Cochains and Homotopy Type - Indiana UniversityA cochain theory is a functor from spaces to chain complexes that satisﬁes a homotopy version of the gluing condition and sends homotopy

Cochains and Homotopy Type

Michael A. Mandell

Indiana University

Geometry SeminarNovember 12, 2009

M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 1 / 20

Overview

Talks

Today: Cochains and Homotopy Type

Cochains and Homotopy Type. Publ. Math. IHES 2006.Cochain Multiplications. Amer. J. Math. 2002.E∞ Algebras and p-Adic Homotopy Theory. Topology 2001.

Tuesday: Introduction to E∞ Algebras

What are E∞ algebras?How do E∞ algebras arise?

Next Thursday: Towards Formality

Current work in progress


Overview

Talks

Today: Cochains and Homotopy TypeCochains and Homotopy Type. Publ. Math. IHES 2006.Cochain Multiplications. Amer. J. Math. 2002.E∞ Algebras and p-Adic Homotopy Theory. Topology 2001.

Tuesday: Introduction to E∞ Algebras

What are E∞ algebras?How do E∞ algebras arise?




Overview

Talks


Tuesday: Introduction to E∞ AlgebrasWhat are E∞ algebras?How do E∞ algebras arise?




Overview

Talks


Tuesday: Introduction to E∞ AlgebrasWhat are E∞ algebras?How do E∞ algebras arise?

Next Thursday: Towards FormalityCurrent work in progress


Overview


AbstractThe cochains of a simplicial complex or singular cochains of a spacecan be viewed as a homotopy theoretic version of the algebra offunctions. The “spectrum” or “variety” of this algebra is a homotopytheoretic completion of the space.

Outline1 Functions and Duality2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type


Overview



Outline

1 Functions and Duality2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type


Overview



Outline1 Functions and Duality

2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type


Overview



Outline1 Functions and Duality2 Functions in Homotopy Theory

3 Rational Homotopy Theory4 Cochains and Homotopy Type


Overview



Outline1 Functions and Duality2 Functions in Homotopy Theory3 Rational Homotopy Theory

4 Cochains and Homotopy Type


Overview



Outline1 Functions and Duality2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type


Functions and Duality

Mathematical Structures and Functions

Category theory

Mathematical structures are determined by their functions.

Yoneda’s Lemma

In a category C, the functor C(X ,−) determines X up to uniqueisomorphism.


A finite dimensional vector space V is determined up to canonicalisomorphism by its vector space of linear functions Hom(V ,C).




Category theory


Yoneda’s Lemma







Category theory


Yoneda’s Lemma







Category theory


Yoneda’s Lemma






Functions and Duality in Topology and Geometry

Gelfand-Naimark Theorem

A compact Hausdorf space is determined up to canonicalisomorphism by its Banach algebra of real valued functions, or itsC∗ algebra of complex valued functions.

Manifolds

Smooth, PL, topological, etc.

Affine Algebraic Sets

The “finite dimensional case”.

Homotopy Theory ???






Manifolds




Homotopy Theory ???






Manifolds




Homotopy Theory ???


Functions in Homotopy Theory


Homotopy Theory

Problem: Find a homotopy theoretic ring of functions thatdetermines the space up to homotopy.

Homotopy classes of functions

?To R?

To Z?




Homotopy Theory


Homotopy classes of functions

?To R?

To Z?




Homotopy Theory


Homotopy classes of functions?To R?

To Z?




Homotopy Theory


Homotopy classes of functions?To R?

To Z?



The Problem: Gluing

U,V ⊂ X open subsets.

Continuous Maps U ∪ V → Y⇐⇒

Continuous Maps U → Y , V → Y that agree on U ∩ V

This does not work for homotopy classes of maps!

Example: Maps from S1 to S1



The Problem: Gluing








The Problem: Gluing








The Solution: Cochain Complexes

Work with chain complexes / differential graded modules.

