;๐/๐0) โโ ฮฉ*(๐ โฒ;F๐[๐1]) induces
an injection on homology in this tridegree.
๐ป*(๐ ;๐/๐0) //
๐ป*(๐ ; ๐โ11 ๐/๐0)
โผ=
๐ป*(๐ โฒ;F๐[๐1]) // ๐ป*(๐ โฒ;F๐[๐ยฑ1
1 ])
We note that ๐ 0๐๐ maps to zero in ฮฉ1(๐ โฒ;F๐[๐1]), and so, because ฮฉ*(๐ ;๐) โโ
ฮฉ*(๐ ;๐/๐0) is surjective, we can find a ๐ค๐ โ ฮฉ0(๐ ;๐) such that ๐๐ค๐ = ๐ 0๐๐ in
ฮฉ1(๐ ;๐/๐0). In particular, ๐ 0๐๐ โ ๐๐ค๐ is divisible by ๐0. Let
๐๐+1 =๐ 0๐๐ โ ๐๐ค๐
๐0.
We claim that ๐๐+1 and ๐๐+1 = ๐ 0๐๐ โ ฮฉ*(๐ ;๐) satisfy the following conditions.
1. ๐๐+1 maps to (โ1)๐+1[๐๐+1]๐๐๐+1โ๐[๐+1]
1 in ฮฉ1(๐ โฒ;F๐[๐1]);
2. ๐๐+1 maps toโ๐โ1
๐=1(โ1)๐โ1
๐[๐๐๐
๐
1 |๐(๐โ๐)๐๐1 ] in ฮฉ2(๐ ;๐/๐0);
3. ๐๐๐+1 = ๐๐๐+1โ๐[๐+1]
0 ๐๐+1.
73
The second condition is clear. To see the last condition, note that ๐๐๐ = ๐๐๐โ๐[๐]
0 ๐๐
implies ๐๐ 0๐๐ = ๐๐๐+1โ๐ยท๐[๐]
0 ๐ 0๐๐ = ๐๐๐+1โ๐[๐+1]+1
0 ๐๐+1, and so
๐๐๐+1 = ๐
(๐ 0๐๐ โ ๐๐ค๐
๐0
)=๐๐ 0๐๐๐0
= ๐๐๐+1โ๐[๐+1]
0 ๐๐+1.
For the first condition, we note that ๐ 0๐๐ will not contribute to the image of ๐๐+1 in
ฮฉ1(๐ โฒ;F๐[๐1]). Moreover, since
๐๐ค๐ = ๐ 0๐๐ = (โ1)๐[๐๐๐]๐๐๐+1โ๐ยท๐[๐]
1
in ๐ โฒโฒ โ F๐[๐1], we see, as in the proof of proposition 5.2.3.1, that the only relevant
term of ๐ค๐ is (โ1)๐๐๐๐+1โ๐[๐+1]
1 ๐๐+1, and that it contributes (โ1)๐+1[๐๐+1]๐๐๐+1โ๐[๐+1]
1
to โ๐โ10 ๐๐ค๐.
The proof is complete by induction and lemma 3.2.4.
5.4 The ๐ธโ-page of the ๐โ11 -BSS
In this subsection we obtain all the nontrivial differentials in the ๐โ11 -BSS. The main
result is simple to prove as long as one has the correct picture in mind; otherwise,
the proof may seem rather opaque. Figure 5-1 on page 77 displays some of Christian
Nassauโs chart [14] for ๐ป*(๐ด) when ๐ = 3. His chart tells us about the object we are
trying to calculate in a range by proposition 4.2.4 and the facts that
๐ป*(๐ ;๐/๐โ0 )/[F๐ [๐0]/๐โ
]= ๐ป*(๐ ;๐)/
[F๐ [๐0]
]and ๐ป*(๐ ;๐) = ๐ป*(๐ด). A ๐0-tower corresponds to a differential in the ๐-BSS. Labels
at the top of towers are the sources of the corresponding Bockstein differentials; labels
at the bottom of towers are the targets of the corresponding Bockstein differentials.
We note that the part of figure 5-1 in gray is not displayed in Nassauโs charts and is
deduced from the results of this chapter.
Recall from corollary 5.1.7 that ๐ป*(๐ ; ๐โ11 ๐/๐0) is an exterior algebra tensored
74
with a polynomial algebra, and so we have a convenient F๐-basis for it given by
monomials in ๐1, the ๐๐โs and the ๐๐โs. We introduce the following notation.
Notation 5.4.1. Suppose given ๐ผ = (๐1, . . . , ๐๐), ๐ฝ = (๐1, . . . , ๐๐ ), ๐พ = (๐1, . . . , ๐๐ )
such that ๐1 > . . . > ๐๐ โฅ 1, ๐1 > . . . > ๐๐ โฅ 1, and ๐๐ โฅ 0 for ๐ โ 1, . . . , ๐ . We
write
1. ๐[๐ผ]๐[๐ฝ,๐พ] for the monomial ๐๐1 ยท ยท ยท ๐๐๐๐๐1๐1 ยท ยท ยท ๐๐๐ ๐๐
;
2. ๐[๐ผ] forโ
๐ ๐๐๐โ1;
3. ๐ผโ for (๐1, . . . , ๐๐โ1) if ๐ โฅ 1;
4. ๐พโ for (๐1, . . . , ๐๐ โ 1) if ๐ โฅ 1 and ๐๐ โฅ 1.
Notice that the indexing of a monomial in the ๐๐โs and ๐๐โs by ๐ผ, ๐ฝ and ๐พ is
unique once we impose the additional condition that ๐๐ โฅ 1 for each ๐ โ 1, . . . , ๐ .
Moreover,๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ]
gives a basis for ๐ป*(๐ ; ๐โ1
1 ๐/๐0).
We have the following corollary to proposition 5.2.1.2 and proposition 5.3.1.1 and
we shall see that it completely describes all the nontrivial differentials in the ๐โ11 -BSS.
Corollary 5.4.2. Suppose given ๐ผ = (๐1, . . . , ๐๐), ๐ฝ = (๐1, . . . , ๐๐ ), ๐พ = (๐1, . . . , ๐๐ )
such that ๐1 > . . . > ๐๐ โฅ 1, ๐1 > . . . > ๐๐ โฅ 1, and ๐๐ โฅ 1 for ๐ โ 1, . . . , ๐ .
Suppose ๐ โฅ 1, that either ๐ = 0, or ๐ โฅ 1 and ๐๐ โค ๐๐ , and that ๐ โ Zโ๐Z. Then
we have the following differential in the ๐โ11 -BSS.
๐๐[๐๐ ]
[๐๐๐
๐๐โ1
1 ๐[๐ผโ]๐[๐ฝ,๐พ]
].
= ๐๐๐๐๐โ1
1 ๐[๐ผ]๐[๐ฝ,๐พ] (5.4.3)
Suppose ๐ โฅ 1, that either ๐ = 0, or ๐ โฅ 1 and ๐๐ > ๐๐ , and that ๐ โ Z. Then we
have the following differential in the ๐โ11 -BSS.
๐๐๐๐ โ1
[๐๐๐
๐๐
1 ๐[๐ผ]๐๐๐ ๐[๐ฝ,๐พโ]
].
= ๐๐๐๐๐
1 ๐[๐ผ]๐[๐ฝ,๐พ] (5.4.4)
Proof. By proposition 5.2.2.1, proposition 5.3.1.1, lemma 2.1.6 and lemma 3.6.4
we see that ๐๐[๐ผ]1 ๐[๐ผ]๐[๐ฝ,๐พ] is a permanent cycle. In the first case lemma 3.5.4
75
gives ๐๐[๐๐ ]๐โ๐[๐ผโ]1 = 0 and so the differential ๐๐[๐๐ ]๐
๐๐๐๐โ1
1.
= ๐๐๐๐๐โ1
1 ๐๐๐ completes the
proof. In the second case lemma 3.5.4 gives ๐๐๐๐ โ1๐โ๐[๐ผ]1 = 0 and so the differential
๐๐๐๐ โ1๐๐๐๐๐
1 ๐๐๐ .
= ๐๐๐๐๐
1 ๐๐๐ completes the proof.
The content of the next proposition is that the previous corollary describes all of
the nontrivial differentials in the ๐โ11 -BSS.
Proposition 5.4.5. The union
1 โช ๐ฅ : ๐ฅ is a source of one of the differentials in corollary 5.4.2
โช ๐ฆ : ๐ฆ is a target of one of the differentials in corollary 5.4.2
is a basis for ๐ป*(๐ ; ๐โ11 ๐/๐0). Moreover, the sources and targets of the differentials
in corollary 5.4.2 are distinct and never equal to 1.
Proof. We note that for any ๐ = 0, ๐๐1 is the source of a differential like the one in
(5.4.3).
Take ๐ผ, ๐ฝ and ๐พ as in (5.4.3). We wish to show that ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the source or
target of one of the differentials in corollary 5.4.2. There are three cases (the second
case is empty if ๐๐ = 1):
1. ๐ = ๐๐๐๐โ1 for some ๐ โ Zโ ๐Z.
2. ๐ = ๐๐๐๐+1โ1 for some ๐ โ Zโ ๐Z and some ๐๐+1 โฅ 1 with ๐๐ > ๐๐+1.
3. ๐ = ๐๐๐๐ for some ๐ โ Z.
In the first case, ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the target of the differential (5.4.3). In the second
case, ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the source of a differential like the one in (5.4.3). In the third
case, ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the source of a differential like the one in (5.4.4).
These cases are highlighted in figure 5-1 when ๐ = 3, ๐ผ = (3), and ๐ฝ and ๐พ are
empty. The three cases are:
1. ๐ = 9๐ for some ๐ โ Zโ 3Z.
2. ๐ = 3๐โ1๐ for some ๐ โ Zโ 3Z and some ๐ with 1 โค ๐ < 3.
76
180 185 190 195 200 205 210 215
15
20
25
30
35
40
45
50
55
๐กโ ๐
๐
๐0
๐451 ๐1
๐451 ๐1
๐451 ๐2
๐451 ๐2
๐451
๐451 ๐3
๐481 ๐1
๐481 ๐1
๐481
๐481 ๐2
๐511
๐511 ๐2
๐541 ๐3
๐541 ๐3
๐541 ๐1
๐541 ๐1
๐541 ๐2
๐541 ๐2
๐541
๐541 ๐4
๐461
๐461 ๐1
๐471
๐471 ๐1
๐461 ๐2
๐471 ๐2
๐481 ๐2
๐511 ๐2
๐481 ๐3
๐491 ๐3
๐501 ๐3
๐511 ๐3
๐491 ๐1
๐491 ๐1๐1
๐501 ๐1
๐501 ๐1๐1
๐491 ๐2
๐491 ๐2๐1
๐501 ๐2
๐501 ๐2๐1
๐511 ๐1๐1
๐511 ๐2๐1
Figure 5-1: The relevant part of ๐ป๐ ,๐ก(๐ด) when ๐ = 3, in the range 175 < ๐กโ ๐ < 219.Vertical black lines indicate multiplication by ๐0. The top and/or bottom of selected๐0-towers are labelled by the source and/or target, respectively, of the correspondingBockstein differential.
77
3. ๐ = 27๐ for some ๐ โ Z.
The first case is highlighted in blue when ๐ = 5; the second case is highlighted in
orange and we see both the cases ๐ = 1 and ๐ = 2 occurring; the last case is highlighted
in red when ๐ = 2.
Take ๐ผ, ๐ฝ and ๐พ as in (5.4.4). We wish to show that ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the source
or target of one of the differentials in corollary 5.4.2. There are two cases:
1. ๐ = ๐๐๐๐ for some ๐ โ Z.
2. ๐ = ๐๐๐๐+1โ1 for some ๐ โ Zโ ๐Z and some ๐๐+1 โฅ 1 with ๐๐+1 โค ๐๐ .
In the first case, ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the target of the differential (5.4.4). In the second
case, ๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ] is the source of a differential like the one in (5.4.3).
These cases are highlighted in figure 5-1 when ๐ = 3, ๐ผ is empty, ๐ฝ = (2) and
๐พ = (1). The two cases are:
1. ๐ = 9๐ for some ๐ โ Z.
2. ๐ = 3๐โ1๐ for some ๐ โ Zโ 3Z and some ๐ with 1 โค ๐ โค 2.
The first case is highlighted in blue when ๐ = 5 and ๐ = 6; the second case is
highlighted in orange and we see both the cases ๐ = 1 and ๐ = 2 occurring.
Since the empty sequences ๐ผ, ๐ฝ and๐พ together with those satisfying the conditions
in (5.4.3) or (5.4.4) make up all choices of ๐ผ, ๐ฝ and ๐พ, and since๐๐1 ๐[๐ผ]๐[๐ฝ,๐พ]
gives a basis for ๐ป*(๐ ; ๐โ1
1 ๐/๐0) (corollary 5.1.7), we have proved the first claim.
Careful inspection of the previous argument shows that this also proves the second
claim.
This proposition allows us to determine an F๐-basis of ๐ธโ(๐โ11 -BSS). We use the
following lemma.
Lemma 5.4.6. Suppose we have an indexing set ๐ด and an F๐-basis
1 โช ๐ฅ๐ผ๐ผโ๐ด โช ๐ฆ๐ผ๐ผโ๐ด
78
of ๐ป*(๐ ; ๐โ11 ๐/๐0) such that each ๐ฅ๐ผ supports a differential ๐๐๐ผ๐ฅ๐ผ = ๐ฆ๐ผ. Then we
have an F๐-basis of ๐ธโ(๐โ11 -BSS) given by the classes of
๐๐ฃ0 : ๐ฃ < 0
โช๐๐ฃ0๐ฅ๐ผ : ๐ผ โ ๐ด, โ๐๐ผ โค ๐ฃ < 0
.
In the above statement, we intend for 1, the ๐ฅ๐ผโs and the ๐ฆ๐ผโs to be distinct as in
proposition 5.4.5.
Proof. Let ๐ฃ < 0. We see make some observations.
1. ๐ธ*,*,*,๐ฃ1 โฉ
โ๐ <๐ im ๐๐ has basis ๐๐ฃ0๐ฆ๐ผ : ๐ผ โ ๐ด, ๐๐ผ < ๐.
2. ๐๐ฃ0๐ฆ๐ผ : ๐ผ โ ๐ด, ๐๐ผ = ๐ is independent in ๐ธ*,*,*,๐ฃ1 /
(๐ธ*,*,*,๐ฃ
1 โฉโ๐ <๐ im ๐๐
).
3. ๐ธ*,*,*,๐ฃ1 โฉ
โ๐ <๐ ker ๐๐ has basis
๐๐ฃ0
โช๐๐ฃ0๐ฅ๐ผ : ๐ผ โ ๐ด, ๐๐ผ โฅ min๐,โ๐ฃ
โช๐๐ฃ0๐ฆ๐ผ : ๐ผ โ ๐ด
.
4. ๐ธ*,*,*,๐ฃโ = (๐ธ*,*,*,๐ฃ
1 โฉโ๐ ker ๐๐ ) / (๐ธ*,*,*,๐ฃ
1 โฉโ๐ im ๐๐ ) has basis
๐๐ฃ0
โช๐๐ฃ0๐ฅ๐ผ : ๐ผ โ ๐ด, ๐๐ผ โฅ โ๐ฃ
.
We see that ๐๐ฃ0 is a basis element for ๐ธ*,*,*,๐ฃโ for all ๐ฃ < 0 and that ๐๐ฃ0๐ฅ๐ผ is a basis
element for ๐ธ*,*,*,๐ฃโ as long as โ๐๐ผ โค ๐ฃ < 0. This completes the proof.
We state the relevant corollary, a description of the ๐ธโ-page in the next section.
Of course, this allows us to find a basis of ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ) if we wish.
5.5 Summary of main results
We have completely calculated the ๐โ11 -BSS.
Theorem 5.5.1. In the ๐โ11 -BSS we have two families of differentials. For ๐ โฅ 1,
1. ๐๐[๐]๐๐๐๐โ1
1.
= ๐๐๐๐โ1
1 ๐๐, whenever ๐ โ Zโ ๐Z;
79
2. ๐๐๐โ1๐๐๐๐
1 ๐๐.
= ๐๐๐๐
1 ๐๐, whenever ๐ โ Z.
Together with the fact that ๐๐1 = 0 for ๐ โฅ 1, these two families of differentials
determine the ๐โ11 -BSS.
Corollary 5.5.2. ๐ธโ(๐โ11 -BSS) has an F๐-basis given by the classes of the following
elements.
