Quillen Cohomology of Operadic Algebras and Obstruction Theory · And assume that the forgetful...

Quillen Cohomology of Operadic Algebras andObstruction Theory

Michael A. Mandell

Indiana University

Banff Workshop on Functor Calculus and OperadsMarch 16, 2011

M.A.Mandell (IU) Obstruction Theory Mar 16 1 / 24

Introduction

Outline

1 Introduction to Quillen homology and cohomologyof operadic algebras

2 Structure of Quillen homology3 Postnikov towers and obstructions4 Application: BP


Introduction

Outline




Introduction

Outline




Introduction

Outline




Introduction

Quillen Homology and Cohomology

General ContextA closed model category M

Assume that subcategory of abelian objects A also forms amodel categoryWith fibrations and weak equivalences as in M

And assume that the forgetful functor has a left adjoint Ab(“abelianization”)

DefinitionQuillen homology is the left derived functor of abelianization. Quillencohomology with coefficients in N ∈ A is [−,N].


Introduction







Introduction







Introduction







Introduction







Introduction

Abelian Objects in Operadic Algebras

Let O be an operad in chain cxs of abelian groups or modules.Assume O(0) = 0.

QuestionWhat is an abelian object in the category of O-algebras?

Must have structure map O(m)⊗ N⊗m → N be zero for m > 1.Just has the structure of an R-module for R = O(1).Equivalently, the structure of an algebra over IR.

AnswerAn IR-algebra = R-module.


Introduction







Introduction







Introduction







Introduction







Introduction







Introduction

Abelianization = Indecomposables

Forgetful functor from abelian O-algebras to O-algebras is “restrictionof scalars” from IR-algebras to O-algebras along O → IR.

Left adjoint is “extension of scalars”

IR ◦O (−)

which is indecomposables:

IR ◦ O ◦ A //// IR ◦ A // QA

R ⊗(⊕O(n)⊗Σn A⊗n) //

// R ⊗ A // QA


Introduction




IR ◦O (−)


IR ◦ O ◦ A //// IR ◦ A // QA

R ⊗(⊕O(n)⊗Σn A⊗n) //

// R ⊗ A // QA


Introduction




IR ◦O (−)


IR ◦ O ◦ A //// IR ◦ A // QA

R ⊗(⊕O(n)⊗Σn A⊗n) //

// R ⊗ A // QA


Introduction




IR ◦O (−)


IR ◦ O ◦ A //// IR ◦ A // QA

R ⊗(⊕O(n)⊗Σn A⊗n) //

// R ⊗ A // QA


Introduction




IR ◦O (−)


IR ◦ O ◦ A //// IR ◦ A // QA

R ⊗(⊕O(n)⊗Σn A⊗n) //

// R ⊗ A // QA


Structure of Quillen Homology

Quillen Homology = Koszul Dual Coalgebra

Quillen homology = Left derived functor of IR ◦O (−)

We know how to do this left derived functor much more generally thanwe know that O-algebras are a closed model category.

Choose a flat right O-module approximation E of IR and look atE ◦O (−).

If R is commutative and O nice, can take E = (DO ◦ O,d).

Then E ◦O A = BOA is the “bar dual” or “Koszul dual” DO-coalgebra ofA.



























Homotopical Origin of Coalgebra Structure

Now don’t assume that R is commutative but do assume thatO-algebras form a closed model category.

Then we have a Quillen adjunction

Q : O-Alg // IR-Alg :Zoo

and a derived adjunction

QL : Ho(O-Alg)// Ho(IR-Alg) :Z Roo

QLA is a coalgebra over the comonad QLZ R.

Goodwillie Calculus: QLZ R(X ) =⊕

(D(n)⊗ X⊗n)hΣn











(D(n)⊗ X⊗n)hΣn











(D(n)⊗ X⊗n)hΣn











(D(n)⊗ X⊗n)hΣn











(D(n)⊗ X⊗n)hΣn



Koszul Duality

(R is commutative, probably a field.)(O is Σ∗ projective.)

If A is connected, you can recover A from DO structure on BOA(= QLA):

COBOA '−→ A

[Getzler-Jones], [Fresse]


Postnikov Towers and Obstructions

Eckmann-Hilton Duality

A1 // A2 // A3 // · · · // A

KA1 KA2oo KA3oo · · ·oo KAoo




A1 A2oo A3oo · · ·oo Aoo

KA1 // KA2 // KA3 // · · · // KA

=⇒ Postnikov Tower




A1 A2oo A3oo · · ·oo Aoo

KA1 // KA2 // KA3 // · · · // KA

=⇒ Postnikov Tower



Postnikov Towers for Operadic Algebras

Now possibly working in spectra with O an operad of spaces.No longer assume O(0) = ∗. Might want unit.

For any O-algebra A, tower of O-modules

A→ · · · → An+1 → An → · · · → A0

withπiA→ πiAn iso for i ≤ nπiAn = 0 for i > n

ProblemBuild as a tower of principal fibrations of O-modules.






