Central Cohomology Operations and K-theory · maps of E, associative but in general not...

OutlineCohomology operations

BP〈n〉 theoriesMain Theorem

Outline of proofOpen questions

Central Cohomology Operations and K -theoryWork in Progress

Imma Galvez-Carrillo (UPC)and

Sarah Whitehouse(Sheffield)

Higher Homotopy in Barcelona 2012IMUB, Universitat de Barcelona

March 24th 2012

Imma Galvez-Carrillo (UPC) and Sarah Whitehouse(Sheffield) Central Cohomology Operations and K -theory




Introduction

Cohomology operations

BP〈n〉 theories

Main Theorem

Outline of proof

Open questions





What is known about centres of cohomology operationsrings

I In [GW05] we proved that for the spectraE = MU(p),BP,KU(p),G

Z (E 0(E )) = A(E )

I In [SW10a, SW10b] similar results are proven for unstableoperations.

I Ever since, we have been trying to fill the gap between thethese two families of theories at both ends of the chromaticspectrum.

I We have now done it for unstable operations for E = BP〈n〉for all chromatic levels n.

I However, there are still many open questions...







Z (E 0(E )) = A(E )











Z (E 0(E )) = A(E )











Z (E 0(E )) = A(E )











Z (E 0(E )) = A(E )









Rings of stable operations in generalized cohomologytheories

I Stable operations are self maps of spectra up to homotopy.For E a ring spectrum, the ring of stable operations inE -theory is given by

E ∗(E ) = [E ,E ]∗

A representative of ϕ a degree operation k will be a map ofspectra, given levelwise by fm : Em → Em+k commuting withsuspensions and such that ϕ = [f ].

I We will be interested in degree 0 operations.

E 0(E ) = [E ,E ]0





Rings of stable operations in generalized cohomologytheories

I Stable operations are self maps of spectra up to homotopy.For E a ring spectrum, the ring of stable operations inE -theory is given by

E ∗(E ) = [E ,E ]∗

A representative of ϕ a degree operation k will be a map ofspectra, given levelwise by fm : Em → Em+k commuting withsuspensions and such that ϕ = [f ].

I We will be interested in degree 0 operations.

E 0(E ) = [E ,E ]0





Rings of stable operations in generalized cohomologytheories, II

I Operations can be added as they are elements in the groupE ∗(E ).

I They also have a product coming from composition of selfmaps of E , associative but in general not commutative.

I The identity map provides an unit. So we get a graded ring.

I This ring acts on the cohomology of any space or spectrum.

I Classical examples: Steenrod algebra, Landweber-Novikovalgebra, K -theory operations...













































Theories with good duality, stable case

I There is also a ring of cooperations, namely the E -homologyof the spectrum E itself.

E∗(E ) = π∗(E ∧ E )

I If E∗(E ) is flat over the coefficient ring E∗, one can get a Hopfalgebroid (or bilateral Hopf algebra) structure defined on it.





Theories with good duality, stable case

I There is also a ring of cooperations, namely the E -homologyof the spectrum E itself.

E∗(E ) = π∗(E ∧ E )

I If E∗(E ) is flat over the coefficient ring E∗, one can get a Hopfalgebroid (or bilateral Hopf algebra) structure defined on it.





Theories with good duality,II

I Any theory E with E∗(E ) free over E∗. Any theory E withE∗(E ) flat over E∗ [MR77].

I The Steenrod algebra and its dual. Case E = HFp, thenHFp∗(HFp) is the classical Milnor’s dual of the Steenroodalgebra.

I The Landweber-Novikov algebra MU∗(MU) and its dualMU∗(MU). Its localised versions at p. Its versions for the splitsummand, BP.

I p-local K -theory and its split summands.
































Theories with good duality,III

I Genera. Complex orientations.

I Landweber exact theories. Conner-Floyd maps.

I What we get in these cases:

E∗(E ) ∼= E∗ ⊗MU∗ MU∗(MU)⊗MU∗ E∗

with the flatness of E∗(E ) over E∗ being then a consequenceof the flatness of MU∗(MU) over MU∗.