๐๐ฃ0 : ๐ฃ < 0
โช
๐๐ฃ0๐
๐๐๐๐โ1
1 ๐[๐ผโ]๐[๐ฝ,๐พ] : ๐ผ, ๐ฝ,๐พ, ๐ satisfy (5.4.3), โ๐[๐๐] โค ๐ฃ < 0
โช
๐๐ฃ0๐
๐๐๐๐
1 ๐[๐ผ]๐๐๐ ๐[๐ฝ,๐พโ] : ๐ผ, ๐ฝ,๐พ, ๐ satisfy (5.4.4), 1โ ๐๐๐ โค ๐ฃ < 0
We have also obtained useful information about the ๐-BSS.
Lemma 5.5.3. The elements
1, ๐2๐๐โ1
1 ๐๐, ๐2๐๐
1 ๐๐ โ ๐ป*(๐ ; ๐โ11 ๐/๐0)
have unique lifts to ๐ป*(๐ ;๐/๐0). The same is true after multiplying by ๐๐1 as long as
๐ โฅ 0.
We give the lifts the same name.
Theorem 5.5.4. Let ๐ โฅ 1. We have the following differentials in the ๐-BSS.
1. ๐๐๐โ1๐๐๐โ1
1.
= โ1,๐โ1;
2. ๐๐[๐]๐๐๐๐โ1
1.
= ๐๐๐๐โ1
1 ๐๐, whenever ๐ โ Zโ ๐Z and ๐ > 1;
3. ๐๐๐โ๐[๐]๐๐๐
1 ๐๐.
= ๐1,๐โ1;
4. ๐๐๐โ1๐๐๐๐
1 ๐๐.
= ๐๐๐๐
1 ๐๐, whenever ๐ โ Z and ๐ > 1.
80
Chapter 6
The localized algebraic Novikov
spectral sequence
In this chapter we calculate the localized algebraic Novikov spectral sequence
๐ป*(๐ ; ๐โ11 ๐/๐โ0 )
๐ก=โ ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐โ).
6.1 Algebraic Novikov spectral sequences
Recall that the coefficient ring of the Brown-Peterson spectrum ๐ต๐ is the polyno-
mial algebra Z(๐)[๐ฃ1, ๐ฃ2, ๐ฃ3, . . .] on the Hazewinkel generators. Moreover, ๐ต๐*๐ต๐ =
๐ต๐*[๐ก1, ๐ก2, ๐ก3, . . .] together with ๐ต๐* defines a Hopf algebroid [13, ยง2].
๐ต๐* admits a filtration by invariant ideals, powers of ๐ผ = ker (๐ต๐* โโ F๐),
and we have ๐ = gr*๐ต๐*. Moreover, this allows us to filter the cobar construction
ฮฉ*(๐ต๐*๐ต๐ ) by setting ๐น ๐กฮฉ๐ (๐ต๐*๐ต๐ ) = ๐ผ ๐กฮฉ๐ (๐ต๐*๐ต๐ ), and we have
gr๐กฮฉ๐ (๐ต๐*๐ต๐ ) = ฮฉ๐ (๐ ;๐๐ก).
In this way we obtain the algebraic Novikov spectral sequence
๐ธ๐ ,๐ก,๐ข1 (alg.NSS) = ๐ป๐ ,๐ข(๐ ;๐๐ก)
๐ก=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ );
81
๐๐ has degree (1, ๐, 0). This makes sense of the terminology โNovikov weight.โ
One motivation for using the algebraic Novikov spectral is to make comparisons
with the Adams spectral sequence, and so we reindex it:
๐ธ๐ ,๐ก,๐ข2 (alg.NSS) = ๐ป๐ ,๐ข(๐ ;๐๐ก)
๐ก=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ )
and the degree of ๐๐ is (1, ๐ โ 1, 0).
๐ โ ๐ต๐* is a ๐ต๐*๐ต๐ -comodule primitive and so ๐ต๐*/๐๐ and ๐โ1๐ต๐* are ๐ต๐*๐ต๐ -
comodules; define ๐ต๐*/๐โ by the following exact sequence of ๐ต๐*๐ต๐ -comodules.
0 // ๐ต๐* // ๐โ1๐ต๐* // ๐ต๐*/๐โ // 0
We find that ๐ฃ๐๐โ1
1 โ ๐ต๐*/๐๐ is a ๐ต๐*๐ต๐ -comodule primitive and so we may define
๐ต๐*๐ต๐ -comodules ๐ฃโ11 ๐ต๐*/๐
๐ and ๐ฃโ11 ๐ต๐*/๐
โ by mimicking the constructions in
section 3.1.
By letting ๐น ๐กฮฉ๐ (๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) = ๐ผ ๐กฮฉ๐ (๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) and reindex-
ing, as above, we obtain the localized algebraic Novikov spectral sequence (loc.alg.NSS)
๐ธ๐ ,๐ก,๐ข2 (loc.alg.NSS) = ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐โ0 ]๐ก)๐ก
=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ).
It has a pairing with the unlocalized algebraic Novikov spectral sequence converging
to the ๐ป*(๐ต๐*๐ต๐ )-module structure map of ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ). Moreover, it
receives a map from the ๐ฃ1-algebraic Novikov spectral sequence (๐ฃ1-alg.NSS)
๐ธ๐ ,๐ก,๐ข2 (๐ฃ1-alg.NSS) = ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐0]๐ก)
๐ก=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐).
6.2 Evidence for the main result
In the introduction, we discussed โprincipal towersโ and their โside towersโ but said
little about the other elements in ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ). Figure 6-1 is obtained from fig-
ure 5-1 by removing principal towers and their side towers. We see that the remaining
82
180 185 190 195 200 205 210 215
30
35
40
45
50
55
๐กโ ๐
๐
๐481 ๐2
๐511 ๐2
๐481 ๐3
๐511 ๐3
๐491 ๐1
๐491 ๐1๐1
๐501 ๐1
๐501 ๐1๐1
๐491 ๐2
๐491 ๐2๐1
๐501 ๐2
๐501 ๐2๐1
๐511 ๐1๐1
๐511 ๐2๐1
Figure 6-1: A part of ๐ป๐ ,๐ก(๐ด) when ๐ = 3, in the range 175 < ๐ก โ ๐ < 219. Verticalblack lines indicate multiplication by ๐0. The top and/or bottom of selected ๐0-towersare labelled by the source and/or target, respectively, of the corresponding Bocksteindifferential.
83
๐0-towers come in pairs, arranged perfectly so that there is a chance that they form an
acyclic complex with respect to ๐2. Moreover, the labelling at the top of the towers
obeys a nice pattern with respect to this arrangement. The pattern of differentials
we hope for can be described by the following equations.
๐481 ๐3 โฆโโ ๐481 ๐2, ๐511 ๐3 โฆโโ ๐511 ๐2, ๐
491 ๐2 โฆโโ ๐491 ๐1, ๐
501 ๐2 โฆโโ ๐501 ๐1, ๐
511 ๐2๐1 โฆโโ ๐511 ๐1๐1.
In each case, this comes from replacing an ๐๐+1 by ๐๐, which resembles a theorem of
Miller.
Theorem 6.2.1 (Miller, [11, 9.19]). In the ๐ฃ1-alg.NSS
๐ป*(๐ ; ๐โ11 ๐/๐0)
๐ก=โ ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐)
we have, for ๐ โฅ 1, ๐2๐๐+1.
= ๐๐.
This is precisely the theorem enabling the calculation of this chapter, which shows
that the ๐2 differentials discussed above do occur in the loc.alg.NSS.
6.3 The filtration spectral sequence (๐0-FILT)
Corollary 5.5.2 describes the associated graded of the ๐ธ2-page of the loc.alg.NSS with
respect to the Bockstein filtration. Since
๐loc.alg.NSS2 : ๐ป๐ ,๐ข(๐ ; [๐โ1
1 ๐/๐โ0 ]๐ก) โโ ๐ป๐ +1,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก+1)
respects the Bockstein filtration, we have a filtration spectral sequence (๐0-FILT)
๐ธ๐ ,๐ก,๐ข,๐ฃ0 (๐0-FILT) = ๐ธ๐ ,๐ก,๐ข,๐ฃ
โ (๐โ11 -BSS)
๐ฃ=โ ๐ธ๐ ,๐ก,๐ข
3 (loc.alg.NSS).
The main result of this section is a calculation of the ๐ธ1-page of this spectral sequence.
Recall corollary 5.5.2.
84
Theorem 6.3.1. ๐ธ1(๐0-FILT) has an F๐-basis given by the following elements.
๐๐ฃ0 : ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โฅ 1, ๐ โ Zโ ๐Z, โ๐[๐] โค ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐
1 ๐๐ : ๐ โฅ 1, ๐ โ Z, 1โ ๐๐ โค ๐ฃ < 0
We prove the theorem via the following proposition.
Proposition 6.3.2. Fix, ๐, ๐ โฅ 1.
๐๐0-FILT0 : ๐ธ๐ ,๐ก,๐ข,๐ฃ
โ (๐โ11 -BSS) โโ ๐ธ๐ +1,๐ก+1,๐ข,๐ฃ
โ (๐โ11 -BSS)
restricts to an operation on the subspaces with bases given by the classes of the ele-
ments ๐๐ฃ0 : ๐ฃ < 0
,
๐๐ฃ0๐๐๐๐๐โ1
1 ๐[๐ผโ]๐[๐ฝ,๐พ] : ๐ผ, ๐ฝ,๐พ, ๐ satisfy (5.4.3), ๐๐ = ๐, โ๐[๐] โค ๐ฃ < 0
,
and๐๐ฃ0๐
๐๐๐๐
1 ๐[๐ผ]๐๐๐ ๐[๐ฝ,๐พโ] : ๐ผ, ๐ฝ,๐พ, ๐ satisfy (5.4.4), ๐๐ = ๐, 1โ ๐๐ โค ๐ฃ < 0
.
Moreover, the respective homology groups have bases given by the elements
๐๐ฃ0 : ๐ฃ < 0
,
๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โ Zโ ๐Z, โ๐[๐] โค ๐ฃ < 0
,
and ๐๐ฃ0๐
๐๐๐
1 ๐๐ : ๐ โ Z, 1โ ๐๐ โค ๐ฃ < 0
.
Proof. Each of the maps in the exact couple defining the ๐โ11 -BSS comes from a map
of algebraic Novikov spectral sequences. This means that if ๐ฅ โ ๐ป*(๐ ; ๐โ11 ๐/๐0) and
๐๐ฃ0๐ฅ โ ๐ธโ(๐โ11 -BSS) then ๐๐0-FILT
0 (๐๐ฃ0๐ฅ) = ๐๐ฃ0๐๐ฃ1-alg.NSS2 ๐ฅ. We understand ๐๐ฃ1-alg.NSS
2 by
85
theorem 6.2.1. For the rest of the proof we write ๐0 for ๐๐0-FILT0 .
๐0(๐๐ฃ0) = 0 and so the claims concerning ๐๐ฃ0 : ๐ฃ < 0 are evident.
First, fix ๐ โฅ 1 and consider
๐ฅ = ๐๐ฃ0๐๐๐๐โ1
1 ๐[๐ผโ]๐[๐ฝ,๐พ]
where ๐ผ, ๐ฝ,๐พ and ๐ satisfy (5.4.3), ๐๐ = ๐, and โ๐[๐] โค ๐ฃ < 0. If ๐ = 1 then ๐0(๐ฅ) = 0
so suppose that ๐ > 1 and let ๐ โ 1, . . . , ๐ โ 1. We wish to show that replacing ๐๐๐
by ๐๐๐โ1 in ๐ฅ gives an element ๐ฅโฒ of the same form as ๐ฅ. This is true because
๐ฅโฒ = ๐๐ฃ0๐๐๐๐โ1
1 ๐[๐ผ โฒโ]๐[๐ฝ โฒ, ๐พ โฒ]
where ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ are determined by the following properties.
1. ๐[๐ผ โฒโ]๐[๐ฝ โฒ, ๐พ โฒ] is obtained from ๐[๐ผโ]๐[๐ฝ,๐พ] by replacing ๐๐๐ by ๐๐๐โ1;
2. ๐โฒ = ๐ โ 1, ๐โฒ1 > . . . > ๐โฒ๐โฒ = ๐;
3. ๐โฒ1 > . . . > ๐โฒ๐ โฒ โฅ 1;
4. ๐โฒ๐ โฅ 1 for all ๐ โ 1, . . . , ๐ โฒ.
In particular, ๐โฒ๐โฒ = ๐ and ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ and ๐ satisfy (5.4.3) because ๐ โฒ โฅ 1, and ๐๐ โฅ ๐๐ = ๐
and ๐๐ > ๐๐ = ๐ implies that ๐โฒ๐ โฒ โฅ ๐ = ๐โฒ๐โฒ . Since ๐0 is a derivation, this observation
shows that ๐0 induces an operation on the second subspace of the proposition. The
claim about the homology is true because the complex(๐ธ[๐๐ : ๐ > ๐]โ F๐[๐๐ : ๐ โฅ ๐] : ๐๐๐+1 = ๐๐
)
has homology F๐.
Second, fix ๐ โฅ 1 and consider
๐ฆ = ๐๐ฃ0๐๐๐๐
1 ๐[๐ผ]๐๐๐[๐ฝ,๐พโ]
86
where ๐ผ, ๐ฝ,๐พ and ๐ satisfy (5.4.4), ๐๐ = ๐, and 1โ ๐๐ โค ๐ฃ < 0.
First, we wish to show the term obtained from applying ๐0 to ๐๐ is trivial. If ๐ = 1
then ๐0(๐๐) = 0 so suppose, for now, that ๐ > 1. Replacing ๐๐ by ๐๐โ1 gives
๐ฆโฒ = ๐๐ฃ0๐(๐๐)๐๐โ1
1 ๐[๐ผ โฒ]๐[๐ฝ โฒ, ๐พ โฒ]
where ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ are determined by the following properties.
1. ๐[๐ผ โฒ]๐[๐ฝ โฒ, ๐พ โฒ] = ๐[๐ผ]๐[๐ฝ,๐พโ]๐๐โ1;
2. ๐ผ โฒ = ๐ผ;
3. ๐โฒ1 > . . . > ๐โฒ๐ โฒ = ๐ โ 1;
4. ๐โฒ๐ โฅ 1 for all ๐ โ 1, . . . , ๐ โฒ.
๐ โฒ โฅ 1 and either ๐ = ๐โฒ = 0, or ๐ = ๐โฒ โฅ 1 and ๐โฒ๐โฒ = ๐๐ > ๐๐ = ๐ > ๐ โ 1 = ๐โฒ๐ โฒ , so
we see that ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ and ๐โฒ = ๐๐ satisfy (5.4.4). This shows that ๐ฆโฒ is the source
of a (5.4.4) ๐โ11 -Bockstein differential, i.e. zero in ๐ธโ(๐โ1
1 -BSS) = ๐ธ0(๐0-FILT). We
deduce that when applying ๐0 the only terms of interest come from applying ๐0 to
the ๐[๐ผ]๐[๐ฝ,๐พโ] part of ๐ฆ.
If ๐ = 0 then ๐0(๐ฆ) = 0 so suppose that ๐ > 0 and let ๐ โ 1, . . . , ๐. We wish to
show that replacing ๐๐๐ by ๐๐๐โ1 in ๐ฆ gives an element ๐ฆโฒโฒ of the same form as ๐ฆ.
๐ฆโฒโฒ = ๐๐ฃ0๐๐๐๐
1 ๐[๐ผ โฒ]๐๐๐[๐ฝ โฒ, ๐พ โฒโ]
where ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ are determined by the following properties.
1. ๐[๐ผ โฒ]๐[๐ฝ โฒ, ๐พ โฒโ] is obtained from ๐[๐ผ]๐[๐ฝ,๐พโ] by replacing ๐๐๐ by ๐๐๐โ1;
2. ๐โฒ = ๐ โ 1 and ๐โฒ1 > . . . > ๐โฒ๐โฒ ;
3. ๐โฒ1 > . . . > ๐โฒ๐ โฒ = ๐;
4. ๐โฒ๐ โฅ 1 for all ๐ โ 1, . . . , ๐ โฒ.
87
๐๐ โฅ ๐๐ > ๐๐ = ๐ ensures that condition (3) can be met. ๐ โฒ โฅ ๐ โฅ 1 and either ๐โฒ = 0 or
๐โฒ โฅ 1 and ๐โฒ๐โฒ โฅ ๐๐ > ๐๐ = ๐ = ๐โฒ๐ โฒ . Thus, ๐ผ โฒ, ๐ฝ โฒ, ๐พ โฒ and ๐ satisfy 5.4.4 and ๐ฆโฒโฒ has the
same form as ๐ฆ. Since ๐0 is a derivation, this shows that ๐0 induces an operation on
the third subspace of the proposition. The claim about the homology is true because(๐ธ[๐๐ : ๐ > ๐]โ F๐[๐๐ : ๐ โฅ ๐] : ๐๐๐+1 = ๐๐
)
has homology F๐.