A→ · · · → An+1 → An → · · · → A0








A→ · · · → An+1 → An → · · · → A0








A→ · · · → An+1 → An → · · · → A0








A→ · · · → An+1 → An → · · · → A0





Principal Fibrations

A→ · · · → An+1 → An → · · · → A0

(For simplicity, assume O augments to comm. operad.)

Work in the category of O-algebras lying over A0 = Hπ0A.

For an A0-module M, have the square-zero O-algebraA0 n M = A0 ∨M.

We will construct the Postnikov tower with

An+1 //

��

A0

��

An // A0 n Σn+2Hπn+1A

homotopy fiber squares.M.A.Mandell (IU) Obstruction Theory Mar 16 12 / 24



A→ · · · → An+1 → An → · · · → A0





An+1 //

��

A0

��





A→ · · · → An+1 → An → · · · → A0





An+1 //

��

A0

��





A→ · · · → An+1 → An → · · · → A0





An+1 //

��

A0

��




Topological (André-)Quillen Cohomology

The mapAn → A0 n Σn+2Hπn+1A

is an element of topological Quillen cohomology

kn+1O ∈ Dn+2(An; Hπn+1A)

Step 1. Have extension of scalars isomorphism

Ho(O-Alg/A0)(C,A0 n M) ∼= Ho(O-A0-Alg/A0)(A0 ∧ C,A0 n M)

Now in context of augmented algebras.





















Step 2. The augmented/non-unital equivalence

Ho(O-A0-Alg/A0)(A0 ∧ C,A0 n M) ∼= Ho(O-A0-Alg)(IR(A0 ∧ C),ZM)

for O the non-unital version of O,

O(n) =

{O(n) n > 0∗ n = 0

Step 3. The indecomposables/zero-multiplication adjunction

Ho(O-A0-Alg)(IR(A0 ∧ C),ZM) ∼= Ho(A0-Mod)(QLIR(A0 ∧ C),M)

∼= π0FA0(QLIR(A0 ∧ C),M).



Step 2. The augmented/non-unital equivalence

Ho(O-A0-Alg/A0)(A0 ∧ C,A0 n M) ∼= Ho(O-A0-Alg)(IR(A0 ∧ C),ZM)

for O the non-unital version of O,

O(n) =

{O(n) n > 0∗ n = 0

Step 3. The indecomposables/zero-multiplication adjunction

Ho(O-A0-Alg)(IR(A0 ∧ C),ZM) ∼= Ho(A0-Mod)(QLIR(A0 ∧ C),M)

∼= π0FA0(QLIR(A0 ∧ C),M).



How do you construct the Postnikov tower?

Theorem (Hurewicz Theorem)Suppose A→ B is n-connected and M is connected. ThenDq(B,A; M) = 0 for q ≤ n and Dn+1(B,A; M) = Hn+1(B,A; M).

Theorem (Universal Coefficient Theorem)There is a natural spectral sequence

Ep,q2 = Extp,qπ∗A0

(D∗(B,A; A0), π∗M)

converging conditionally to Dp+q(B,A; M).

Apply to the (n + 1)-connected map A→ An+1






Ep,q2 = Extp,qπ∗A0

(D∗(B,A; A0), π∗M)








Ep,q2 = Extp,qπ0A(D∗(B,A; A0), π∗M)








Ep,q2 = Extp,qπ0A(D∗(B,A; A0), π∗M)





How do you construct the Postnikov tower? (cont.)

Applying Hurewicz to the (n + 1)-connected map A→ An+1, we get

Dq(An+1,A; A0) = 0, q ≤ n + 1Dn+2(An+1,A; A0) = Hn+2(An+1,A; A0) = Hn+2(An+1,A;π0A)

= π0A⊗ πn+1A

Applying Universal Coefficient, we get

Dn+2(An,A; Hπn+1A) = Homπ0A(π0A⊗ πn+1A, πn+1A)

= Hom(πn+1A, πn+1A).






= π0A⊗ πn+1A









= π0A⊗ πn+1A









= π0A⊗ πn+1A









= π0A⊗ πn+1A







Dn+2(An,A; Hπn+1A) = Hom(πn+1A, πn+1A).

Choosing identity element, we get a (homotopy class of) diagram

A //

��

A0

��


Construct An+1 as homotopy pullback.Get A→ An+1 → An.






A //

��

A0

��








A

++WWWWWWWWWWWWWWWWWWW

��22

2222

2222

""

An+1 //

��

A0

��





Obstruction Theory

A→ · · · → An+1 → An → · · · → A0

TheoremA map of O-algebras f : B → An lifts (in the homotopy category) to amap of O-algebras B → An+1 if and only if f ∗kn+1

O = 0 inDn+2(B;πn+1A). When a lift exists, the set of lifts has a free transitiveaction of Dn+1(B;πn+1A).