Insist on a chain homotopy version of the gluing condition

Work up to chain homotopy equivalence or “quasi-isomorphism”Homotopy equivs give quasi-isomorphisms of complexes

Definition (Cochain Theory)A cochain theory is a functor from spaces to chain complexes thatsatisfies a homotopy version of the gluing condition and sendshomotopy equivalences to quasi-isomorphisms.

=⇒ cohomology








=⇒ cohomology








=⇒ cohomology








=⇒ cohomology



Cochain Theories

“Ordinary” = cohomology of a point concentrated in degree zero.

TheoremAny two ordinary cochain theories with the same coefficients arenaturally quasi-isomorphic (for “nice” spaces).

More about this on Tuesday (if there is interest)

Up to natural quasi-isomorphism, maps between ordinary cochaintheories in one-to-one correspondence with maps betweencoefficients.

For commutative ring coefficients, expect some kind of ring structure.



Cochain Theories








Cochain Theories








Algebra Cochain Theories


E∞ algebra

(more about what this is on Tuesday).

For a commutative ring k containing Q,E∞ k -algebra ' comm. diff. graded k algebra (k -CDGA).

For a commutative ring k not containing Q, no cochain theory can takevalues in k -CDGAs. (More about this on Tuesday.)





E∞ algebra








E∞ algebra








E∞ algebra







TheoremAny two ordinary E∞ k-algebra cochain theories are naturallyquasi-isomorphic as functors to E∞ k-algebras.

Any two ordinary k-CDGA cochain theories are naturallyquasi-isomorphic as functors to k-CDGAs.

The natural quasi-isomorphism is essentially unique.

ExampleThe De Rham complex is a cochain functor (on smooth manifolds withboundary) to R-CDGAs. Any other one is naturally quasi-isomorphic.




TheoremAny two ordinary E∞ k-algebra cochain theories are naturallyquasi-isomorphic as functors to E∞ k-algebras.

Any two ordinary k-CDGA cochain theories are naturallyquasi-isomorphic as functors to k-CDGAs.

The natural quasi-isomorphism is essentially unique.

ExampleThe De Rham complex is a cochain functor (on smooth manifolds withboundary) to R-CDGAs. Any other one is naturally quasi-isomorphic.


Rational Homotopy Theory

Variants on the De Rham Complex

Use forms that are piecewise smooth.

Use forms piecewise polynomial in coordinates on a triangulation.

Use forms that are polynomial on any face of a triangulation.

Use forms that are rational polynomial on any face of a triangulation.

Thom-Sullivan Rational De Rham complex

On a q-simplex t0 + · · ·+ tq = 1, use forms∑aI t

n11 · · · t

nqq dtj1 ∧ · · · ∧ dtjr aI ∈ Q

For any simplicial complex or simplicial set X , the rational De Rhamcomplex Ω∗X is the Q-CDGA where an element consists of a form oneach simplex that agree under the face inclusions.










n11 · · · t












n11 · · · t












n11 · · · t












n11 · · · t






Theorem (Quillen / Sullivan)The quasi-isomorphism class of the Q-CDGA Ω∗X determines allrational homotopy information about simply connected spaces.

ExampleIf X is simply connected, you can recover π∗X ⊗Q from Ω∗X as thereduced André-Quillen cohomology of Ω∗X , or (equivalently) as theprimitive elements of the Hopf algebra Ext∗Ω∗X (Q,Q)




Theorem (Quillen / Sullivan)The quasi-isomorphism class of the Q-CDGA Ω∗X determines allrational homotopy information about simply connected spaces.

ExampleIf X is simply connected, you can recover π∗X ⊗Q from Ω∗X as thereduced André-Quillen cohomology of Ω∗X , or (equivalently) as theprimitive elements of the Hopf algebra Ext∗Ω∗X (Q,Q)



The Spacial Realization of a Q-CDGA

Let A be a Q-CDGA.

Look at Q-CDGA maps from A to Q.