6.4 The ๐ธโ-page of the loc.alg.NSS
One knows that ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) is nonzero, only in cohomological degree 0
and 1. ๐ป0(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) is generated as an abelian group by the elements
1
๐๐: ๐ โฅ 1
โช
๐ฃ๐๐
๐โ1
1
๐๐: ๐ โฅ 1, ๐ โ Zโ ๐ Z
.
These are detected in the loc.alg.NSS by the following elements of ๐ป*(๐ ; ๐โ11 ๐/๐โ0 ).
1
๐๐0: ๐ โฅ 1
โช๐๐๐
๐โ1
1
๐๐0: ๐ โฅ 1, ๐ โ Zโ ๐ Z
An element of order ๐ in ๐ป1(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ) = Z/๐โ is given by the class of
โ๐โ1๐ฃโ11 [๐ก1] โ ฮฉ1(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐โ)
in ๐ป1(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ), which is detected by ๐โ10 ๐1 in the loc.alg.NSS. Theo-
rem 6.3.1, degree considerations, and the fact that each ๐๐ฃ0 is a permanent cycle in
the loc.alg.NSS, allow us to see that there are permanent cycles in the loc.alg.NSS,
which are not boundaries, which are detected in the ๐0-FILT spectral sequence by the
elements ๐๐ฃ0๐๐ : ๐ โฅ 1, 1โ ๐๐ โค ๐ฃ < 0
.
88
These elements must detect the elements of ๐ป1(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ). In summary,
we have the following proposition.
Proposition 6.4.1. ๐ธโ(loc.alg.NSS) has an F๐-basis given by the following elements.
๐๐ฃ0 : ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โฅ 1, ๐ โ Zโ ๐Z, โ๐ โค ๐ฃ < 0
โช
๐๐ฃ0๐๐ : ๐ โฅ 1, 1โ ๐๐ โค ๐ฃ < 0
Here, ๐๐ฃ0๐๐ denotes the element of ๐ธ3(loc.alg.NSS) representing ๐๐ฃ0๐๐ โ ๐ธ1(๐0-FILT).
Using theorem 6.3.1 together with this result, we see that the only possible pattern
for the differentials between a principal tower and its side towers, in the loc.alg.NSS,
is the one drawn in figure 1-1.
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90
Chapter 7
Some permanent cycles in the ASS
Our calculation of the ๐โ11 -BSS gives information about the Adams ๐ธ2-page via the
zig-zag (1.4.5). Our calculation of the loc.alg.NSS gives information about Adams ๐2
differentials in a similar way (figure 1-2). We would like to learn about higher Adams
differentials, but first, we say what we can about some permanent cycles in the Adams
spectral sequence. We show that for each ๐ โฅ 0, ๐๐๐โ๐โ1
0 โ1,๐ is a permanent cycle in
the Adams spectral sequence and we give a homotopy class representing it. This is
the odd primary analogue of a result of Davis and Mahowald appearing in [6].
7.1 Maps between stunted projective spaces
The maps we construct to represent the classes ๐๐๐โ๐โ1
0 โ1,๐ make use of maps we have
between skeletal subquotients of (ฮฃโ๐ตฮฃ๐)(๐). The analog of these spectra at ๐ = 2 are
the stunted projective spaces R๐๐๐ and so we use the same terminology. Throughout
this thesis we write ๐ป for ๐ปF๐, the mod ๐ Eilenberg-Mac Lane spectrum.
In [1] Adams shows that there is a CW spectrum ๐ต with one cell in each positive
dimension congruent to 0 or โ1 modulo ๐ = 2๐ โ 2 such that ๐ต โ (ฮฃโ๐ตฮฃ๐)(๐). In
particular, ๐ต is built up from many copies of the mod ๐ Moore spectrum ๐/๐. The
maps we construct between stunted projective spaces all come from the fact that
multiplication by ๐ is zero on ๐/๐ (since ๐ is odd). For this reason, we emphasize the
filtration by the copies of ๐/๐ over the skeletal filtration, and writing a superscript in
91
square brackets to denote the skeletal filtration, we use the following notation.
Notation 7.1.1. Write ๐ต for the spectrum of [1, 2.2]. For ๐ โฅ 0 let ๐ต๐ = ๐ต[๐๐] and
for 1 โค ๐ โค ๐ let ๐ต๐๐ = ๐ต๐/๐ต๐โ1. Notice that ๐ต0 = * and so ๐ต๐ = ๐ต๐
1 . For ๐ > ๐
let ๐ต๐๐ = *.
We now proceed to construct compatible maps between stunted projective spaces
of Adams filtration one. All proofs will be deferred until the end of the section.
Lemma 7.1.2. For each ๐ โฅ 1 there exists a unique map ๐ : ๐ต๐ โโ ๐ต๐โ1 such that
the left diagram commutes. Moreover, the center diagram commutes so that the right
diagram commutes.
๐ต๐
๐
""
๐
๐ต๐โ1 ๐ // ๐ต๐
๐ต๐ ๐ //
๐
""
๐ต๐+1
๐
๐ต๐
๐ต๐ ๐ //
๐
๐ต๐+1
๐
๐ต๐โ1 ๐ // ๐ต๐
For 1 โค ๐ โค ๐ the filler for the diagram
๐ต๐ ๐ //
๐
๐ต๐+1 ๐ //
๐
๐ต๐+1๐+1
๐ต๐โ1 ๐ // ๐ต๐ ๐ // ๐ต๐
๐
is unique and we call it ๐ . The collection of such ๐ are compatible.
For 1 โค ๐ โค ๐ the filler for the diagram
๐ต๐โ1 ๐ //
๐
๐ต๐ ๐ //
๐
๐ต๐๐
๐ต๐โ1 ๐ // ๐ต๐ ๐ // ๐ต๐
๐
is unique and so equal to ๐.
92
The following diagrams commute for the appropriate values of ๐ and ๐.
๐ต๐+1๐+1
๐
๐
๐ต๐๐
๐๐ // ๐ต๐+1๐+1
๐ต๐๐
๐๐ //
๐
๐ต๐+1๐+1
๐
๐ต๐๐
๐ต๐๐+1
๐
๐ // ๐ต๐๐+1
๐ต๐+1๐+1
๐ // ๐ต๐๐
๐
OO๐ต๐+1๐+1
๐ // ๐ต๐๐
๐
๐ต๐+1๐
๐
OO
๐ // ๐ต๐+1๐
We wish to analyze the Adams filtrations of the maps that we have just con-
structed. First, we describe spectra which are more convenient than those appearing
in the relevant ๐ป-canonical Adams towers, and for this we need to recall the structure
of ๐ป*(๐ต+) = ๐ป*(๐ตฮฃ๐).
Proposition 7.1.3 ([1, 2.1]). Let ๐ : ๐ถ๐ โโ ฮฃ๐ be the inclusion of a Sylow subgroup.
1. ๐ป*(๐ต๐ถ๐) = ๐ธ[๐ฅ]โ F๐[๐ฆ] where |๐ฅ| = 1, |๐ฆ| = 2 and ๐ฝ๐ฅ = ๐ฆ.
2. ๐ป*(๐ตฮฃ๐) = ๐ธ[๐ฅ๐โ1]โ F๐[๐ฆ๐] where (๐ต๐)*(๐ฅ๐โ1) = ๐ฅ๐ฆ๐โ2 and (๐ต๐)*(๐ฆ๐) = ๐ฆ๐โ1.
Notation 7.1.4. For ๐ โฅ 1 write ๐๐ for ๐ฅ๐โ1๐ฆ๐โ1๐ โ ๐ป๐๐โ1(๐ต). Use the same notation
for the corresponding elements in ๐ป*(๐ต๐๐ ).
Definition 7.1.5. For 1 โค ๐ โค ๐ define ๐ต๐๐ โจ1โฉ by the following cofibration sequence.
๐ต๐๐ โจ1โฉ // ๐ต๐
๐
(๐๐,...,๐๐) //โ๐๐ ฮฃ๐๐โ1๐ป
Let ๐ต0โจ1โฉ = * and for ๐ โฅ 1 let ๐ต๐โจ1โฉ = ๐ต๐1 โจ1โฉ.
For 1 โค ๐ โค ๐, we have the following square of cofibration sequences.
๐ต๐โ1โจ1โฉ ๐ //
๐ต๐โจ1โฉ ๐ //
๐ต๐๐ โจ1โฉ
๐ต๐โ1 ๐ //
๐ต๐ ๐ //
๐ต๐๐
โ๐โ11 ฮฃ๐๐โ1๐ป //
โ๐1 ฮฃ๐๐โ1๐ป //
โ๐๐ ฮฃ๐๐โ1๐ป
93
The purpose of the spectra just defined is highlighted by the following lemma.
Lemma 7.1.6. A map to ๐ต๐๐ can be factored through ๐ต๐
๐ โจ1โฉ if and only if it can be
factored through ๐ป โง๐ต๐๐ = fib(๐ต๐
๐ โโ ๐ป โง๐ต๐๐ ).
The following lemma shows that the maps we have constructed between stunted
projective spaces have Adams filtration one.
Lemma 7.1.7. For each ๐ โฅ 1 there exists a unique map ๐ : ๐ต๐ โโ ๐ต๐โ1โจ1โฉ such
that the left diagram commutes. Moreover, the right diagram commutes.
๐ต๐
๐
๐
zz๐ต๐โ1โจ1โฉ // ๐ต๐โ1
๐ต๐ ๐ //
๐
๐ต๐+1
๐
๐ต๐โ1โจ1โฉ ๐ // ๐ต๐โจ1โฉ
For 1 โค ๐ โค ๐ the filler for the diagram
๐ต๐ ๐ //
๐
๐ต๐+1 ๐ //
๐
๐ต๐+1๐+1
๐ต๐โ1โจ1โฉ ๐ // ๐ต๐โจ1โฉ ๐ // ๐ต๐
๐ โจ1โฉ
is unique and we call it ๐. For 1 โค ๐ โค ๐ the following diagram commutes.
๐ต๐+1๐+1
๐
๐
๐ต๐๐ โจ1โฉ // ๐ต๐
๐
Before proving all the lemmas above, we make a preliminary calculation.
Lemma 7.1.8. For ๐,๐ โฅ 1 [ฮฃ๐ต๐โ1, ๐ต๐๐ ] = 0, [ฮฃ๐ต๐, ๐ต๐
๐ ] = 0, [ฮฃ๐ต๐, ๐ต๐๐ โจ1โฉ] = 0.
Proof. The results are all obvious if ๐ < ๐ so suppose that ๐ โฅ ๐.
The first follows from cellular approximation; the third does too, although we will
give a different proof.
94
Cellular approximation gives [ฮฃ๐ต๐, ๐ต๐๐ ] = [ฮฃ๐ต๐
๐ , ๐ต๐๐ ] = [ฮฃ๐/๐, ๐/๐]. We have an
exact sequence
๐2(๐/๐) โโ [ฮฃ๐/๐, ๐/๐] โโ ๐1(๐/๐)
and ๐1(๐/๐) = ๐2(๐/๐) = 0, which gives the second identification.
Since [ฮฃ๐ต๐,โ๐๐ ฮฃ๐๐โ2๐ป] = 0, [ฮฃ๐ต๐, ๐ต๐
๐ โจ1โฉ] โโ [ฮฃ๐ต๐, ๐ต๐๐ ] is injective and this
completes the proof.
Proof of lemma 7.1.2. ๐ exists because the composite๐ต๐ ๐โโ ๐ต๐ โโ ๐ต๐๐ = ฮฃ๐๐โ1๐/๐
is null. ๐ is unique because [๐ต๐,ฮฃโ1๐ต๐๐ ] = 0.
Since [๐ต๐,ฮฃโ1๐ต๐+1๐+1 ] = 0 the map ๐* : [๐ต๐, ๐ต๐] โโ [๐ต๐, ๐ต๐+1] is injective and so
commutativity of the following diagram gives commutativity of the second diagram
in the lemma.
๐ต๐ ๐ //
๐
๐ต๐+1
๐
๐
๐ต๐
๐
๐ต๐ ๐ // ๐ต๐+1
Uniqueness of the fillers is given by the facts [ฮฃ๐ต๐, ๐ต๐๐ ] = 0 and [ฮฃ๐ต๐โ1, ๐ต๐
๐ ] = 0,
respectively.
We turn to compatibility of the collection ๐ : ๐ต๐+1๐+1 โโ ๐ต๐
๐ . We already have
compatibility of the collection ๐ : ๐ต๐ โโ ๐ต๐โ1, i.e. the following diagram in the
homotopy category commutes.
๐ต1 //
๐
๐ต2 //
๐
. . . // ๐ต๐ ๐ //
๐
๐ต๐+1 //
๐
. . .
* // ๐ต1 // . . . // ๐ต๐โ1 ๐ // ๐ต๐ // . . .
For concreteness, suppose that we a have pointset level model for this diagram in
which each representative ๐ : ๐ต๐โ1 โโ ๐ต๐ is a cofibration between cofibrant spectra.
By a spectrum, we mean an ๐-module [7], and so every spectrum is fibrant. The
95
โhomotopy extension propertyโ that ๐-modules satisfy says that we can make any of
the squares strictly commute at the cost of changing the right map to a homotopic
one. By proceeding inductively, starting with the left most square, we can assume
that the representative ๐ โs are chosen so that each square strictly commutes. Let
๐ : ๐ต๐+1๐+1 โโ ๐ต๐
๐ be obtained by taking strict cofibers of the appropriate diagram.
The homotopy class of ๐ provides a filler for the diagram in the lemma and so is equal
to ๐ . It is clear that the ๐ โs are compatible and so the ๐ โs are compatible.
The deductions that each of the final four diagrams commute are similar and rely
on the uniqueness of the second filler. Weโll need the fourth diagram so we show this
in detail. We have a commuting diagram.
๐ต๐โ1 //
๐
๐ต๐+1 //
=
๐ต๐+1๐
๐
๐ต๐ //
๐
๐ต๐+1 //
๐
๐ต๐+1๐+1
๐
๐ต๐โ1 //
=
๐ต๐ //
๐
๐ต๐๐
๐
๐ต๐โ1 // ๐ต๐+1 // ๐ต๐+1
๐
The vertical composites in the first two columns are ๐ and so the third is too.
Proof of lemma 7.1.6. ๐ป*(๐ต๐๐ ) is free over ๐ธ[๐ฝ] with basis ๐๐, . . . , ๐๐. This basis
allowed us to construct the top map in the following diagram.
๐ต๐๐
(๐๐,...,๐๐) //
โ๐๐ ฮฃ๐๐โ1๐ป
โ๐๐ ฮฃ๐๐โ1(1,๐ฝ)
๐ป โง๐ต๐๐
โ //โ๐๐
(ฮฃ๐๐โ1๐ป โจ ฮฃ๐๐๐ป
)
We have a map (1, ๐ฝ) : ๐ป โโ ๐ป โจ ฮฃ๐ป which is used to construct the map on the
96
right. Since the target of this map is an ๐ป-module we obtain the bottom map and
one can check that this is an equivalence. Thus, we obtain the map of cofibration
sequences displayed at the top of the following diagram.
๐ต๐๐ โจ1โฉ //
๐ต๐๐
(๐๐,...,๐๐) //
=
โ๐๐ ฮฃ๐๐โ1๐ป
โโ
[โ๐๐ ฮฃ๐๐โ1(1,๐ฝ)
]
๐ป โง๐ต๐๐
//
๐ต๐๐
//
=
๐ป โง๐ต๐๐[โ๐๐ ฮฃ๐๐โ1(1,*)
]โโ
๐ต๐๐ โจ1โฉ // ๐ต๐
๐
(๐๐,...,๐๐) //โ๐๐ ฮฃ๐๐โ1๐ป
The bottom right square is checked to commute and so we obtain the map of cofibra-
tion sequences displayed at the bottom. This diagram shows that a map to ๐ต๐๐ can
be factored through ๐ต๐๐ โจ1โฉ if and only if it can be factoring through ๐ป โง๐ต๐
๐ ; this is
also clear if one uses the more general theory of Adams resolutions discussed in [11].
[One sees that (๐๐, . . . , ๐๐) is an ๐ป*-isomorphism in dimensions which are strictly
less than (๐ + 1)๐ โ 1 so ๐ต๐๐ โจ1โฉ is ((๐ + 1)๐ โ 3)-connected and hence (๐๐ + 1)-
connected.]
Proof of lemma 7.1.7. We have [๐ต๐,ฮฃ๐๐โ2๐ป] = 0 and so the map
[๐ต๐,
๐โ1โ1
ฮฃ๐๐โ1๐ป
]โโ
[๐ต๐,
๐โ1
ฮฃ๐๐โ1๐ป
]
is injective. Since ๐๐ = ๐ and ๐ = 0 on ๐ป, the following diagram proves the existence
of ๐.