I When there is good duality,

E ∗(E ) ∼= homE∗(E∗(E ),E∗)

I That means that in these good cases, operations aredetermined by the action they induce on coefficients.









E∗(E ) ∼= E∗ ⊗MU∗ MU∗(MU)⊗MU∗ E∗



E ∗(E ) ∼= homE∗(E∗(E ),E∗)










E∗(E ) ∼= E∗ ⊗MU∗ MU∗(MU)⊗MU∗ E∗



E ∗(E ) ∼= homE∗(E∗(E ),E∗)










E∗(E ) ∼= E∗ ⊗MU∗ MU∗(MU)⊗MU∗ E∗



E ∗(E ) ∼= homE∗(E∗(E ),E∗)










E∗(E ) ∼= E∗ ⊗MU∗ MU∗(MU)⊗MU∗ E∗



E ∗(E ) ∼= homE∗(E∗(E ),E∗)






Generalized unstable cohomology operations

I E cohomology theory, E k the k-th space in a Ω-spectrumrepresenting E .

I The unstable bidegree (0, 0) operations of E-theory are

E 0(E 0) ∼= [E 0,E 0]

I Not all operations are now additive. Non additive operationsare a very different business. We concentrate in additiveoperations, known as PE 0(E 0), which we will name A(E ).

I Algebraic structure: products. Hopf ring structure.








E 0(E 0) ∼= [E 0,E 0]










E 0(E 0) ∼= [E 0,E 0]










E 0(E 0) ∼= [E 0,E 0]







Generalized unstable cohomology operations, II

I Sending an operation to its action on coefficients gives

E 0(E 0)→ End(π∗(E 0))

φ 7→ φ∗.

I The restriction of this map to the additive E -operations A(E )is a ring homomorphism βE :

βE : A(E )→ End(π∗(E 0))

φ 7→ φ∗.





Theories with good duality, unstable case

I The theories considered by [SW10a] all have good duality forthe unstable case. Again, operations are dual to cooperationsunder the isomorphism of E∗-modules

E ∗(E 0) ∼= homE∗(E∗(E 0),E∗)

On the left-hand side, there is the profinite topology, on theright the dual-finite topology.

I This isomorphism restricted to the additive operations A(E )gives

A(E ) = PE ∗(E0) ∼= homE∗(QE∗(E 0),E∗)

where QE∗(E 0) is the quotient of the indecomposable quotientof the cooperations for the ?-product in the Hopf ring E∗(E 0).





Theories with good duality, unstable case

I The theories considered by [SW10a] all have good duality forthe unstable case. Again, operations are dual to cooperationsunder the isomorphism of E∗-modules

E ∗(E 0) ∼= homE∗(E∗(E 0),E∗)

On the left-hand side, there is the profinite topology, on theright the dual-finite topology.

I This isomorphism restricted to the additive operations A(E )gives

A(E ) = PE ∗(E0) ∼= homE∗(QE∗(E 0),E∗)

where QE∗(E 0) is the quotient of the indecomposable quotientof the cooperations for the ?-product in the Hopf ring E∗(E 0).





Stable Adams operations in some complex oriented theories

Proposition

Let E = MU, BP, KU or G . A stable multiplicative cohomologyoperation θ : E → E is uniquely determined by its value θ(xE ) onthe orientation class xE ∈ E 2(CP∞).

To have stable Adams operations, we need to work p-locally, for aprime p.





Stable Adams operations in some complex orientedtheories, II

DefinitionLet E = MU(p), BP, KU(p) or G . A stable Adams operation Ψα

E is

defined for each α ∈ Z×(p) as the unique multiplicative operationgiven on the orientation class xE by

ΨαE (xE ) =

[α]E (xE )

α∈ E ∗(CP∞) = E∗[[xE ]] ,

where [α]E denotes the formal sum.

LemmaΨα

E acts as multiplication by αn on E2n.





Stable Adams operations in some complex orientedtheories, II

DefinitionLet E = MU(p), BP, KU(p) or G . A stable Adams operation Ψα

E is

defined for each α ∈ Z×(p) as the unique multiplicative operationgiven on the orientation class xE by

ΨαE (xE ) =

[α]E (xE )

α∈ E ∗(CP∞) = E∗[[xE ]] ,

where [α]E denotes the formal sum.

