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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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 2 / 20
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?
Next Thursday: Towards Formality
Current work in progress
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 2 / 20
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∞ AlgebrasWhat are E∞ algebras?How do E∞ algebras arise?
Next Thursday: Towards Formality
Current work in progress
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 2 / 20
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∞ AlgebrasWhat are E∞ algebras?How do E∞ algebras arise?
Next Thursday: Towards FormalityCurrent work in progress
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 2 / 20
Overview
Cochains and Homotopy Type
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
Overview
Cochains and Homotopy Type
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.
Outline
1 Functions and Duality2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
Overview
Cochains and Homotopy Type
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 Duality
2 Functions in Homotopy Theory3 Rational Homotopy Theory4 Cochains and Homotopy Type
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
Overview
Cochains and Homotopy Type
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 Theory
3 Rational Homotopy Theory4 Cochains and Homotopy Type
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
Overview
Cochains and Homotopy Type
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 Theory
4 Cochains and Homotopy Type
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
Overview
Cochains and Homotopy Type
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 3 / 20
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.
Functions and Duality
A finite dimensional vector space V is determined up to canonicalisomorphism by its vector space of linear functions Hom(V ,C).
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 4 / 20
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.
Functions and Duality
A finite dimensional vector space V is determined up to canonicalisomorphism by its vector space of linear functions Hom(V ,C).
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 4 / 20
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.
Functions and Duality
A finite dimensional vector space V is determined up to canonicalisomorphism by its vector space of linear functions Hom(V ,C).
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 4 / 20
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.
Functions and Duality
A finite dimensional vector space V is determined up to canonicalisomorphism by its vector space of linear functions Hom(V ,C).
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 4 / 20
Functions and Duality
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 ???
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 5 / 20
Functions and Duality
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 ???
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 5 / 20
Functions and Duality
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 ???
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 5 / 20
Functions in 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?
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 6 / 20
Functions in 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?
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 6 / 20
Functions in 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?
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 6 / 20
Functions in 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?
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 6 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 7 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 7 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 7 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 8 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 8 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 8 / 20
Functions in Homotopy Theory
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
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 8 / 20
Functions in Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 9 / 20
Functions in Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 9 / 20
Functions in Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 9 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
For commutative ring coefficients, expect some kind of ring structure.
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.)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 10 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
For commutative ring coefficients, expect some kind of ring structure.
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.)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 10 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
For commutative ring coefficients, expect some kind of ring structure.
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.)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 10 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
For commutative ring coefficients, expect some kind of ring structure.
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.)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 10 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 11 / 20
Functions in Homotopy Theory
Algebra Cochain Theories
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 11 / 20
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 12 / 20
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 12 / 20
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 12 / 20
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 12 / 20
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 12 / 20
Rational Homotopy Theory
Rational Homotopy Theory
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)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 13 / 20
Rational Homotopy Theory
Rational Homotopy Theory
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)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 13 / 20
Rational Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 14 / 20
Rational Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 14 / 20
Rational Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 14 / 20
Rational Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 14 / 20
Rational Homotopy Theory
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 14 / 20
Rational Homotopy Theory
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 ).)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 15 / 20
Rational Homotopy Theory
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 ).)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 15 / 20
Rational Homotopy Theory
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 ).)
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 15 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 16 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 16 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 16 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 16 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 17 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 17 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 17 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 18 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 18 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 18 / 20
Cochains and Homotopy Type
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.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 18 / 20
Cochains and Homotopy Type
The Arithmetic Square
Fiber squareZ //
∏Z∧p
Q // (∏
Z∧p )⊗Q
Homotopy type determined by rational homotopy type, p-adichomotopy types, and patching data.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 19 / 20
Cochains and Homotopy Type
The Arithmetic Square
Fiber squareZ //
∏Z∧p
Q // (∏
Z∧p )⊗Q
Homotopy type determined by rational homotopy type, p-adichomotopy types, and patching data.
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 19 / 20
Cochains and Homotopy Type
Cochains and Homotopy Type
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 .
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 20 / 20
Cochains and Homotopy Type
Cochains and Homotopy Type
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 .
M.A.Mandell (IU) Cochains and Homotopy Type Nov 12 20 / 20