๐ต๐
๐
๐ต๐โ1โจ1โฉ // ๐ต๐โ1 //
๐
โ๐โ11 ฮฃ๐๐โ1๐ป
๐ต๐ //
โ๐1 ฮฃ๐๐โ1๐ป
97
Uniqueness of ๐ is given by the fact that [๐ต๐,โ๐โ1
1 ฮฃ๐๐โ2๐ป] = 0.
Since [๐ต๐,โ๐
1 ฮฃ๐๐โ2๐ป] = 0 the map [๐ต๐, ๐ต๐โจ1โฉ] โโ [๐ต๐, ๐ต๐] is injective and so
commutativity of the following diagram gives commutativity of the second diagram.
๐ต๐
๐
**๐
๐
๐ต๐+1 ๐ //
๐
๐ต๐โจ1โฉ
๐ต๐โ1โจ1โฉ //
๐
))
๐ต๐โ1
๐
))๐ต๐โจ1โฉ // ๐ต๐
The filler is unique because, by lemma 7.1.8, [ฮฃ๐ต๐, ๐ต๐๐ โจ1โฉ] = 0. The final diagram
commutes because we have the following commutative diagram and a uniqueness
condition on ๐ as a filler.
๐ต๐ ๐ //
๐
๐
๐ต๐+1
๐
๐
~~
๐ต๐โ1โจ1โฉ
๐ // ๐ต๐โจ1โฉ
๐ต๐โ1 ๐ // ๐ต๐
7.2 Homotopy and cohomotopy classes in stunted
projective spaces
Throughout this thesis we write ๐ด = ๐ป*๐ป for the dual Steenrod algebra, and ๐ด* =
๐ป*๐ป for the Steenrod algebra.
To construct the homotopy class representing ๐๐๐โ๐โ1
0 โ1,๐ in the Adams spectral
98
sequence we make use of a homotopy class in ๐๐๐๐โ1(๐ต๐๐
๐๐โ๐). First, we analyze the
algebraic picture and identify the corresponding ๐ด-comodule primitive. Recall 7.1.4.
Notation 7.2.1. For ๐ โฅ 1 write ๐๐ for the class in ๐ป๐๐โ1(๐ต) dual to ๐๐ โ ๐ป๐๐โ1(๐ต).
Use the same notation for the corresponding elements in ๐ป*(๐ต๐๐ ).
Lemma 7.2.2. For each ๐ โฅ 0, ๐๐๐ โ ๐ป๐๐๐โ1(๐ต) is an ๐ด-comodule primitive.
Proof. The result is clear when ๐ = 0, since ๐ป๐(๐ต) = 0 for ๐ < ๐ โ 1. Assume from
now on that ๐ > 0.
Since the (co)homology of ๐ต is concentrated in dimensions which are 0 or โ1
congruent to ๐, the dual result is that ๐ ๐๐๐ = 0 whenever ๐, ๐ > 0 and ๐+ ๐ = ๐๐.
Let ๐ : ๐ถ๐ โโ ฮฃ๐ be the inclusion of a Sylow subgroup. Since (๐ต๐)* is injective
it is enough to show that the equation is true after applying (๐ต๐)*. Writing this out
explicitly, we must show that
๐ ๐(๐ฅ๐ฆ(๐โ1)๐โ1) = 0 whenever ๐, ๐ > 0 and ๐+ ๐ = ๐๐.
Writing ๐ for the total reduced ๐-th power we have ๐ (๐ฅ) = ๐ฅ and ๐ (๐ฆ) = ๐ฆ + ๐ฆ๐ =
๐ฆ(1 + ๐ฆ๐โ1). Suppose that ๐, ๐ > 0 and that ๐+ ๐ = ๐๐. Then
๐ (๐ฅ๐ฆ(๐โ1)๐โ1) = ๐ฅ๐ฆ(๐โ1)๐โ1(1+๐ฆ๐โ1)(๐โ1)๐โ1 = ๐ฅ๐ฆ(๐โ1)๐โ1
(๐โ1)๐โ1โ๐=0
((๐โ 1)๐ โ 1
๐
)๐ฆ(๐โ1)๐
which gives
๐ ๐(๐ฅ๐ฆ(๐โ1)๐โ1) =
((๐โ 1)๐ โ 1
๐
)๐ฅ๐ฆ(๐โ1)๐๐โ1
as long as ๐ โค (๐ โ 1)๐ โ 1 and ๐ ๐(๐ฅ๐ฆ(๐โ1)๐โ1) = 0 otherwise. We just need to show
that
๐
((๐โ 1)(๐๐ โ ๐)โ 1
๐
)whenever 0 < ๐ โค (๐ โ 1)(๐๐ โ ๐) โ 1. The largest value of ๐ for which we have
๐ โค (๐ โ 1)(๐๐ โ ๐) โ 1 is (๐ โ 1)๐๐โ1 โ 1 so write ๐ = ๐ ๐๐ for 0 โค ๐ < ๐ and ๐ โก 0
(mod ๐). Let ๐ = (๐ โ 1)(๐๐ โ ๐) โ 1 so that we are interested in(๐๐
). ๐ โ ๐ โก โ1
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(mod ๐๐+1) and so when we add ๐โ ๐ to ๐ in base ๐ there is a carry. An elementary
fact about binomial coefficients completes the proof.
The relevant topological result is given by the following proposition.
Proposition 7.2.3. For each ๐ โฅ 0, ๐๐๐ โ ๐ป๐๐๐โ1(๐ต๐๐
๐๐โ๐) is in the image of the
Hurewicz homomorphism.
Proof. The result is clear for ๐ = 0, since we have the map ๐๐โ1 โโ ฮฃ๐โ1๐/๐ = ๐ต11 .
For ๐ โฅ 1, setting ๐ = 0, ๐ = ๐+ 1, ๐ = ๐๐โ ๐โ 1 and ๐ = ๐๐โ 1 in [4, 2.9(๐ฃ)] shows
that
๐ = ๐ต[๐๐๐โ1]/๐ต[(๐๐โ๐โ1)๐โ1]
has reductive top cell and we have an โinclude-collapseโ map ๐ โโ ๐ต๐๐
๐๐โ๐.
To construct the homotopy class representing ๐๐๐โ๐โ1
0 โ1,๐ in the Adams spectral
sequence we also make use of the transfer map.
Definition 7.2.4. Write ๐ก : ๐ตโ1 โโ ๐0 for the transfer map of [1, 2.3(๐)] and let ๐ถ
be the cofiber of ฮฃโ1๐ก.
We need to analyze the affect of ๐ก algebraically.
Notation 7.2.5. We have a cofibration sequence ๐โ1 โโ ๐ถ โโ ๐ต. Abuse notation
and write ๐๐ and ๐๐ for the elements in ๐ป*(๐ถ) and ๐ป*(๐ถ) which correspond to the
elements of the same name in ๐ป*(๐ต) and ๐ป*(๐ต). Write ๐ and ๐ข for the dual classes
in ๐ป*(๐ถ) and ๐ป*(๐ถ) corresponding to generators of ๐ปโ1(๐โ1) and ๐ปโ1(๐โ1).
Lemma 7.2.6. Suppose ๐ โฅ 0. Then ๐๐๐ โ ๐ป๐๐๐โ1(๐ถ) is mapped to 1โ ๐๐๐.
+ ๐๐๐
1 โ๐ข
under the ๐ด-coaction map.
Proof. First, letโs introduce some notation which will be useful for the proof. Write
Sq๐๐๐ and Sq๐๐+1๐ for ๐ ๐ and ๐ฝ๐ ๐, respectively. Recall that the Steenrod algebra ๐ด*
has a F๐-vector space basis given by admissible monomials
โฌ = Sq๐1๐ ยท ยท ยท Sq๐๐๐ : ๐๐ โฅ ๐๐๐+1, ๐๐ โก 0 or 1 (mod ๐).
100
We claim that Sq๐๐๐๐ ๐
.= ๐๐
๐ , and that ๐๐ = 0 for any ๐ โ โฌ of length greater than 1.
Here, length greater than one means that ๐ > 1 and ๐๐ > 0.
By lemma 7.2.2 we know that ๐๐๐ is mapped, under the coaction map, to 1โ๐๐๐ +
๐ โ ๐ข for some ๐ โ ๐ด. If we can prove the claim above then we will deduce that
๐.
= ๐๐๐
1 .
Take an element ๐ = Sq๐1๐ ยท ยท ยท Sq๐๐๐ โ โฌ of length greater than one. Let ๐ = โ๐๐โ1/๐โ
so that either ๐๐โ1 = ๐๐ or ๐๐ + 1 and Sq๐๐โ1๐ = ๐ ๐ or ๐ฝ๐ ๐. We have
๐๐โ1 โฅ ๐๐๐ =โ ๐๐โ1 โ 1 โฅ ๐๐๐ โ 1 > (๐โ 1)๐๐ โ (๐โ 1) = ๐(๐๐ โ 1)/2
and so 2๐ โฅ 2(๐๐โ1โ1)/๐ > ๐๐โ1 = |Sq๐๐๐ ๐ |. Since Sq๐๐๐ ๐ comes from the cohomology
of a space we deduce that ๐ ๐Sq๐๐๐ ๐ = 0. Thus, Sq๐๐โ1๐ Sq๐๐๐ ๐ = 0 and ๐๐ = 0, which
verifies the second part of the claim.
We are left to prove that ๐ ๐๐๐.
= ๐๐๐ for each ๐ โฅ 0. First, we prove the ๐ = 0
case ๐ 1๐.
= ๐1. This statement is equivalent to the claim that
๐๐โ1 = ๐ต11 โโ ๐ตโ
1๐กโโ ๐0
is detected by a unit multiple of โ1,0 in the Adams spectral sequence. By cellular
approximation a generator of ๐๐โ1(๐ตโ1 ) is given by ๐๐โ1 = ๐ต1
1 โโ ๐ตโ1 . By definition
๐ก : ๐ตโ1 โโ ๐0 is an isomorphism on ๐๐โ1. By low dimensional calculations a generator
of ๐๐โ1(๐0) is detected by โ1,0. This completes the proof of the ๐ = 0 case.
To prove that ๐ ๐๐๐.
= ๐๐๐ it is enough to show that ๐ฝ๐ ๐๐๐
.= ๐ฝ๐๐
๐ . Notice that
|๐ฝ๐1| = ๐ and so
๐ฝ๐๐๐
= (๐ฝ๐1)๐๐
= ๐ ๐๐โ1๐/2 ยท ยท ยท๐ ๐๐/2๐ ๐/2๐ฝ๐1.
= ๐ ๐๐โ1๐/2 ยท ยท ยท๐ ๐๐/2๐ ๐/2๐ฝ๐ 1๐.
We are left with proving that ๐ ๐๐โ1๐/2 ยท ยท ยท๐ ๐/2๐ฝ๐ 1๐.
= ๐ฝ๐ ๐๐๐ . We induct on ๐, the
101
result being trivial for ๐ = 0. Suppose it is proven for some ๐ โฅ 0. Then we have
๐ ๐๐๐/2 ยท ยท ยท๐ ๐/2๐ฝ๐ 1๐.
= ๐ ๐๐๐/2๐ฝ๐ ๐๐๐
.= (๐ฝ๐ ๐๐+๐๐๐/2 + elements of โฌ of length greater than 1)๐
= ๐ฝ๐ ๐๐+1
๐,
which completes the inductive step and the proof of the lemma.
7.3 A permanent cycle in the ASS
We are now ready to prove the main result of the chapter.
Theorem 7.3.1. The element ๐๐๐โ๐โ1
0 โ1,๐.
= [๐0]๐๐โ๐โ1[๐๐
๐
1 ] โ ๐ป๐๐โ๐,๐๐(๐+1)โ๐โ1(๐ด)
is a permanent cycle in the Adams spectral sequence represented by the map
๐ผ : ๐๐๐๐โ1 ๐ // ๐ต๐๐
๐๐โ๐๐ // ๐ต๐๐โ1
๐๐โ๐โ1// . . . // ๐ต๐+2
2
๐ // ๐ต๐+11
๐ก // ๐0.
Here, ๐ comes from proposition 7.2.3, ๐ comes from lemma 7.1.2, and ๐ก is the restric-
tion of the transfer map.
Proof. By lemma 7.1.2 the following diagram commutes.
๐๐๐๐โ1
๐$$
๐ผ
++๐ต๐๐
๐๐โ๐๐ // ๐ต๐๐โ1
๐๐โ๐โ1// . . . // ๐ต๐+2
2
๐ // ๐ต๐+11
//
๐
๐0
๐ต๐๐
1
๐
OO
๐๐๐โ๐โ1
// ๐ต๐๐
1
๐
๐ตโ
1
๐ก
MM
We look at the maps induced on ๐ธ2-pages.
By definition, ๐ : ๐๐๐๐โ1 โโ ๐ต๐๐
๐๐โ๐ is represented in the Adams spectral sequence
102
by ๐๐๐ โ ๐ป0,๐๐๐โ1(๐ด;๐ป*(๐ต๐๐
๐๐โ๐)), and by lemma 7.2.2, this element is the image of
๐๐๐ โ ๐ป0,๐๐๐โ1(๐ด;๐ป*(๐ต๐๐
1 )). Moreover,
๐๐๐โ๐โ1
0 ยท ๐๐๐ โ ๐ป๐๐โ๐โ1,๐๐(๐+1)โ๐โ2(๐ด;๐ป*(๐ตโ1 )).
๐ก* : ๐ธ2(๐ตโ1 ) โโ ๐ธ2(๐
0) is described by the geometric boundary theorem. The cofi-
bration sequence ๐โ1 โโ ๐ถ โโ ๐ต induces a short exact sequence of ๐ด-comodules.
The boundary map obtained by applying ๐ป*(๐ด;โ) is the map induced by ๐ก.
0 // ฮฉ*(๐ด;๐ป*(๐โ1)) // ฮฉ*(๐ด;๐ป*(๐ถ)) // ฮฉ*(๐ด;๐ป*(๐ต)) // 0
[๐0]๐๐โ๐โ1๐๐๐
//_
ยท
[๐0]๐๐โ๐โ1๐๐๐
[๐0]๐๐โ๐โ1[๐๐
๐
1 ] // [๐0]๐๐โ๐โ1[๐๐
๐
1 ]๐ข
Thus, by using lemma 7.2.6, we see that
๐ก*(๐๐๐โ๐โ10 ยท ๐๐๐)
.= ๐๐
๐โ๐โ10 โ1,๐ โ ๐ป๐๐โ๐,๐๐(๐+1)โ๐โ1(๐ด).
This almost completes the proof. There is a subtlety, however. A map of filtration
degree ๐ only gives a well-defined map on ๐ธ๐+1 pages. To complete the proof we break
the rectangle appearing in the first diagram up into (๐๐ โ ๐โ 1)2 squares. We have
demonstrated this for the case when ๐ = 5 and ๐ = 1 below.
๐ต54
๐ // ๐ต43
๐ // ๐ต32
๐ // ๐ต21
๐
๐5 // ? // ? // ?_
๐ต5
3
๐ //
๐
OO
๐ต42
๐ //
๐
OO
๐ต31
5 //
๐
๐
OO
๐ต31
๐
๐5 //_
OO
? //_
OO
? //_
_
OO
?_
๐ต5
2
๐ //
๐
OO
๐ต41
5 //
๐
OO
๐
๐ต41
5 //
๐
๐ต41
๐
๐5 //_
OO
? //_
OO
_
? //_
?_
๐ต5
15 //
๐
OO
๐ต51
5 // ๐ต51
5 // ๐ต51 ๐5
//_
OO
๐0 ยท ๐5 // ๐20 ยท ๐5 // ๐30 ยท ๐5
103
Each square involves two maps of Adams filtration zero in the vertical direction and
two maps of Adams filtration one in the horizontal direction. Each square commutes
by lemma 7.1.2 and the maps induced on ๐ธ2-pages are well-defined. This completes
the proof.
104
Chapter 8
Adams spectral sequences
In this chapter we set up and calculate the localized Adams spectral sequence for
the ๐ฃ1-periodic sphere. Along the way we construct Adams spectral sequences for
calculating the homotopy of the mod ๐๐ Moore spectrum ๐/๐๐, the Prรผfer sphere
๐/๐โ = hocolim(๐/๐๐ // ๐/๐2
๐ // ๐/๐3 // . . .),
and we prove the final theorem stated in the introduction.
8.1 Towers and their spectral sequences
In this section we introduce some essential concepts and constructions: towers (defi-
nition 8.1.4), the smash product of towers and the spectral sequences associated with
them. We provide important examples, which will be useful for the construction of
the modified Adams spectral sequence for ๐/๐๐ and for verifying its properties. We
also recall the main properties of the Adams spectral sequence.