LemmaΨα

E acts as multiplication by αn on E2n.





The spectra BP〈n〉( [Wil75], [JW73])

I For each n ≥ 0, there is a connective commutative ringspectrum BP〈n〉 with coefficient groups

BP〈n〉∗ = Z(p)[v1, v2, . . . , vn] = BP∗/(vn+1, vn+2, . . . ) = BP∗/Jn

BP∗ = Z(p)[v1, v2, . . . ], vi the Hazewinkel generators,Jn = (vn+1, vn+2, . . . ).

I They form a tower of BP-module spectra:

BP // . . . // BP〈n〉 // BP〈n − 1〉 // . . .

. . . // BP〈2〉 // . . . // BP〈1〉 // BP〈0〉





The spectra BP〈n〉( [Wil75], [JW73])

I For each n ≥ 0, there is a connective commutative ringspectrum BP〈n〉 with coefficient groups

BP〈n〉∗ = Z(p)[v1, v2, . . . , vn] = BP∗/(vn+1, vn+2, . . . ) = BP∗/Jn

BP∗ = Z(p)[v1, v2, . . . ], vi the Hazewinkel generators,Jn = (vn+1, vn+2, . . . ).

I They form a tower of BP-module spectra:

BP // . . . // BP〈n〉 // BP〈n − 1〉 // . . .

. . . // BP〈2〉 // . . . // BP〈1〉 // BP〈0〉





The Wilson splitting of BP〈n〉 from BP

I In [Wil75], Wilson constructs unstable splittings of the form

BPk∼= BP〈n〉k ×

∏j>n

BP〈j〉k+2(pj−1)

for k ≤ 2(pn + · · ·+ p + 1) and for k < 2(pn + · · ·+ p + 1)this decomposition is as irreducibles.





The Wilson splitting of BP〈n〉 from BP : maps from it

I Let n ≥ 0. The map above is one of BP-module spectra(indeed map of ring spectra):

πn : BP → BP〈n〉

induces one between the 0-th spaces of the spectra

πn : BP0 → BP〈n〉0

and a ring map on homotopy groups which is the canonicalprojection

(πn)∗ : BP∗ → BP〈n〉∗











(πn)∗ : BP∗ → BP〈n〉∗











(πn)∗ : BP∗ → BP〈n〉∗





The Wilson splitting of BP〈n〉 from BP : maps from it, II

I In the other direction, the map is not so good.It is only a H-spaces map, not giving a map at the level ofspectra.It does not induces the obvious map on coefficients.

I For all n ≥ 0, there is an H-space splittingθn : BP〈n〉

0→ BP0 of πn.

I Then en = θnπn is the corresponding additive idempotentBP-operation.

I Choices can be made compatibly so that enem = emen = emfor m < n








0→ BP0 of πn.










0→ BP0 of πn.










0→ BP0 of πn.







The Wilson splitting of BP〈n〉 from BP : maps from it, III

LemmaWe have maps

in : A(BP〈n〉) A(BP) : pn

such that

1. in pn : A(BP)→ A(BP) is given by [f ] 7→ [en f en];

2. pn splits in (so in is injective and pn is surjective);

3. in is a ring homomorphism;

4. pn is an additive group homomorphism.





The Wilson splitting of BP〈n〉 from BP : maps from it, IV

That is,

[θn − πn] : BP〈n〉0(BP〈n〉0)→ BP0(BP0)

[f ] 7→ [θn f πn]

[πn − θn] : BP0(BP0)→ BP〈n〉0(BP〈n〉0)

[f ] 7→ [πn f θn].

πn and θn are H-space maps, so they restrict to the additiveoperations, where we call them in and pn.(1) follows from θnπn = en.(2),(3),(4) from πnθn ' id and from θn being a map of H-spaces.






That is,

[θn − πn] : BP〈n〉0(BP〈n〉0)→ BP0(BP0)

[f ] 7→ [θn f πn]

[πn − θn] : BP0(BP0)→ BP〈n〉0(BP〈n〉0)

[f ] 7→ [πn f θn].