Homotopy theoretically, this forms a space.

Q→ Ω∗∆[n] is a quasi-isomorphismQ→ Ω∗∆[•] is a simplicial resolution.

V•(A) = Hom(A,Ω∗∆[•]) is a simplicial set

A “derived” version of this V L gives a functor from Q-CDGAs to thehomotopy category of spaces.




Let A be a Q-CDGA.









Let A be a Q-CDGA.









Let A be a Q-CDGA.









Let A be a Q-CDGA.








The Quillen-Sullivan Theorem

If X is simply connected, then X → V L(Ω∗X ) is a rational equivalence.

V L(Ω∗X ) is the rationalization of X(A kind of completion or “localization”)

If Ω∗X and Ω∗Y are quasi-isomorphic Q-CDGAs, then V L(Ω∗X ) andV L(Ω∗Y ) are homotopy equivalent spaces.

There are “rational equivalences”

X → Z ← Y

(with Z = V L(Ω∗X ) ' V L(Ω∗Y ).)








X → Z ← Y

(with Z = V L(Ω∗X ) ' V L(Ω∗Y ).)








X → Z ← Y

(with Z = V L(Ω∗X ) ' V L(Ω∗Y ).)



Cochain Theories in Other Characteristics

In characteristic p, no CDGA cochain theory.

Look at singular or simplicial cochains C∗.CqX functions on q-simplices of X .

C∗ is an E∞ algebra cochain theory.

Analogy with De Rham complex

Think of Cq(∆[n]) as the q-forms on ∆[n].Then an element of CqX consists of a q-form on each simplexthat agree under the face inclusions.



























Spacial Realization

Fix a comm. ring k and look at cochains with coefficients in k .

k → C∗∆[n] is a quasi-isomorphismk → C∗∆[•] is a simplicial resolution

For an E∞ k -algebra ASpacial realization of maps from A to kVA = Hom(A,C∗∆[•])

Derived version V L a functor from E∞ k -algebras to homotopycategory of spaces.



Spacial Realization







Spacial Realization







p-Adic Homotopy Theory

Theory works differently in characteristic p than in characteristic zero.

For k = Z/p, X → VC∗X is not a p-adic equivalence.VC∗X is more like the free loop space on X .

For k = Fp, X → VC∗X is p-completion when X is simply connected.

Quasi-isomorphism C∗(X ; Fp)⊗ Fp → C∗(X ; Fp).

Consequence

All p-adic homotopy information determined by C∗(X ; Fp).

If C∗(X ; Fp) and C∗(Y ; Fp) are quasi-isomorphic as E∞Fp-algebras, then C∗(X ; Fp) and C∗(Y ; Fp) are quasi-isomorphicas E∞ Fp-algebras, and X and Y are p-equivalent.








Consequence










Consequence










Consequence





The Arithmetic Square

Fiber squareZ //

∏Z∧p

Q // (∏

Z∧p )⊗Q

Homotopy type determined by rational homotopy type, p-adichomotopy types, and patching data.



The Arithmetic Square

Fiber squareZ //

∏Z∧p

Q // (∏

Z∧p )⊗Q

Homotopy type determined by rational homotopy type, p-adichomotopy types, and patching data.




Let k = Z.

TheoremIf X and Y are simply connected and the integral cochains C∗X andC∗Y are quasi-isomorphic E∞ algebras, then X and Y are homotopyequivalent.

Future / Past

Use E∞ structure on C∗X to obtain homotopy information on X .




Let k = Z.

TheoremIf X and Y are simply connected and the integral cochains C∗X andC∗Y are quasi-isomorphic E∞ algebras, then X and Y are homotopyequivalent.

Future / Past

Use E∞ structure on C∗X to obtain homotopy information on X .


Documents

Cochains and Homotopy Type - Indiana UniversityA cochain theory is a functor from spaces to chain complexes that satisﬁes a homotopy version of the gluing condition and sends homotopy