Notation 8.1.1. We write S for the stable homotopy category.
Definition 8.1.2. Write Ch(S ) for the category of non-negative cochain complexes
in S . An object ๐ถโ of this category is a diagram
๐ถ0 ๐ // ๐ถ1 // . . . // ๐ถ๐ ๐ // ๐ถ๐ +1 // . . .
105
in S with ๐2 = 0. An augmentation ๐ โโ ๐ถโ of a cochain complex ๐ถโ โ Ch(S ) is
a map of cochain complexes from ๐ โโ * โโ . . . โโ * โโ * โโ . . . to ๐ถโ.
Notation 8.1.3. Let Z denote the category with the integers as objects and hom-sets
determined by: |Z(๐,๐)| = 1 if ๐ โฅ ๐, and |Z(๐,๐)| = 0 otherwise. Write Zโฅ0 for
the full subcategory of Z with the non-negative integers as objects.
Definition 8.1.4. An object ๐๐ of the diagram category S Zโฅ0 is called a sequence.
A system of interlocking cofibration sequences
๐0
. . .oo ๐๐ โ1oo
๐๐ oo
๐๐ +1oo
. . .oo
๐ผ0
<<
๐ผ๐ โ1
;;
๐ผ๐
;;
๐ผ๐ +1
;;
in S is called a tower and we use the notation ๐, ๐ผ. Notice that a tower ๐, ๐ผ has
an underlying sequence ๐๐ and an underlying augmented cochain complex ๐0 โโ
ฮฃโ๐ผโ.
A map of towers ๐, ๐ผ โโ ๐, ๐ฝ is a compatible collection of maps
๐๐ โโ ๐๐ โช ๐ผ๐ โโ ๐ฝ๐ .
The following tower is important for us.
Definition 8.1.5. We write ๐0, ๐/๐ for the tower in which each of the maps in the
underlying sequence is ๐ : ๐0 โโ ๐0. The underlying augmented cochain complex
is ๐0 โโ ฮฃโ๐/๐, where each differential is given by a suspension of the Bockstein
๐ฝ : ๐/๐ โโ ๐1 โโ ฮฃ๐/๐.
Definition 8.1.6. Suppose that ๐, ๐ผ is a tower. Then by changing ๐, ๐ผ up to an
isomorphism we can find a pointset model in which each ๐๐ +1 โโ ๐๐ is a cofibration
between cofibrant ๐-modules [7] and ๐ผ๐ is the strict cofiber of this map. We say that
such a pointset level model is cofibrant.
106
By taking a cofibrant pointset level model ๐0, ๐/๐, we can construct another
tower by collapsing the ๐-th copy of ๐0.
Definition 8.1.7. Let ๐/๐๐โ*, ๐/๐ be the tower obtained in this way.
๐/๐๐
๐/๐๐โ1๐oo
๐/๐๐โ2๐oo
. . .oo ๐/๐oo
*oo
*oo
. . .oo
๐/๐
==
๐/๐
<<
๐/๐
>>
๐/๐
AA
*
CC
*
BB
To construct the multiplicative structure on the modified Adams spectral sequence
for ๐/๐๐ we need to make sense of smashing towers together. A modern version of [5,
definition 4.2] is as follows.
Definition 8.1.8. Suppose ๐, ๐ผ and ๐, ๐ฝ are towers and that we have chosen
cofibrant models for them. Let
๐๐ = colim๐+๐โฅ๐ 0โค๐,๐โค๐
๐๐ โง ๐๐.
The indexing category in this colimit is a full subcategory of ZรZ and the notation
only indicates the objects. Let
๐พ๐ =โ๐+๐=๐
0โค๐,๐โค๐
๐ผ ๐ โง ๐ฝ ๐.
We have maps ๐๐ +1 โโ ๐๐ and ๐๐ โโ ๐พ๐ , which give a cofibrant model for the smash
product of towers ๐, ๐ผ โง ๐, ๐ฝ = ๐,๐พ. Moreover, the underlying augmented
cochain complex of ๐, ๐ผโง๐, ๐ฝ is the tensor product of the underlying augmented
cochain complexes of ๐, ๐ผ and ๐, ๐ฝ.
Note that the definition of ๐, ๐ผ โง ๐, ๐ฝ depends on the choice of cofibrant
models for ๐, ๐ผ and ๐, ๐ฝ, but the following proposition shows that it is well-
defined up to isomorphism.
107
Proposition 8.1.9. Suppose we have maps of towers
๐, ๐ผ โโ ๐ณ , โ, ๐, ๐ฝ โโ ๐ด ,๐ฅ .
Then there exists a map of towers ๐, ๐ผโง ๐, ๐ฝ โโ ๐ณ , โโง ๐ด ,๐ฅ such that the
underlying map on augmented cochain complexes is the tensor product[(๐0 โ ฮฃโ๐ผโ
)โโ
(๐ณ0 โ ฮฃโโโ
)]โง
[(๐0 โ ฮฃโ๐ฝโ
)โโ
(๐ด0 โ ฮฃโ๐ฅ โ
)].
Writing down the proof of the proposition carefully is a lengthy detour. I assure
the reader that I have done this. Indeed, in the draft of my thesis all details were
included and this will remain available on my website. However, I do not want the
content of my thesis to be concerned with such technical issues. The point is that a
map of towers restricts to a map of sequences. On the cofibrant pointset level models,
we only know that each square commutes up to homotopy, but each homotopy is
determined by the map on the respective cofiber and this is part of the data of the
map of towers. Since the ๐๐ appearing in definition 8.1.8 is, in fact, a homotopy
colimit, we can define maps ๐๐ โโ ๐ต๐ using these homotopies. One finds that this
provides a map of sequences compatible with the tensor product of the underlying
maps of augmented cochain complexes.
As an example of a smash product of towers and a map of towers, we would like
to construct a map (recall definition 8.1.5)
๐0, ๐/๐ โง ๐0, ๐/๐ โโ ๐0, ๐/๐ (8.1.10)
extending the multiplication ๐0 โง ๐0 โโ ๐0. Using the terminology of [5, 11], we
see that ๐0, ๐/๐ is the ๐/๐-canonical resolution of ๐0. Moreover, [5, 4.3(b)] tells us
that ๐0, ๐/๐ โง ๐0, ๐/๐ is an ๐/๐-Adams resolution. Thus, the following lemma,
108
which is given in [11], means that it is enough to construct a map
(๐0 โ ฮฃโ๐/๐
)โง(๐0 โ ฮฃโ๐/๐
)โโ
(๐0 โ ฮฃโ๐/๐
). (8.1.11)
Lemma 8.1.12. Suppose ๐, ๐ผ and ๐, ๐ฝ are ๐ธ-Adams resolutions. Then any
map of augmented cochain complexes(๐0 โ ฮฃโ๐ผโ
)โโ
(๐0 โ ฮฃโ๐ฝโ
)extends to a
map of towers.
The following lemma shows that we can construct the map (8.1.11) by using the
multiplication ๐ : ๐/๐โง๐/๐ โโ ๐/๐ on every factor appearing in the tensor product.
Lemma 8.1.13. The following diagram commutes, where ๐ : ๐/๐ โง ๐/๐ โโ ๐/๐ is
the multiplication on the ring spectrum ๐/๐.
๐/๐ โง ๐/๐ (๐ฝโง๐/๐,๐/๐โง๐ฝ) //
๐
(ฮฃ๐/๐ โง ๐/๐) โจ (๐/๐ โง ฮฃ๐/๐)
ฮฃ(๐,๐)
๐/๐
๐ฝ // ฮฃ๐/๐
Proof. ๐/๐ โง ๐/๐ = ๐/๐ โจ ฮฃ๐/๐ and so to check commutativity of the diagram it is
enough to restrict to each factor. We are then comparing maps in [ฮฃ๐/๐,ฮฃ๐/๐] =
[๐/๐, ๐/๐] and [๐/๐,ฮฃ๐/๐]. Both groups are cyclic of order ๐ and generated by 1 and
๐ฝ, respectively. Since 1 and ๐ฝ are homologically non-trivial, the lemma follows from
the fact that the diagram commutes after applying homology.
Our motivation for constructing the map (8.1.10) was, in fact, to prove the fol-
lowing lemma.
Lemma 8.1.14. The exists a map of towers
๐/๐๐โ*, ๐/๐ โง ๐/๐๐โ*, ๐/๐ โโ ๐/๐๐โ*, ๐/๐
(recall definition 8.1.7) compatible with the map of (8.1.10).
109
Proof. Take the cofibrant pointset level model for ๐0, ๐/๐ which was used to define
๐/๐๐โ*, ๐/๐ and consider the underlying map of sequences of (8.1.10). We use the
โhomotopy extension propertyโ that ๐-modules satisfy, just like in the proof of lemma
7.1.2. It says that we can make any of the squares in the map of sequences strictly
commute at the cost of changing the left map to a homotopic one. The homotopy
we extend should be the one determined by the map on cofibers. By starting at the
(2๐ โ 1)-st position, we can make the first (2๐ โ 1) squares commute strictly. One
obtains the map of the lemma by collapsing out the ๐-th copy of ๐0 in ๐0, ๐/๐.
For us, the purpose of a tower is to a construct spectral sequence.
Definition 8.1.15. The ๐, ๐ผ-spectral sequence is the spectral sequence obtained
from the exact couple got by applying ๐*(โ) to a given tower ๐, ๐ผ. For ๐ โฅ 0, it
has
๐ธ๐ ,๐ก1 (๐, ๐ผ) = ๐๐กโ๐ (๐ผ
๐ ) = ๐๐ก(ฮฃ๐ ๐ผ๐ )
and ๐ธโ,๐ก1 = ๐๐ก(ฮฃ
โ๐ผโ) as chain complexes. It attempts to converge to ๐๐กโ๐ (๐0).
The filtration is given by ๐น ๐ ๐*(๐0) = im(๐*(๐๐ ) โโ ๐*(๐0)). Given an element
in ๐น ๐ ๐*(๐0) we can obtain a permanent cycle by lifting to ๐*(๐๐ ) and mapping this
lift down to ๐*(๐ผ๐ ).
Smashing together towers enables us to construct pairings of such spectral se-
quences.
Proposition 8.1.16 ([5, 4.4]). We have a pairing of spectral sequences
๐ธ๐ ,๐ก๐ (๐, ๐ผ)โ ๐ธ๐ โฒ,๐กโฒ
๐ (๐, ๐ฝ) โโ ๐ธ๐ +๐ โฒ,๐ก+๐กโฒ
๐ (๐, ๐ผ โง ๐, ๐ฝ).
At the ๐ธ1-page the pairing is given by the natural map
๐๐กโ๐ (๐ผ๐ )โ ๐๐กโฒโ๐ โฒ(๐ฝ๐
โฒ) โง // ๐(๐ก+๐กโฒ)โ(๐ +๐ โฒ)(๐ผ
๐ โง ๐ฝ๐ โฒ) // ๐(๐ก+๐กโฒ)โ(๐ +๐ โฒ)(๐พ๐ +๐ โฒ),
where ๐พ is as in definition 8.1.8. If all the spectral sequences converge then the pairing
converges to the smash product โง : ๐*(๐0)โ ๐*(๐0) โโ ๐*(๐0 โง ๐0).
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People often only talk about the Adams spectral sequence for a spectrum ๐ from
the ๐ธ2-page onwards. Our definition gives a functorial construction from the ๐ธ1-page.
Recall again, from [5, 11], the definition of the ๐ป-canonical resolution.
Notation 8.1.17. We write ๐ปโง*, ๐ป [*] for the ๐ป-canonical resolution of ๐0. Here
we mimic the notation used in [2], and intend for ๐ป [๐ ] to mean ๐ป โง ๐ปโง๐ . The ๐ป-
canonical resolution for a spectrum ๐ is obtained by smashing with the tower whose
underlying augmented cochain complex has augmentation ๐ โโ ๐ถโ given by the
identity; we write ๐ปโง*, ๐ป [*] โง๐.
Definition 8.1.18. Suppose ๐ is any spectrum. The Adams spectral sequence for ๐
is the ๐ปโง*, ๐ป [*] โง๐-spectral sequence.
The ๐ธ1-page of the Adams spectral sequence can be identified with the cobar
complex ฮฉโ(๐ด;๐ป*(๐)) and there exists a map of towers
๐ปโง*, ๐ป [*] โง ๐ปโง*
, ๐ป [*] โโ ๐ปโง*, ๐ป [*]
such that the induced pairing on ๐ธ1-pages is the multiplication on ฮฉโ(๐ด). This gives
the following properties of the Adams spectral sequence for ๐, which we list as a
proposition.
Proposition 8.1.19. The Adams spectral sequence is functorial in ๐ and it has
๐ธ1-page given by ฮฉโ(๐ด;๐ป*(๐)). We have a pairing of Adams spectral sequences
๐ธ๐ ,๐ก๐ (๐)โ ๐ธ๐ โฒ,๐กโฒ
๐ (๐ ) โโ ๐ธ๐ +๐ โฒ,๐ก+๐กโฒ
๐ (๐ โง ๐ )
which, at the ๐ธ1-page, agrees with the following multiplication (see [10, pg. 76]).
ฮฉโ(๐ด;๐ป*(๐))โ ฮฉโ(๐ด;๐ป*(๐ )) โโ ฮฉโ(๐ด;๐ป*(๐)โฮ ๐ป*(๐ )) = ฮฉโ(๐ด;๐ป*(๐ โง ๐ ))
Providing ๐ is ๐-complete the Adams spectral sequence for ๐ converges to ๐*(๐) in
the sense of definition 2.2.2, case 1. If each Adams spectral sequence converges then
the pairing above converges to the smash product โง : ๐*(๐)โ ๐*(๐ ) โโ ๐*(๐ โง ๐ ).
111
8.2 The modified Adams spectral sequence for ๐/๐๐
When one starts to calculate ๐ป*(๐ด;๐ป*(๐/๐)), the Adams ๐ธ2-page for the mod ๐
Moore spectrum, the first step is to describe the ๐ด-comodule ๐ป*(๐/๐). It is the subal-
gebra of ๐ด in ๐ด-comodules given by ๐ธ[๐0]. In particular, it has a nontrivial ๐ด-coaction.
For ๐ โฅ 2, ๐ป*(๐/๐๐) has trivial ๐ด-coaction which means that ๐ป*(๐ด;๐ป*(๐/๐
๐)) is two
copies of the Adam ๐ธ2-page for the sphere. We would like the ๐ธ2-page to reflect that
fact that the multiplication by ๐๐-map is zero on ๐/๐๐. This is the case when we set
up the modified Adams spectral sequence for ๐/๐๐.
Recall 8.1.17 and definition 8.1.7.
Definition 8.2.1. The modified Adams spectral sequence for ๐/๐๐ (MASS-๐) is the
๐ปโง*, ๐ป [*] โง ๐/๐๐โ*, ๐/๐-spectral sequence.
Smashing the maps of towers (recall lemma 8.1.14)
๐ปโง*, ๐ป [*] โง ๐ปโง*
, ๐ป [*] โโ ๐ปโง*, ๐ป [*],
๐/๐๐โ*, ๐/๐ โง ๐/๐๐โ*, ๐/๐ โโ ๐/๐๐โ*, ๐/๐
and composing with the swap map, we obtain a map of towers
[๐ปโง*
, ๐ป [*] โง ๐/๐๐โ*, ๐/๐]โง2โโ ๐ปโง*
, ๐ป [*] โง ๐/๐๐โ*, ๐/๐
extending the multiplication ๐/๐๐โง๐/๐๐ โโ ๐/๐๐. By proposition 8.1.16, the MASS-
๐ is multiplicative.
We turn to the structure of the ๐ธ1-page. First, we note that the underlying chain
complex of ๐/๐๐โ*, ๐/๐ is a truncated version of ฮฃโ๐/๐:
๐/๐๐ฝ // ฮฃ๐/๐
๐ฝ // ฮฃ2๐/๐ // . . . // ฮฃ๐โ1๐/๐ // * // * // . . .
Definition 8.2.2. Write Bโ for ๐ป*(ฮฃโ๐/๐) and B(๐)โ for the homology of the com-
plex just noted. Write 1๐, ๐0,๐ for the F๐-basis elements of B๐, and also for their images
112
in B(๐)๐. Note that 1๐ and ๐0,๐ are zero in B(๐) for ๐ โฅ ๐.
Since ฮฃโ๐/๐ and its truncation are ring objects in Ch(S ) we see that Bโ and
B(๐)โ are DG algebras over ๐ด. Moreover, using the same identification used for the
Adams ๐ธ1-page, we see that ๐ธโ,*1 (MASS-๐) = ฮฉโ(๐ด;B(๐)โ), as DG algebras. This
cobar complex has coefficients in a DG algebra. Such a set up is described in [10].