πn and θn are H-space maps, so they restrict to the additiveoperations, where we call them in and pn.

(1) follows from θnπn = en.(2),(3),(4) from πnθn ' id and from θn being a map of H-spaces.






That is,

[θn − πn] : BP〈n〉0(BP〈n〉0)→ BP0(BP0)

[f ] 7→ [θn f πn]

[πn − θn] : BP0(BP0)→ BP〈n〉0(BP〈n〉0)

[f ] 7→ [πn f θn].

πn and θn are H-space maps, so they restrict to the additiveoperations, where we call them in and pn.(1) follows from θnπn = en.(2),(3),(4) from πnθn ' id and from θn being a map of H-spaces.





The Wilson splitting of BP〈n〉 from BP : maps from it, V

RemarkHence we may identify as subrings of A(BP)

A(BP〈n〉) ∼= enA(BP)en





Theorem: the centre of unstable additive (0, 0)-degreeoperations in BP〈n〉 comes from K -theory

I Using the unstable BP splittings θn, we define an injectivering homomorphism

ιn : A(g)→ A(BP〈n〉)

I This ι is different from the one abovei1 : A(BP〈1〉) = A(g)→ A(BP).

I Our main result: ιn(A(g)) = Z (A(BP〈n〉))I This builds on the previous result:

Theorem ( [SW10a])

There is an injective ring homomorphism ι : A(g)→ A(BP) suchthat the image is precisely the centre of the ring A(BP).










Theorem ( [SW10a])










I Our main result: ιn(A(g)) = Z (A(BP〈n〉))

I This builds on the previous result:

Theorem ( [SW10a])











Theorem ( [SW10a])






Outline of the proof: faithfulness

Proposition

For all n ≥ 0, the ring homomorphism

βBP〈n〉 : A(BP〈n〉)→ End(π∗(BP〈n〉0))

is injective.

Proof.For φ ∈ A(BP〈n〉) such that βBP〈n〉(φ) = φ∗ = 0,

we haveβBP(in(φ)) = (in(φ))∗ = (θnφπn)∗ = (θn)∗φ∗(πn)∗ = 0.But from[SW10a] we know that βBP is injective. in is injective too. Soφ = 0.






Proposition



is injective.

Proof.For φ ∈ A(BP〈n〉) such that βBP〈n〉(φ) = φ∗ = 0, we haveβBP(in(φ)) = (in(φ))∗ = (θnφπn)∗ = (θn)∗φ∗(πn)∗ = 0.

But from[SW10a] we know that βBP is injective. in is injective too. Soφ = 0.






Proposition



is injective.

Proof.For φ ∈ A(BP〈n〉) such that βBP〈n〉(φ) = φ∗ = 0, we haveβBP(in(φ)) = (in(φ))∗ = (θnφπn)∗ = (θn)∗φ∗(πn)∗ = 0.But from[SW10a] we know that βBP is injective.

in is injective too. Soφ = 0.






Proposition



is injective.

Proof.For φ ∈ A(BP〈n〉) such that βBP〈n〉(φ) = φ∗ = 0, we haveβBP(in(φ)) = (in(φ))∗ = (θnφπn)∗ = (θn)∗φ∗(πn)∗ = 0.But from[SW10a] we know that βBP is injective. in is injective too.

Soφ = 0.






Proposition



is injective.

Proof.For φ ∈ A(BP〈n〉) such that βBP〈n〉(φ) = φ∗ = 0, we haveβBP(in(φ)) = (in(φ))∗ = (θnφπn)∗ = (θn)∗φ∗(πn)∗ = 0.But from[SW10a] we know that βBP is injective. in is injective too. Soφ = 0.





Outline of the proof: Unstable Adams operations forBP〈n〉

DefinitionThe unstable Adams operations for BP〈n〉 are the images of thecorresponding BP operations under the map pn:

ΨkBP〈n〉 := pn(Ψk

BP)

for k ∈ Z(p).

This gives unstable Adams operations for BP〈n〉 such thatΨk

BP〈n〉(z) = k(p−1)rz , for z ∈ BP〈n〉2(p−1)r .