TheoremAn O-algebra structure on An lifts to an O-algebra structure on An+1 ifand only if the spectrum-level k-invariant kn+1 ∈ Hn+2(An, πn+1A) liftsto an element of Dn+2(An, πn+1A).



Obstruction Theory

A→ · · · → An+1 → An → · · · → A0

TheoremA map of O-algebras f : B → An lifts (in the homotopy category) to amap of O-algebras B → An+1 if and only if f ∗kn+1

O = 0 inDn+2(B;πn+1A). When a lift exists, the set of lifts has a free transitiveaction of Dn+1(B;πn+1A).

TheoremAn O-algebra structure on An lifts to an O-algebra structure on An+1 ifand only if the spectrum-level k-invariant kn+1 ∈ Hn+2(An, πn+1A) liftsto an element of Dn+2(An, πn+1A).


Application: BP

Application: BP

Theorem (Basterra-Mandell)BP has an E4 ring spectrum structure. It is unique up to automorphismin the homotopy category of E4 ring spectra.

Existence:Compute topological Quillen (co)homology in a range forPostnikov section BPn.Play off of MU

Uniqueness:

Compute topological Quillen (co)homology.Obstructions for constructing an E4 map BP → BP ′ are zero.Any map of spectra BP → BP is either zero on an equivalence.


Application: BP

Application: BP



Uniqueness:



Application: BP

Application: BP



Uniqueness:



Application: BP

Application: BP



Uniqueness:



Application: BP

Application: BP



Uniqueness:



Application: BP

Application: BP



Uniqueness:



Application: BP

Computing topological Quillen homology of BP

Need some facts:H∗BP = Z(p)[ξ1, ξ2, . . .]

For augmented/non-unital En HZ(p)-algebras, topological Quillenhomology can be computed as an iterated bar construction.

D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)

We compute that this is free and concentrated in even degrees.

It follows that D∗(BP; HZ(p)) is concentrated in even degrees.

Obstruction for lifting a map BP → BP ′n to BP ′n+1 is an element ofDn+2(BP; Hπn+1BP).


Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP




D∗(BP; HZ(p)) ∼= π∗+4B4(HZ(p) ∧ BP)





Application: BP

The Computation

Want to show π∗B4(HZ(p) ∧ BP) is free and concentrated in evendegrees.

Suffices to compute homotopy groups of

B4(HZp ∧ BP) ∧HZp HZ/p ∼= B4(HZ/p ∧ BP)

Start with

π∗(HZ/p ∧ BP) = H∗(BP;Z/p) = Z/p[ξ1, ξ2, . . .], |ξi | = 2pi − 2.

Spectral sequence to compute π∗ of B(HZ/p ∧ BP) collapses at E2and you get π∗B(Z/p ∧ BP) is exterior on odd degree classes σξi indegree 2pi − 1.


Application: BP

The Computation




Start with




Application: BP

The Computation




Start with




Application: BP

The Computation




Start with




Application: BP

Spectral sequence to compute π∗ of B2(HZ/p ∧ BP) collapses at E2and you get E∞-term is truncated polynomial on classes γjσ

2ξi indegree 2pi+j , for i = 1,2, . . ., and j = 0,1, . . .

Need to figure out multiplicative extensions in order to get the fullcomputation for π∗B2(Z/p ∧ BP)

Map BP → HZ/p is an E4 map, so can read of the Dyer-Lashofoperations (that exist on the homology of E4 ring spectra) on H∗BPfrom the Dyer-Lashof operations for H∗ The Dyer-Lashof operation

Qpiξi = ξi+1 + decomposables

implies that

(γjσ2ξi)

p =

Qpi+jγjσ

2ξi = γjσ2ξi+1.

But this is now the p-th power operation.


Application: BP






implies that

(γjσ2ξi)

p =

Qpi+jγjσ

2ξi = γjσ2ξi+1.



Application: BP






implies that

(γjσ2ξi)

p =

Qpi+jγjσ

2ξi = γjσ2ξi+1.



Application: BP






implies that(γjσ

2ξi)p = Qpi+j

γjσ2ξi = γjσ

2ξi+1.



Application: BP

Associated graded was truncated poly on γjσ2ξi ; now know

(γjσ2ξi)

p = γjσ2xi+1. So we get

π∗B2(HZ/p ∧ BP) = Z/p[γ0σ2x1, γ1σ

2x1, γ2σ2x1, . . .]

polynomial on classes in degrees 2pj+1 for j = 0,1, . . .

Looking at the spectral sequence, we get π∗B3(HZ/p ∧ BP) is exterioron odd degree classes in degrees 2pj+1 + 1.

Looking at the spectral sequence, we get π∗B4(HZ/p ∧ BP) isconcentrated in even degrees.


Application: BP


(γjσ2ξi)



2x1, γ2σ2x1, . . .]





Application: BP


(γjσ2ξi)



2x1, γ2σ2x1, . . .]





Application: BP


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