To describe the ๐ธ2-page we need the following theorem and lemma.
Theorem 8.2.3 ([10, pg. 80]). For any differential ๐ด-comodule Mโ which is bounded
below we have a homology isomorphism
ฮฉโ(๐ด;Mโ) โโ ฮฉโ(๐ ;๐โ๐ Mโ).
Here, ๐ is a twisting homomorphism ๐ธ โโ ๐; ๐ธ is the exterior part of ๐ด and ๐ takes
1 โฆโโ 0, ๐๐ โฆโโ ๐๐, and ๐๐1 ยท ยท ยท ๐๐๐ โฆโโ 0 when ๐ > 1.
Lemma 8.2.4. We have a homology isomorphism
ฮฉโ(๐ ;๐โ๐ B(๐)โ) โโ ฮฉโ(๐ ;๐/๐๐0 ).
Moreover, this is a map of differential algebras.
Proof. A short calculation in ๐โ๐ B(๐)โ shows that
๐(๐ โ 1๐) = 0 and ๐(๐ โ ๐0,๐) = ๐0๐ โ 1๐ โ ๐ โ 1๐+1.
[A sign might be wrong here but the end result will still be the same.] Define a map
๐โ๐ B(๐)โ โโ ๐/๐๐0
by ๐ โ 1๐ โฆโโ ๐๐0๐ and ๐ โ ๐0,๐ โฆโโ 0. This is a map of differential algebras over ๐ ,
where the target has a trivial differential. In addition, it is a homology isomorphism
and so the Eilenberg-Moore spectral sequence completes the proof.
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We should keeping track of the gradings under the maps we use:
๐ธ๐,๐1 (MASS-๐) =
โจ๐+๐=๐
ฮฉ๐,๐(๐ด;B(๐)๐)
โโโจ๐+๐=๐๐ +ฮ=๐๐ข+ฮ=๐
ฮฉ๐ ,๐ข(๐ ;๐ฮ โ๐ B(๐)๐)
โโโจ๐ +๐ก=๐๐ข+๐ก=๐
ฮฉ๐ ,๐ข(๐ ; [๐/๐๐0 ]๐ก).
We summarize what we have proved.
Proposition 8.2.5. The modified Adams spectral sequence for ๐/๐๐ (MASS-๐) is a
multiplicative spectral sequence with ๐ธ1-page ฮฉโ(๐ด;B(๐)โ) and
๐ธ๐,๐2 (MASS-๐) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐/๐๐0 ]๐ก).
We also make note of a modified Adams spectral sequence for ๐ต๐ โง ๐/๐๐, which
receives the ๐ต๐ -Hurewicz homomorphism from the MASS-๐.
Definition 8.2.6. The modified Adams spectral sequence for ๐ต๐ โง๐/๐๐ (MASS-BP-
๐) is the ๐ปโง*, ๐ป [*]โง๐ต๐ โง๐/๐๐โ*, ๐/๐-spectral sequence, where ๐ต๐ denotes the
tower whose underlying augmented cochain complex has augmentation ๐ต๐ โโ ๐ถโ
given by the identity.
In the identification of the ๐ธ1 and ๐ธ2-page of this spectral sequence Bโ is replaced
by ๐ป*(๐ต๐ โง ฮฃโ๐/๐) = ๐ โฮ Bโ. By using a shearing isomorphism, we obtain the
following proposition.
Proposition 8.2.7. The modified Adams spectral sequence for ๐ต๐ โง ๐/๐๐ (MASS-
BP-๐) is a multiplicative spectral sequence with ๐ธ1-page ฮฉโ(๐ด;๐ โฮ B(๐)โ) and
๐ธ๐,๐2 (MASS-BP-๐) = ๐ธ๐,๐
โ (MASS-BP-๐) = [๐/๐๐0 ]๐,๐โ๐.
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8.3 The modified Adams spectral sequence for ๐/๐โ
The cleanest way to define our modified Adams spectral sequence for the Prรผfer sphere
involves defining a reindexed MASS-๐. To make sense of the reindexing geometrically,
we extend our definition of sequences and towers.
Definition 8.3.1. An object ๐๐ of the diagram category S Z is called a Z-sequence.
A system of interlocking cofibration sequences
. . . ๐๐ โ1oo
๐๐ oo
๐๐ +1oo
. . .oo
๐ผ๐ โ1
;;
๐ผ๐
;;
๐ผ๐ +1
;;
in S , where ๐ โ Z, is called a Z-tower and we use the notation ๐, ๐ผ. A Z-tower
is said to be bounded below if there is an ๐ โ Z such that ๐ผ๐ = * for ๐ < ๐ . Notice
that a Z-tower ๐, ๐ผ has an underlying Z-sequence ๐๐ .
A map of Z-towers ๐, ๐ผ โโ ๐, ๐ฝ is a compatible collection of maps
๐๐ โโ ๐๐ โช ๐ผ๐ โโ ๐ฝ๐ .
We can still smash together bounded below Z-towers and they still give rise to a
spectral sequence.
Definition 8.3.2. Let ๐/๐minโ*,๐, ๐/๐ be the bounded below Z-tower obtained
from ๐/๐๐โ*, ๐/๐ by shifting it ๐ positions to the left.
Definition 8.3.3. The reindexed Adams spectral sequence for ๐/๐๐ (RASS-๐) is the
๐ปโง*, ๐ป [*] โง ๐/๐minโ*,๐, ๐/๐-spectral sequence.
Recall definition 3.1.4. We see immediately from proposition 8.2.5 that
๐ธ๐,๐2 (RASS-๐) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐๐]๐ก).
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Moreover, there are maps of bounded below Z-towers
๐/๐minโ*,๐, ๐/๐ โโ ๐/๐minโ*,๐+1, ๐/๐,
which give maps of spectral sequences from the RASS-๐ to the RASS-(๐+1). Chasing
through the identification of the ๐ธ2-pages one see that the map at ๐ธ2-pages is induced
by the inclusion ๐๐ โโ๐๐+1.
Definition 8.3.4. The modified Adams spectral sequence for ๐/๐โ (MASS-โ) is the
colimit of the reindexed Adams spectral sequences for ๐/๐๐. It has
๐ธ๐,๐2 (MASS-โ) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐/๐โ0 ]๐ก).
There are some technicalities to worry about when taking the colimit of spectral
sequences. We resolve such issues in appendix B.
By definition, we have a map of spectral sequences from the RASS-๐ to the MASS-
โ. Lemma A.4 provides the following corollary to lemma 4.1.10.
Corollary 8.3.5. The map ๐ธ๐,๐โ (RASS-(๐ + 1)) โโ ๐ธ๐,๐
โ (MASS-โ) is surjective
when ๐โ ๐ = ๐๐๐ and ๐ โฅ ๐๐ โ ๐โ 1.
8.4 A permanent cycle in the MASS-(๐ + 1)
In order to localize the MASS-(๐+ 1) we need to find a permanent cycle detecting a
๐ฃ1-self map
๐ฃ๐๐
1 : ๐/๐๐+1 โโ ฮฃโ๐๐๐๐/๐๐+1.
The following theorem provides such a permanent cycle.
Theorem 8.4.1. The element ๐๐๐
1 in ๐ป0,๐๐๐(๐ ; [๐/๐๐+10 ]๐
๐) is a permanent cycle in
the MASS-(๐+ 1).
Proof. By definition, the RASS-(๐+1) is obtained by reindexing the MASS-(๐+1) and
so it is equivalent to prove that ๐๐๐
1 /๐๐+10 โ ๐ป0,๐๐๐(๐ ; [๐๐+1]
๐๐โ๐โ1) is a permanent
116
cycle in the RASS-(๐+1). Lemmas 4.1.10 and A.4 show that it is enough to prove that
๐๐๐
1 /๐๐+10 โ ๐ป0,๐๐๐(๐ ; [๐/๐โ0 ]๐
๐โ๐โ1) is a permanent cycle in the MASS-โ. The map
of spectral sequences induced by ฮฃโ1๐/๐โ โโ ๐0 (proposition A.1) is an isomor-
phism in this range and so we are left with showing that ๐(๐๐๐
1 /๐๐+10 ) = ๐๐
๐โ๐โ10 โ1,๐
is a permanent cycle in the ASS, but this is the content of theorem 7.3.1.
Pick a representative for ๐๐๐
1 in ๐๐๐๐(๐/๐๐+1). Using the map of spectral sequences
from the MASS-(๐+1) to the MASS-BP-1 and the fact that ๐๐๐
1 in๐/๐0 has the highest
monomial weight, i.e. modified Adams filtration, among elements of the same internal
degree, we see that the image of the chosen representative in ๐ต๐*(๐/๐) is ๐ฃ๐๐
1 . Thus,
tensoring up any representative for ๐๐๐
1 to a self-map ๐ฃ๐๐
1 : ๐/๐๐+1 โโ ฮฃโ๐๐๐๐/๐๐+1
defines a ๐ฃ1 self-map. Corollary 8.3.5 tells us that it is possible to refine our choice of a
representative for ๐๐๐
1 so that it maps to the ๐ผ of theorem 7.3.1 under ฮฃโ1๐/๐๐+1 โ ๐0.
This completes the proof of the final theorem stated in the introduction.
8.5 The localized Adams spectral sequences
Since ๐๐๐โ1
1 is a permanent cycle in the MASS-๐, multiplication by ๐๐๐โ1
1 defines a
map of spectral sequences, which enables us to make the following definition.
Definition 8.5.1. The localized Adams spectral sequence for ๐ฃโ11 ๐/๐๐ (LASS-๐) is
the colimit of the following diagram of spectral sequences.
๐ธ*,** (MASS-๐)
๐๐๐โ1
1 // ๐ธ*,** (MASS-๐)
๐๐๐โ1
1 // ๐ธ*,** (MASS-๐)
๐๐๐โ1
1 // . . .
It has
๐ธ๐,๐2 (LASS-๐) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐๐0 ]๐ก).
Since the MASS-๐ is multiplicative, the differentials in the LASS-๐ are derivations.
117
The following diagram commutes when ๐ = 2.
๐ธ*,*๐ (RASS-๐)
๐๐๐
1 //
๐ธ*,*๐ (RASS-๐)
๐ธ*,*๐ (RASS-(๐+ 1))
๐๐๐
1 // ๐ธ*,*๐ (RASS-(๐+ 1))
Taking homology, we see, inductively, that it commutes for all ๐ โฅ 2. This means
that we have maps of spectral sequences between reindexed localized Adams spectral
sequences for ๐ฃโ11 ๐/๐๐ and so we can make the following definition.
Definition 8.5.2. The localized Adams spectral sequence for the ๐ฃ1-periodic sphere
(LASS-โ) is the colimit of the apparent reindexed localized Adams spectral sequences
for ๐ฃโ11 ๐/๐๐. It has
๐ธ๐,๐2 (LASS-โ) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก).
8.6 Calculating the LASS-โ
Our calculation of the LASS-โ imitates that of the loc.alg.NSS. First, we note some
permanent cycles in the MASS-โ and LASS-โ.
Proposition 8.6.1. For ๐ โฅ 1 and ๐ โฅ 0, ๐๐๐๐โ1
1 /๐๐0 is a permanent cycle in the
MASS-โ. For ๐ โฅ 1 and ๐ โ Z, ๐๐๐๐โ1
1 /๐๐0 is a permanent cycle in the LASS-โ.
Proof. In the first case, ๐๐๐๐โ1
1 is permanent cycle in the MASS-๐ and so ๐๐๐๐โ1
1 /๐๐0
is a permanent cycle in the RASS-๐ and the MASS-โ. In the second case, ๐๐๐๐โ1
1 is
permanent cycle in the LASS-๐ and the same argument gives the result.
Corollary 5.5.2 describes the associated graded of the ๐ธ2-page of the LASS-โ
with respect to the Bockstein filtration and we claim that
๐2 : ๐ธ๐,๐2 (LASS-โ) โโ ๐ธ๐+2,๐+1
2 (LASS-โ)
118
respects the Bockstein filtration.
Note that ๐0 โ ๐ป*(๐ ; ๐โ11 ๐/๐๐0 ) is a permanent cycle in the LASS-๐. Because ๐2
is a derivation in the LASS-๐, multiplication by ๐0 commutes with ๐2. Thus, we find
the same in the reindexed localized Adams spectral sequences for ๐ฃโ11 ๐/๐๐ and hence,
in the LASS-โ, too. This verifies the claim.
We conclude that we have a filtration spectral sequence (๐0-FILT2)
๐ธ๐,๐,๐ฃ0 (๐0-FILT2) =
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ1
1 -BSS)๐ฃ
=โ ๐ธ๐,๐3 (LASS-โ).
Our calculation of this spectral sequence comes down to the calculation of the ๐ธ1-page
of the ๐0-FILT spectral sequence made in proposition 6.3.1.
In appendix A we show that each of the maps in the exact couple defining the
๐โ11 -BSS comes from a map of localized Adams spectral sequences. This means that
if ๐ฅ โ ๐ป*(๐ ; ๐โ11 ๐/๐0) and ๐๐ฃ0๐ฅ โ ๐ธโ(๐โ1
1 -BSS) then ๐๐0-FILT20 (๐๐ฃ0๐ฅ) = ๐๐ฃ0๐
LASS-12 ๐ฅ. The
๐/๐ analog of theorem 1.4.7 therefore tells us that
๐๐0-FILT20 :
โจ๐ โฅ๐ ๐ +๐ก=๐๐ข+๐ก=๐
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ1
1 -BSS) โโโจ๐ โฅ๐ +1๐ +๐ก=๐+2๐ข+๐ก=๐+1
๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ1
1 -BSS)
and that if ๐ฅ lies in a single trigrading, then
๐๐0-FILT20 (๐๐ฃ0๐ฅ) โก ๐๐ฃ0๐
๐ฃ1-alg.NSS2 ๐ฅ = ๐๐0-FILT
0 (๐๐ฃ0๐ฅ)
up to terms with higher ๐ -grading. Thus, using a filtration spectral sequence with
respect to the ๐ -grading we deduce from proposition 6.3.1 that there is an F๐-vector
space isomorphism
๐ธ๐,๐,๐ฃ1 (๐0-FILT2) โผ=
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ธ๐ ,๐ก,๐ข,๐ฃ1 (๐0-FILT).
In appendix B the LASS-โ is shown to converge to ๐*(๐ฃโ11 ๐/๐โ). Moreover, since
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the localized Adams-Novikov spectral for ๐*(๐ฃโ11 ๐/๐โ) is degenerate and convergent
(the height 1 telescope conjecture is true), we know precisely what group the LASS-โ
converges to. From the bound on the size of the ๐ธ3-page given by our knowledge of
๐ธ1(๐0-FILT2) we can deduce the following proposition.
Proposition 8.6.2. For ๐ โฅ 3, there exist isomorphisms
๐ธ๐,๐๐ (LASS-โ) โผ=
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ธ๐ ,๐ก,๐ข๐ (loc.alg.NSS)
compatible with differentials.
๐ธโ(LASS-โ) has an F๐-basis given by the classes of the following elements.
๐๐ฃ0 : ๐ฃ < 0
โช๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โฅ 1, ๐ โ Zโ ๐Z, โ๐ โค ๐ฃ < 0
โช
๐๐ฃ0๐๐ : ๐ โฅ 1, 1โ ๐๐ โค ๐ฃ < 0
Here, ๐๐ฃ0๐๐ denotes an element of ๐ธ3(LASS-โ) corresponding to the element of the
same name in ๐ธ3(loc.alg.NSS) (see proposition 6.4.1).
8.7 The Adams spectral sequence
The following two corollaries show that our calculation of the LASS-โ has implica-
tions for the Adams spectral sequence.
First, lemma A.4 provides the following corollary to proposition 4.3.3.
Corollary 8.7.1. The localization map ๐ธ๐,๐3 (MASS-โ) โโ ๐ธ๐,๐
3 (LASS-โ)
1. is a surjection if ๐ < ๐(๐โ 1)(๐ + 1)โ 2, i.e. ๐โ ๐ < (๐2 โ ๐โ 1)(๐ + 1)โ 1;
2. is an isomorphism if ๐โ 1 < ๐(๐โ 1)(๐ โ 1)โ 2,
i.e. ๐โ ๐ < (๐2 โ ๐โ 1)(๐ โ 1)โ 2.
120
Using the map of spectral sequences induced by ฮฃโ1๐/๐โ โโ ๐0 (proposition
A.1) we obtain the following corollary.