Outline of the proof: Unstable Adams operations forBP〈n〉

DefinitionThe unstable Adams operations for BP〈n〉 are the images of thecorresponding BP operations under the map pn:

ΨkBP〈n〉 := pn(Ψk

BP)

for k ∈ Z(p).

This gives unstable Adams operations for BP〈n〉 such thatΨk

BP〈n〉(z) = k(p−1)rz , for z ∈ BP〈n〉2(p−1)r .





Outline of the proof: the comparison map ι is a ring map.

I We know from [SW10a] that

ι(Ψkg ) = Ψk

BP

I From ΨkBP〈n〉 = pn(Ψk

BP)

and the description of A(g) in terms

of Adams operations, one gets a map ιn determined byI mapping the g Adams operations to the corresponding BP〈n〉

Adams operationsI extending to suitable infinite linear combinations.

I Our main result will be that the analogue of the result for BPholds for

ιn : A(g)→ A(BP〈n〉)I ιn is a ring homomorphism (even though pn is not).







ι(Ψkg ) = Ψk

BP


BP) and the description of A(g) in terms











ι(Ψkg ) = Ψk

BP







I ιn is a ring homomorphism (even though pn is not).







ι(Ψkg ) = Ψk

BP











Outline of the proof: the comparison map ι is a ringmap,II.

Proposition

For all n ≥ 1, the map ιn : A(g)→ A(BP〈n〉) is an injective unitalring homomorphism whose image is contained in the centre ofA(BP〈n〉).





Outline of the proof: compatibility of maps.

We have this commutative diagram of abelian groups, for m ≤ n.

A(BP〈n〉)∼=in// enA(BP)en

em−em

A(g)

ιm99

ι //

ιn %%

A(BP)

pn

OO

pm

A(BP〈m〉)∼=im// emA(BP)em





Outline of the proof: Diagonal operations

Unstable diagonal operations for BP were defined in [SW10a]. Thesame can be done for BP〈n〉.

DefinitionLet D(BP〈n〉) be the subring of A(BP〈n〉) consisting of operationswhose action on each π2(p−1)r (BP〈n〉

0) is multiplication by some

µr of Z(p).

We call elements of D(BP〈n〉) unstable diagonaloperations.






Unstable diagonal operations for BP were defined in [SW10a]. Thesame can be done for BP〈n〉.

DefinitionLet D(BP〈n〉) be the subring of A(BP〈n〉) consisting of operationswhose action on each π2(p−1)r (BP〈n〉

0) is multiplication by some

µr of Z(p). We call elements of D(BP〈n〉) unstable diagonaloperations.






LemmaD(BP〈n〉) ⊆ Z (A(BP〈n〉)).

The key point is to prove

Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)

Our proof of this amounts to finding enough operations to force acentral operation to act diagonally.We construct stable BP operations that act as we want,we then view these as additive unstable BP operationsand then we project them to A(BP〈n〉).








Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)









Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)

Our proof of this amounts to finding enough operations to force acentral operation to act diagonally.

We construct stable BP operations that act as we want,we then view these as additive unstable BP operationsand then we project them to A(BP〈n〉).








Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)

Our proof of this amounts to finding enough operations to force acentral operation to act diagonally.We construct stable BP operations that act as we want,

we then view these as additive unstable BP operationsand then we project them to A(BP〈n〉).








Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)

Our proof of this amounts to finding enough operations to force acentral operation to act diagonally.We construct stable BP operations that act as we want,we then view these as additive unstable BP operations

and then we project them to A(BP〈n〉).








Proposition

Z (A(BP〈n〉)) ⊆ D(BP〈n〉)






Outline of the proof: Congruences

Sg= subring of∏∞

i=0 Z(p) of sequences (µi )i≥0 satisfying thesystem of congruences which characterizes the action on coefficientgroups of an element of A(g).





Outline of the proof: Congruences,II

Description of the congruences [SW10a, Section 4]:

I G the periodic Adams summand, G∗ = Z(p)[u±1].

I Q the indecomposable quotient for the ?-product, fn | n ≥ 0a Z(p)-basis for QG0(G 0).

I fn’s can be written as rational polynomials in the variablew = u−1v .