Corollary 8.7.2. ๐ธ๐,๐3 (ASS) โผ= ๐ธ๐โ1,๐
3 (LASS-โ) if ๐โ ๐ < (๐2 โ ๐โ 1)(๐ โ 2)โ 3
and ๐โ ๐ > 0.
The line of the corollary just stated is drawn in green in figure 1-1.
One has to be careful when discussing higher differentials in the Adams spectral
sequence. Here is what we know:
โ There are permanent cycles at the top of each principal tower, the images of the
following elements under the map ๐ : ๐ป*(๐ ;๐/๐โ0 ) โโ ๐ป*(๐ ;๐) โผ= ๐ป*(๐ด).
๐๐ฃ0๐
๐๐๐โ1
1 : ๐ โฅ 1, ๐ โ Zโ ๐Z, ๐ โฅ 1, โ๐ โค ๐ฃ < 0
Every other element in a principal tower supports a nontrivial differential.
โ An element of a side tower in the Adams spectral sequence cannot be hit by a
shorter differential than the corresponding element of the LASS-โ.
โ A non-permanent cycle in a principal tower in the Adams spectral sequence
above the line of the previous corollary supports a differential of the expected
length.
โ A non-permanent cycle in a principal tower in the Adams spectral sequence
cannot support a longer nontrivial differential than the corresponding element
of the LASS-โ, but perhaps it supports a shorter one than expected, leaving
an element of a side tower to detect a nontrivial homotopy class.
Looking at the charts of Nassau [14], we cannot find an example of the final
phenomenon, but the class ๐1,0 โ ๐ธ2,๐๐2 (ASS) gives a related example. [Similarly, one
could consider the potential ๐ = 3 Arf invariant elements.] We describe this example
presently.
121
Under the isomorphism ๐ธ2,๐๐2 (ASS) โผ= ๐ธ1,๐๐
2 (MASS-โ), ๐1,0 is mapped to an ele-
ment detected by ๐โ(๐โ1)0 ๐๐1๐1 in the ๐โ0 -BSS. Under the localization map this element
maps to an element of ๐ธ1,๐๐2 (LASS-โ) detected by ๐โ(๐โ1)
0 ๐๐1๐1 in the ๐โ11 -BSS.
The element
๐ฅ = ๐โ๐โ10 ๐๐1 โ ๐โ1
0 ๐โ11 ๐2 โ ๐ป0,๐๐(๐ ; [๐โ1
1 ๐/๐โ0 ]โ1) โ ๐ธโ1,๐๐โ12 (LASS-โ)
is detected by ๐โ๐โ10 ๐๐1 in the ๐โ1
1 -BSS. Our calculation of the LASS-โ shows that
๐2๐ฅ โ ๐ธ1,๐๐2 (LASS-โ) is nonzero. Moreover, we find that ๐ธ1,๐๐
2 (LASS-โ) = F๐ so
that a unit multiple of ๐ฅ maps via ๐2 to the localization of ๐1,0. This demonstrates
the well-known fact that ๐ฝ โ ๐๐๐โ2(๐0) is not ๐ฃ1-periodic.
122
Appendix A
Maps of spectral sequences
The most difficult result of this appendix is the following proposition.
Proposition A.1. There are maps of spectral sequences
RASS-๐ //
ASS
=
ฮฃโ1๐/๐๐ //
๐0
=
MASS-โ //
=
ASS
induced by ฮฃโ1๐/๐โ //
=
๐0
MASS-โ //MASS-1 ฮฃโ1๐/๐โ // ๐/๐.
At ๐ธ2-pages we get the maps by taking the connecting homomorphisms in the long
exact sequences got by applying ๐ป*(๐ ;โ) to the following short exact sequences of
๐ -comodules (recall definition 3.1.4).
0 // ๐ //
=
๐โจ๐โ๐0 โฉ //
๐๐//
0
0 // ๐ //
๐โ10 ๐ //
/๐0
๐/๐โ0 //
=
0
0 // ๐/๐0 // ๐/๐โ0๐0 // ๐/๐โ0 // 0
123
The previous proposition was required in the proof of theorem 8.4.1, which was
necessary to construct the LASS-๐ and the LASS-โ. We record, for completeness,
the following lemma.
Lemma A.2. There are maps of spectral sequences
MASS-(๐+ 1) โโ MASS-๐ induced by ๐/๐๐+1 โโ ๐/๐๐
MASS-๐ โโ MASS-BP-๐ induced by ๐/๐๐ โโ ๐ต๐ โง ๐/๐๐
RASS-๐ โโ RASS-(๐+ 1) induced by ๐/๐๐๐โโ ๐/๐๐+1
RASS-๐ โโ MASS-โ induced by ๐/๐๐ โโ ๐/๐โ
MASS-(๐+ 1) โโ MASS-(๐+ 1) induced by ๐/๐๐+1๐ฃ๐
๐
1โโ ฮฃโ๐๐๐๐/๐๐+1
MASS-๐ โโ LASS-๐ induced by ๐/๐๐ โโ ๐ฃโ11 ๐/๐๐
RASS-(๐+ 1) โโ RASS-(๐+ 1) induced by ๐/๐๐+1๐ฃ๐
๐
1โโ ฮฃโ๐๐๐๐/๐๐+1
MASS-โ โโ LASS-โ induced by ๐/๐โ โโ ๐ฃโ11 ๐/๐โ
To calculate the LASS-โ we used the ๐0-FILT2 spectral sequence. To calculate
๐ธ1(๐0-FILT2) we required the following proposition.
Proposition A.3. There are maps of spectral sequences induced by the following
cofibration sequences.
๐/๐ โโ ๐/๐โ โโ ๐/๐โ
๐ฃโ11 ๐/๐ โโ ๐ฃโ1
1 ๐/๐โ โโ ๐ฃโ11 ๐/๐โ
At ๐ธ2-pages the maps are the ones in the exact couples defining the ๐โ0 -BSS and the
๐โ11 -BSS, respectively.
Often we have a map of spectral sequences and we know that on a given page,
at various bidegrees, we have surjections and injections. The following lemma tells
us that if we have a map of spectral sequences, a ๐๐-differential, and that the map
on the ๐ธ๐-pages is a surjection at the source of the differential and an injection at
the target of the differential, then we have a surjection and an injection at the same
positions on the ๐ธ๐+1-page. The proof is a diagram chase.
124
Lemma A.4. Suppose ๐ถโ โโ ๐ทโ is a map of cochain complexes (in abelian groups),
that ๐ถ๐ โโ ๐ท๐ is surjective and ๐ถ๐+1 โโ ๐ท๐+1 is injective. Then ๐ป๐(๐ถโ) โโ
๐ป๐(๐ทโ) is surjective and ๐ป๐+1(๐ถโ) โโ ๐ป๐+1(๐ทโ) is injective.
We now turn to the proofs of the two propositions.
Proof of proposition A.1. The map RASS-๐ โโ MASS-โ is given by definition. The
map ASS โโ MASS-1 is just a normal map of Adams spectral sequences induced by
๐0 โโ ๐/๐. The difficult map to construct is the one induced by ฮฃโ1๐/๐๐ โโ ๐0.
We turn to this presently.
The idea is to start with the connecting homomorphism corresponding to the short
exact sequence of ๐ -comodules 0 โโ ๐ โโ ๐โจ๐โ๐0 โฉ โโ๐๐ โโ 0, and try to realize
it geometrically.
Consider the following short exact sequence of cochain complexes in ๐ด-comodules.
0 //
0 //
0 //
. . . // 0 //
F๐ //
0
0 //
ฮฃโ๐๐ธ[๐0,โ๐] //
ฮฃโ๐+1๐ธ[๐0,โ๐+1]//
. . . // ฮฃโ1๐ธ[๐0,โ1]//
F๐ //
0
0 // ฮฃโ๐๐ธ[๐0,โ๐] // ฮฃโ๐+1๐ธ[๐0,โ๐+1]
// . . . // ฮฃโ1๐ธ[๐0,โ1]// 0 // 0
The suspensions indicate cohomological degree. The first complex is concentrated in
cohomological degree 0 and is just F๐. The last complex is a shifted version of B(๐)โ,
which we call Bโ(๐)โ. We call the middle complex C(๐)โ. We will show that the
connecting homomorphism of interest is the same as the connecting homomorphism
corresponding to the short exact sequence of differential ๐ด-comodules
0 โโ F๐ โโ C(๐)โ โโ Bโ(๐)โ โโ 0.
Recall lemma 8.2.4 which helped us to identify the ๐ธ2-page of the MASS-๐. We
125
used a map
๐โ๐ B(๐)โ โโ ๐/๐๐0
defined by ๐ โ 1๐ โฆโโ ๐๐0๐ and ๐ โ ๐0,๐ โฆโโ 0. Similarly, we have maps making the
following diagram commute.
0 // ๐โ๐ F๐ //
๐โ๐ C(๐)โ //
๐โ๐ Bโ(๐)โ //
0
0 // ๐ // ๐โจ๐โ๐0 โฉ //๐๐// 0
Theorem 8.2.3 was also important in identifying the ๐ธ2-page of the MASS-๐. Using it
again, together with the maps just defined, we find that we have a diagram of cochain
complexes.
0 // ฮฉโ(๐ด;F๐) //
ฮฉโ(๐ด;C(๐)โ) //
ฮฉโ(๐ด;Bโ(๐)โ) //
0
0 // ฮฉโ(๐ ;๐) // ฮฉโ(๐ ;๐โจ๐โ๐0 โฉ) // ฮฉโ(๐ ;๐๐) // 0
Each of the vertical maps is a homology isomorphism and so we can calculate the
connecting homomorphism of interest using, instead, the connecting homomorphism
associated with 0 โโ F๐ โโ C(๐)โ โโ Bโ(๐)โ โโ 0.
The connecting homomorphism for this short exact sequence can be described
even more explicitly than is usual. Lifting under the map C(๐)โ Bโ(๐)โ can done
using the unique ๐ด-comodule splitting Bโ(๐)โ โห C(๐)โ, which puts a zero in the F๐spot. Similarly, the map F๐ โห C(๐)โ has a unique ๐ด-comodule splitting C(๐)โ F๐.
The diagram on the next page realizes the cochain complexes F๐, C(๐)โ, and
Bโ(๐)โ geometrically. We note that the first cochain complex comes from the tower
๐0 whose underlying Z-sequence is ๐0 in nonpositive degrees, * in positive degrees,
with identity structure maps where possible. The last cochain complex is the under-
lying cochain complex of ๐/๐minโ*,๐, ๐/๐, one of the towers used to construct the
126
RASS-๐.
* //
* //
* //
. . . // * //
๐0 //
*
* //
ฮฃโ๐๐/๐ //
ฮฃโ๐+1๐/๐ //
. . . // ฮฃโ1๐/๐ //
๐0 //
*
* // ฮฃโ๐๐/๐ // ฮฃโ๐+1๐/๐ // . . . // ฮฃโ1๐/๐ // * // *
Label these cochain complexes in S by the same names as the cochain complexes
obtained by applying ๐ป*(โ) and recall the underlying cochain complex ฮฃโ๐ป [โ] of the
canonical ๐ป-resolution of ๐0. The snake lemma for calculating the connecting the
homomorphism is realized geometrically by the following composite.
[ฮฃโ๐ป [โ] โงBโ(๐)โ
]๐๐ //
[ฮฃโ๐ป [โ] โง C(๐)โ
]๐๐ //
[ฮฃโ๐ป [โ] โง C(๐)โ
]๐+1๐ //
[ฮฃโ๐ป [โ]
]๐+1
Here, ๐ and ๐ denote the respective splittings at the level of underlying spectra, ๐
is the differential in the cochain complex ฮฃโ๐ป [โ] โง C(๐)โ, and we have used that
ฮฃโ๐ป [โ] โง F๐ = ฮฃโ๐ป [โ].
To get the map of spectral sequences we just need to define a map of Z-towers
๐/๐minโ*,๐, ๐/๐ โโ ๐0, which pairs with ๐ปโง*, ๐ป [*] to give the composite above.
Such a map of Z-towers has nonzero degree: it raises cohomological degree by 1. The
underlying map of Z-sequences takes ฮฃโ1๐/๐๐ to ๐0 via the composite
ฮฃโ1๐/๐๐ โโ ฮฃโ1๐/๐โ โโ ๐0
and the map on underlying cochain complexes is the map Bโ(๐)โ โโ F๐ which, on
homology, takes ๐0,โ1 to 10. This completes the construction of the map of spectral
sequences RASS-๐ โโ ASS induced by ฮฃโ1๐/๐๐ โโ ๐0.
127
Since the maps of towers we use are compatible with the maps
๐/๐minโ*,๐, ๐/๐ โโ ๐/๐minโ*,๐+1, ๐/๐,
the maps just constructed pass to the colimit to give the map MASS-โ โโ ASS
induced by ฮฃโ1๐/๐โ โโ ๐0. The map MASS-โ โโ MASS-1 can be obtained by
composition with the map ASS โโ MASS-1.
Proof of proposition A.3. The map of spectral sequences induced by the connecting
map ฮฃโ1๐/๐โ โโ ๐/๐ was constructed in the previous proposition. The map in-
duced by ๐/๐ โโ ๐/๐โ is the map MASS-1 โผ= RASS-1 โโ MASS-โ.
The maps MASS-(๐ + 1) โโ MASS-๐ induced by ๐/๐๐+1 โโ ๐/๐๐ can be rein-
dexed to give maps RASS-(๐ + 1) โโ RASS-๐ of nonzero degree. Taking a colimit
we obtain the map MASS-โ โโ MASS-โ induced by ๐ : ๐/๐โ โโ ๐/๐โ.
We turn to the localized version. The map induced by ๐ฃโ11 ๐/๐ โโ ๐ฃโ1
1 ๐/๐โ is
obtained in an identical manner to the unlocalized one, passing through a reindexed
localized Adams spectral sequence for ๐ฃโ11 ๐/๐. Similarly, for the map ๐ : ๐ฃโ1
1 ๐/๐โ โโ
๐ฃโ11 ๐/๐โ, the maps RASS-(๐ + 1) โโ RASS-๐ localize to give maps of reindexed
localized Adams spectral sequences, and we can take a colimit.
Finally, for the connecting homomorphism we recall that the map RASS-๐ โโ
MASS-1 is constructed using the map of towers ๐/๐minโ*,๐, ๐/๐ โโ ๐0 โโ ๐/๐,
which makes use of the connecting homomorphism ฮฃโ1๐/๐โ โโ ๐/๐. Each of the
maps RASS-๐ โโ MASS-1 localizes. The collection of localized maps is compatible
and so defines the requisite map LASS-โ โโ LASS-1.
128
Appendix B
Convergence of spectral sequences
In this appendix we check that each of the spectral sequences used in this thesis
converges in the sense of definition 2.2.2. In particular, we describe how to deal with
the technicalities associated with taking the colimits of spectral sequences that appear
in the definition of the MASS-โ, the LASS-๐, and the LASS-โ.
In definition 8.1.15 we define the filtration of ๐*(๐0) that is relevant for the ๐, ๐ผ-
spectral sequence, and we describe a detection map
๐น ๐ ๐๐กโ๐ (๐0)/๐น๐ +1๐๐กโ๐ (๐0) โโ ๐ธ๐ ,๐ก
โ (๐, ๐ผ).
This takes care of the filtration and detection map for the MASS-๐ and we will verify
case 1 of definition 2.2.2 to prove the following proposition.
Proposition B.1. The MASS-๐ converges to ๐*(๐/๐๐).
Moreover, the filtration associated with the RASS-๐ is obtained by reindexing the
filtration associated with the MASS-๐, and our proof will verify case 3 of definition
2.2.2 to give the following corollary.
Corollary B.2. The RASS-๐ converges to ๐*(๐/๐๐).
Delaying the proof of these results for now, we note that we have not even defined
the filtration or detection map for the MASS-โ, the LASS-๐, and the LASS-โ. We
129
carefully discuss the situation for the MASS-โ by turning straight to the proof of
the following proposition.
Proposition B.3. The MASS-โ converges to ๐*(๐/๐โ).
Proof. The purpose of a convergent spectral sequence is to identify the associated
graded of an abelian group with respect to some convergent filtration. Thus, the
most immediate aspects of the MASS-โ to address are the associated filtration, the
๐ธโ-page and the relationship between the two.
We have injections ๐น ๐๐*(๐/๐๐) โโ ๐*(๐/๐
๐), where the ๐น denotes the filtration
associated with the RASS-๐. Since, the maps ๐ : ๐/๐๐ โโ ๐/๐๐+1 used to define
๐/๐โ are compatible with these filtrations, and filtered colimits preserve exactness,
we obtain an injection colim๐๐น๐๐*(๐/๐
๐) โโ colim๐๐*(๐/๐๐) = ๐*(๐/๐
โ). We define
๐น ๐๐*(๐/๐โ) = im
(colim๐ ๐น
๐๐*(๐/๐๐) โโ ๐*(๐/๐
โ)
).