I The n-th congruence is the condition that the rational linearcombination of the µi obtained from fn by sending w i to µilies in Z(p).

I Different choices of basis lead to equivalent systems ofcongruences with the same solution set Sg .

















































Outline of the proof: Congruences, III

We have strenghtened the congruences result in of [SW10a] sothat it goes down to BP〈n〉.

Proposition

Fix n ≥ 1. Suppose that an operation ϕ ∈ A(BP) is such that itsaction on homotopy ϕ∗ : BP∗ → BP∗ satisfies the followingconditions: for each i ≥ 0, there is some µi ∈ Z(p) such that

1. ϕ∗(x) ≡ µix mod Jn if x /∈ Jn, |x | = 2(p − 1)i .

2. ϕ∗(x) = 0 if x ∈ Jn.

Then (µi )i≥0 ∈ Sg .






Proposition

Let n ≥ 1 and φ ∈ D(BP〈n〉). Then in(φ) ∈ A(BP) satisfies thehypotheses of the proposition above.





End of the proof

TheoremFor all n ≥ 1, the image of the injective ring homomorphismιn : A(g) → A(BP〈n〉) is the centre Z (A(BP〈n〉)) of A(BP〈n〉).

A(g)∼=ιn//

∼=

Im(ιn) // Z (A(BP〈n〉))

= // D(BP〈n〉) _

Sg

= // Sg





Open questions

I For which theories is this true?

Our guess is that, both in the stable and unstable case, thecenter of the ring of stable operations of degree 0 is the imageof the one for K -theory.Similarly for the corresponding unstable version.But have not succeeded in proving it by a general argumentfor a wide enough class of theories.

I What is known for n=2 theories? We have some partialresults for elliptic cohomology using Adams and Heckeoperators, but we run into other problems.

I Relation to other BP-related work.





Open questions

I For which theories is this true?Our guess is that, both in the stable and unstable case, thecenter of the ring of stable operations of degree 0 is the imageof the one for K -theory.

Similarly for the corresponding unstable version.But have not succeeded in proving it by a general argumentfor a wide enough class of theories.







Open questions

I For which theories is this true?Our guess is that, both in the stable and unstable case, thecenter of the ring of stable operations of degree 0 is the imageof the one for K -theory.Similarly for the corresponding unstable version.

But have not succeeded in proving it by a general argumentfor a wide enough class of theories.







Open questions

I For which theories is this true?Our guess is that, both in the stable and unstable case, thecenter of the ring of stable operations of degree 0 is the imageof the one for K -theory.Similarly for the corresponding unstable version.But have not succeeded in proving it by a general argumentfor a wide enough class of theories.







More open questions

I What are Adams operations, really? The original Adamsdefinitions for unstable ones in periodic K -theory, wasfollowed by definitions for some stable ones in diverse contextsby Novikov and others. These ones got unstable versions aswell. Despite some general arguments, construccions are verymuch ad hoc. One would like to have a general enoughdefinition for them.





Imma Galvez and Sarah Whitehouse, Infinite sums of Adamsoperations and cobordism, Math. Z. 251 (2005), no. 3,475–489. MR 2190339 (2007k:55031)

David Copeland Johnson and W. Stephen Wilson, Projectivedimension and Brown-Peterson homology, Topology 12(1973), 327–353. MR 0334257 (48 #12576)

Haynes R. Miller and Douglas C. Ravenel, Morava stabilizeralgebras and the localization of Novikov’s E2-term, DukeMath. J. 44 (1977), no. 2, 433–447. MR 0458410 (56#16613)

M.-J. Strong and Sarah Whitehouse, Infinite sums of unstableAdams operations and cobordism, J. Pure Appl. Algebra 214(2010), no. 6, 910–918. MR 2580668 (2011i:55024)





, Integer-valued polynomials and K -theory operations,Proc. Amer. Math. Soc. 138 (2010), no. 6, 2221–2233. MR2596063 (2011e:55022)

W. Stephen Wilson, The Ω-spectrum for Brown-Petersoncohomology. II, Amer. J. Math. 97 (1975), 101–123. MR0383390 (52 #4271)


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