When we say that the MASS-โ is the colimit of the reindexed Adams spectral
sequences for ๐/๐๐ we mean that
๐ธ๐,๐๐ (MASS-โ) = colim๐ ๐ธ
๐,๐๐ (MASS-๐)
for each ๐ โฅ 2. Since filtered colimits commute with homology we have identifications
๐ป๐,๐(๐ธ*,*๐ (MASS-โ), ๐๐) = ๐ธ๐,๐
๐+1(MASS-โ) for each ๐ โฅ 2, which justifies calling the
MASS-โ a spectral sequence.
Staying true to definition 2.1.9, the ๐ธโ-page of the MASS-โ is given by the
permanent cycles modulo the boundaries. However, we had another choice for the
definition of the ๐ธโ-page:
๐ธ๐,๐โ (MASS-โ) = colim๐ ๐ธ
๐,๐โ (MASS-๐).
We show that the two definitions coincide presently.
The vanishing line of corollary 4.1.9 ensures that, for large ๐ depending only on
(๐, ๐), not on ๐, we have maps ๐ธ๐,๐๐ (RASS-๐) โโ ๐ธ๐,๐
๐+1(RASS-๐). Moreover, ๐ is
130
allowed to be โ. Thus,
๐ธ๐,๐โ (MASS-โ) = colim๐>>0 ๐ธ
๐,๐๐ (MASS-โ)
= colim๐>>0 colim๐ ๐ธ๐,๐๐ (RASS-๐)
= colim๐ colim๐>>0 ๐ธ๐,๐๐ (RASS-๐)
= colim๐ ๐ธ๐,๐โ (RASS-๐).
The vanishing line makes sure that an element of the RASS-๐ cannot support longer
and longer differentials as it is mapped forward into subsequent reindexed Adams
spectral sequences, without eventually becoming a permanent cycle.
This observation is what allows us to make an identification
๐น ๐๐๐โ๐(๐/๐โ)/๐น ๐+1๐๐โ๐(๐/๐โ) = ๐ธ๐,๐โ (MASS-โ).
Providing we have proved the previous corollary, that the RASS-๐ converges, we have
the following short exact sequence.
0 โโ ๐น ๐+1๐๐โ๐(๐/๐๐) โโ ๐น ๐๐๐โ๐(๐/๐๐+1) โโ ๐ธ๐,๐โ (MASS-๐) โโ 0 (B.4)
Taking colimits gives another short exact sequence. By our definition of ๐น ๐๐*(๐/๐โ),
and the fact that the right term can be identified with ๐ธ๐,๐โ (MASS-โ), that short
exact sequence gives the requisite identification.
We are just left with showing thatโ๐ ๐น
๐๐*(๐/๐โ) = ๐*(๐/๐
โ) and that for each
๐ข, there exists a ๐ with ๐น ๐๐๐ข(๐/๐โ) = 0. For the first part we note that
โ๐๐น ๐๐*(๐/๐
โ) = im
(colim๐ colim๐ ๐น
๐๐*(๐/๐๐) โโ ๐*(๐/๐
โ)
)= im
(colim๐ colim๐ ๐น
๐๐*(๐/๐๐) โโ ๐*(๐/๐
โ)
)= im
(colim๐ ๐*(๐/๐
๐) โโ ๐*(๐/๐โ)
)= ๐*(๐/๐
โ).
131
For the second part, we use both the vanishing line of corollary 4.1.9 and the con-
vergence of the RASS-๐ again. They tell us that ๐น ๐๐๐โ๐(๐/๐๐) is zero for ๐ > ๐พ
where
๐พ =(๐โ ๐) + 1
๐โ 1.
๐พ is independent of ๐, so ๐น ๐๐๐โ๐(๐/๐โ) = 0 for ๐ > ๐พ.
The vanishing line makes sure that if an element of ๐*(๐/๐๐) has infinitely many
filtration shifts as it is mapped forward into subsequent Moore spectra, then it must
map to zero in ๐*(๐/๐โ).
We have proved that the MASS-โ convergences in accordance with definition
2.2.2, case 3.
Since the convergence of the LASS-๐ and LASS-โ are similar we address them
presently.
Proposition B.5. The LASS-(๐+ 1) converges to ๐*(๐ฃโ11 ๐/๐๐+1).
Proof. This is essentially the same proof as just given for the MASS-โ. We just need
to make a few remarks.
First, we have not said precisely what we mean by ๐ฃโ11 ๐/๐๐+1. Theorem 8.4.1 tells
us that ๐๐๐
1 is a permanent cycle in the MASS-(๐+1). Thus it detects some homotopy
class, which we call
๐ฃ๐๐
1 : ๐0 โโ ฮฃโ๐๐๐๐/๐๐+1.
Since ๐/๐๐+1 is a ring spectrum, we can โtensor upโ to obtain a ๐ฃ1 self-map, which
we give the same name
๐ฃ๐๐
1 : ๐/๐๐+1 โโ ฮฃโ๐๐๐๐/๐๐+1.
๐ฃโ11 ๐/๐๐+1 is the homotopy colimit of the diagram
๐/๐๐+1๐ฃ๐
๐
1 // ฮฃโ๐๐๐๐/๐๐+1๐ฃ๐
๐
1 // ฮฃโ2๐๐๐๐/๐๐+1๐ฃ๐
๐
1 // ฮฃโ3๐๐๐๐/๐๐+1 // . . .
By construction we have ๐*(๐ฃโ11 ๐/๐๐+1) = (๐ฃ๐
๐
1 )โ1๐*(๐/๐๐+1). This is what allows us
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to use the multiplicative structure of the MASS-n to localize the spectral sequence as
opposed to constructing maps of towers.
Let [๐, ๐] = ๐ + ๐๐๐ and [๐, ๐] = ๐ + ๐๐(๐ + 1)๐. The vanishing line of corollary
4.1.9 is parallel to the multiplication-by-๐๐๐
1 -line and this ensures that for large ๐,
depending only on (๐, ๐), not ๐, we have maps
๐ธ[๐,๐],[๐,๐]๐ (MASS-(๐+ 1)) โโ ๐ธ
[๐,๐],[๐,๐]๐+1 (MASS-(๐+ 1)).
Thus, just as in the previous proof we have an identification
๐ธ๐,๐โ (LASS-(๐+ 1)) = colim๐ ๐ธ
[๐,๐],[๐,๐]โ (MASS-(๐+ 1)).
where the maps in the system are multiplication by ๐๐๐
1 .
The proof of convergence is now the same as for the MASS-โ. We define
๐น ๐๐๐ข(๐ฃโ11 ๐/๐๐+1) = im
(colim๐ ๐น
[๐,๐]๐๐ข+๐๐๐๐(๐/๐๐+1) โโ ๐๐ข(๐ฃ
โ11 ๐/๐๐+1)
),
where the ๐น on the right hand side of the equation denotes the MASS-(๐+1) filtration.
We verify convergence in accordance with definition 2.2.2, case 3.
Proposition B.6. The LASS-โ converges to ๐*(๐ฃโ11 ๐/๐โ).
Proof. The proof is exactly the same as for the MASS-โ. We define
๐น ๐๐*(๐ฃโ11 ๐/๐โ) = im
(colim๐ ๐น
๐๐*(๐ฃโ11 ๐/๐๐) โโ ๐*(๐ฃ
โ11 ๐/๐โ)
),
where the ๐น on the right hand side of the equation denotes a reindexed localized
Adams filtration; we use the convergence and vanishing lines of the reindexed localized
Adams spectral sequences for ๐ฃโ11 ๐/๐๐ instead of the convergence and vanishing lines
for the RASS-๐. The only subtlety is taking the colimit of the short exact sequence
analogous to (B.4). This is intertwined with the issue of defining of ๐ฃโ11 ๐/๐โ.
Corollary 3.8 of [8] tells us that there exists integers ๐1, ๐2, ๐3, . . . such that the
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following diagrams commute.
๐/๐๐
[๐ฃ๐
๐โ1
1
]๐๐๐//
๐
ฮฃโ๐๐๐๐๐๐/๐๐
๐
๐/๐๐+1
[๐ฃ๐
๐
1
]๐๐// ฮฃโ๐๐๐๐๐๐/๐๐+1
This means that ๐ : ๐/๐๐ โโ ๐/๐๐+1 induces a map ๐ : ๐ฃโ11 ๐/๐๐ โโ ๐ฃโ1
1 ๐/๐๐+1.
๐ฃโ11 ๐/๐โ is the homotopy colimit of the diagram
๐ฃโ11 ๐/๐
๐ // ๐ฃโ11 ๐/๐2 // . . . // ๐ฃโ1
1 ๐/๐๐๐ // ๐ฃโ1
1 ๐/๐๐+1 // . . .
Moreover, the diagram below commutes, where the filtrations are those of the RASS-๐
and RASS-(๐+ 1).
๐น ๐๐*(๐/๐๐)
[๐ฃ๐
๐โ1
1
]๐๐๐//
๐
๐น ๐+๐๐๐๐๐*(๐/๐๐)
๐
๐น ๐๐*(๐/๐
๐+1)
[๐ฃ๐
๐
1
]๐๐// ๐น ๐+๐๐๐๐๐*(๐/๐
๐+1)
Thus, ๐ : ๐ฃโ11 ๐/๐๐ โโ ๐ฃโ1
1 ๐/๐๐+1 respects the reindexed localized Adams filtrations.
The proof convergence of the loc.alg.NSS follows the same chain of ideas as for
the LASS-โ.
Lemma B.7. The loc.alg.NSS converges to ๐ป*(๐ต๐*๐ต๐ ; ๐ฃโ11 ๐ต๐*/๐
โ).
Proof. This is essentially the same proof as for the LASS-โ. The following algebraic
Novikov spectral sequence converges in accordance with definition 2.2.2, case 1.
๐ป๐ ,๐ข(๐ ; [๐/๐๐0 ]๐ก)๐ก
=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ ;๐ต๐*/๐๐)
This is because the ๐ผ-adic filtration of ฮฉ*(๐ต๐*๐ต๐ ;๐ต๐*/๐๐) is finite in a fixed internal
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degree: ๐น โ๐ข/๐โ+๐ฮฉ*,๐ข(๐ต๐*๐ต๐ ;๐ต๐*/๐๐) = 0. Because we have a vanishing line parallel
to the multiplication-by-๐๐๐โ1
1 -line, we deduce that each
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐๐0 ]๐ก)
๐ก=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐๐)
converges in accordance with definition 2.2.2, case 3.
After the reindexing that occurs in constructing
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐โ0 ]๐ก)
๐ก=โ ๐ป๐ ,๐ข(๐ต๐*๐ต๐ ; ๐ฃโ1
1 ๐ต๐*/๐โ)
the vanishing lines for these spectral sequences become independent of ๐ and so we
conclude convergence in accordance with definition 2.2.2, case 3.
We now go back to the proof of the first proposition.
Proof of proposition B.1. We are in case 1 of definition 2.2.2. We need to check the
following conditions.
โ The map ๐น ๐๐*(๐/๐๐)/๐น ๐+1๐*(๐/๐
๐) โโ ๐ธ๐,*โ (MASS-๐) appearing in definition
8.1.15 is an isomorphism.
โโ๐ ๐น
๐๐*(๐/๐๐) = 0 and the map ๐*(๐/๐๐) โโ lim๐ ๐*(๐/๐
๐)/๐น ๐๐*(๐/๐๐) is an
isomorphism.
We will appeal to [15, theorem 3.6] but first we need to relate our construction of the
MASS-๐ to the one given there.
Suppose ๐, ๐ผ and ๐, ๐ฝ are towers and that we have chosen cofibrant models
for them. Write ๐น (โ,โ) for the internal hom functor in ๐-modules [7] and ๐ for a
cofibrant replacement functor. Then we obtain a zig-zag
colim๐+๐โฅ๐0โค๐,๐โค๐
๐๐ โง ๐๐โผโโ hocolim๐+๐โฅ๐
0โค๐,๐โค๐๐๐ โง ๐๐ โโ hocolim๐+๐โฅ๐
0โค๐,๐โค๐๐น (๐๐น (๐๐, ๐
0), ๐๐).
As long as each ๐๐ is finite, this will be an equivalence. Taking ๐, ๐ผ to be ๐ปโง*, ๐ป [*]
and ๐, ๐ฝ to be ๐/๐๐โ*, ๐/๐, we that our MASS-๐ is the same as the one in [15,
135
definition 2.2] when one uses the dual of ๐/๐๐โ*, ๐/๐ in the source.
๐๐ is zero on ๐/๐๐ and so [15, proposition 1.2(a)] tells us ๐/๐๐ is ๐-adically cocom-
plete. Moreover, each ๐/๐๐ is finite and connective. Thus, [15, theorem 3.6] applies
(after suspending once): the first bullet point holds andโ๐ ๐น
๐๐*(๐/๐๐) = 0. More-
over, the vanishing line of corollary 4.1.9 gives a vanishing line for the MASS-๐, and
so we see that for each ๐ข, there exists a ๐ with ๐น ๐๐๐ข(๐/๐๐) = 0. We conclude that
the map ๐*(๐/๐๐) โโ lim๐ ๐*(๐/๐๐)/๐น ๐๐*(๐/๐
๐) is an isomorphism, as required.
Proof of corollary B.2. We are in case 3 of definition 2.2.2 by the previous argument.
It is far easier to show that the other spectral sequences we use converge.
Lemma B.8. The ๐-BSS, the ๐โ0 -BSS and the ๐โ11 -BSS converge.
Proof. The relevant filtrations are given in 3.2.1, 3.3.1 and 3.5.2, as are the identifi-
cations ๐ธ๐ฃโ = ๐น ๐ฃ/๐น ๐ฃ+1. For the ๐-BSS we are in case 1 of definition 2.2.2:
๐น 0๐ป*(๐ ;๐) = ๐ป*(๐ ;๐) and ๐น ๐ก+1๐ป๐ ,๐ข(๐ ;๐๐ก) = 0
and so the requisite conditions hold. For the ๐โ0 -BSS and the ๐โ11 -BSS we are in case
2 of definition 2.2.2.
Corollary B.9. The ๐0-FILT and ๐0-FILT2 spectral sequence converge to the ๐ธ3-
pages of the loc.alg.NSS and the LASS-โ, respectively.
Proof. Since the ๐โ11 -BSS converges in accordance with definition 2.2.2, case 2, one
finds that this means the ๐0-FILT and ๐0-FILT2 spectral sequences do, too.
For our final convergence proof we need the following lemma.
Lemma B.10. Fix (๐, ๐). There are finitely many (๐ , ๐ก, ๐ข) with ๐ + ๐ก = ๐, ๐ข+ ๐ก = ๐
and ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก) nonzero.
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Proof. Multiplication by ๐1 defines an isomorphism
โจ๐ +๐ก=๐๐ข+๐ก=๐
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก) โโโจ
๐ +๐ก=๐+1๐ข+๐ก=๐+๐+1
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก).
Thus, it is equivalent to ask the question for (๐+ ๐, ๐+ ๐(๐+ 1)). By corollary 4.3.2
we can choose ๐ so that
โจ๐ +๐ก=๐+๐
๐ข+๐ก=๐+๐(๐+1)
๐ป๐ ,๐ข(๐ ; [๐/๐0]๐ก) โโ
โจ๐ +๐ก=๐+๐
๐ข+๐ก=๐+๐(๐+1)
๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐ก)
is an isomorphism. Nonzero elements in the left hand side have ๐ , ๐ก โฅ 0. There are
finitely many (๐ , ๐ก, ๐ข) with ๐ + ๐ก = ๐ + ๐ and ๐ , ๐ก โฅ 0 and so the result follows.
This shows that the spectral sequence argument alluded to in the proof of [11,
theorem 4.8] is valid. It also allows us to prove the following lemma.
Proposition B.11. The ๐ -filtration spectral sequence of section 8.6 converges to
๐ธ1(๐0-FILT2).
Proof. To show that the spectral sequence converges in accordance with definition
2.2.2, case 1, we just need to show that for each (๐, ๐, ๐ฃ), there are finitely many
(๐ , ๐ก, ๐ข) with ๐ + ๐ก = ๐, ๐ข+ ๐ก = ๐ and ๐ธ๐ ,๐ก,๐ข,๐ฃโ (๐โ1
1 -BSS) nonzero. This follows from the
fact that for each (๐, ๐, ๐ฃ), there are finitely many (๐ , ๐ก, ๐ข) with ๐ + ๐ก = ๐, ๐ข + ๐ก = ๐
and ๐ป๐ ,๐ข(๐ ; [๐โ11 ๐/๐0]
๐กโ๐ฃ) nonzero.
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