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LECTURE 1: OVERVIEW One of the Principal Aims of Algebraic Topology

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LECTURE 1: OVERVIEW One of the Principal Aims of Algebraic Topology LECTURE 1: OVERVIEW One of the principal aims of algebraic topology is to study topological spaces X by means of algebraic invariants, such as their homology and cohomology groups. Ideally, one would like invariants which have both of the following features: a They should be rich and interesting, with the potential to capture useful information about X. (b) They should be simple and easy to work with, amenable to study by methods of algebra. These( ) requirements are in tension with one another: it is generally hard to satisfy both of them simultaneously. Nevertheless, there are some invariants which do a good job of reconciling a with b . Let us consider some examples which arise in the study of rational homotopy theory. We begin by reviewing some definitions. ( ) ( ) Definition 1. Let f X Y be a map between simply connected topological spaces. We will say that f is a rational homotopy equivalence if the pullback map f H Y ; Q H X∶; Q →is an isomorphism. ∗More∗ generally,∗ we say that a pair of (simply connected) topological spaces X and∶ Y (are rationally) → ( homotopy) equivalent if they are related by a chain of rational homotopy equivalences. Roughly speaking, the goal of rational homotopy theory is to classify (simply connected) topological spaces up to rational homotopy equivalence. It follows immediately from the definitions that if X and Y are rationally homotopy equiv- alent, then the rational cohomology rings H X; Q and H Y ; Q are isomorphic. One can ask if the converse holds: ∗ ∗ ( ) ( ) Question 2. Let X and Y be simply connected topological spaces, and suppose that the cohomology rings H X; Q and H Y ; Q are isomorphic. Does it follow that X and Y are rationally∗ homotopy equivalent?∗ ( ) ( ) The answer to Question 2 is \no" in general: it is possible for spaces X and Y to have the same rational cohomology ring \by accident" (that is, an isomorphism H X; Q H Y ; Q which is not induced by a map of topological spaces, or a chain∗ of such).∗ Put another way, it is hopeless to try to reconstruct a space X (or any( approximation) ≃ ( to)X) from the information of the cohomology ring H X; Q alone. ∗ One can attempt to remedy the situation by replacing the cohomology( ring) H X; Q by a somewhat richer invariant: the singular cochain complex C X; Q . Note∗ that the cup product on cohomology is defined at the cochain level:∗ that is, ( ) 1 ( ) 2 LECTURE 1: OVERVIEW there are maps s t s t C X; Q Q C X; Q C X; Q which endow C X; Q with the structure of a differential+ graded algebra over ∪ ∶ ( ) ⊗ ( ) → ( ) Q. We can therefore∗ regard the construction X C X; Q as a contravariant functor ( ) ∗ ↦ ( ) Topological spaces Differential graded algebras over Q : Essentially{ by definition, this} functor→ { carries rational homotopy equivalences} to quasi-isomorphisms between differential graded algebras. We can therefore pose the following refinement of Question 2: Question 3. Let X and Y be simply connected topological spaces, and sup- pose that the differential graded algebras C X; Q and C Y ; Q are quasi- isomorphic. Does it follow that X and Y are∗ rationally homotopy∗ equivalent? ( ) ( ) Question 3 still has a negative answer, but the reason is more subtle. For any topological space X, the cohomology ring H X; Q is a graded-commutative algebra: that is, we have ∗ x x 1 stx ( x ) for x Hs X; Q and x Ht X; Q ′. However,′ this commutativity is not visible ∪ = (− ) ∪ s at the cochain level. If′ x and x are represented by cocycles x C X; Q and t x C∈ X;(Q , then) we can∈ write( ′ ) ′ st ∈ ( ) ∈ ( ) x x 1 x x dσ x; x ; ′ ′ ′ s t 1 for some cochain σ x; x ∪ C= (− )X; Q∪, which+ ( can) be explicitly chosen. If α C X; Q C Y ; Q′ is a quasi-isomorphism+ − of differential graded algebras, then∗ the difference∗ α( σ x;) x∈ σ α( x ; α) x is an s t 1 -cycle which rep- s t 1 resents∶ ( a cohomology) → ( class) ′η H Y ; Q .′ If this cohomology class does not vanish, then it is impossible( ( )) for−α +(to−( arise) ( from)) a rational( + homotopy− ) equivalence of spaces. ∈ ( ) Put more informally: Question 3 has a negative answer because the differential graded algebra C X; Q fails to encode information about the commutativity of the cup product∗ on cohomology. To remedy this, Sullivan introduced a variant of the singular cochain( ) complex C X; Q , in which multiplication is graded- commutative \on the nose": ∗ ( ) Construction 4 (Sullivan). For each n 0, let n ∆ x0; : : : ; xn R≥ 0 x0 xn 1 ≥ denote the standard n-simplex= {( and let) ∈ S + ⋯ + = } n ∆ x0; : : : ; xn R x0 xn 1 + = {( ) ∈ S + ⋯ + = } LECTURE 1: OVERVIEW 3 denote the affine space containing it. We will say that a differential form ! on ∆n is polynomial it it belongs to the subalgebra of Ω ∆n generated (over the rational numbers) by the functions xi and their differentials∗ dxi. +Let X be an arbitrary topological space and let m 0( be+ an) integer. A singular m-form on X is function ! Continuous maps ∆n X Polynomial≥ m-forms on ∆n σ ! ∶ { → } → σ{ +} which satisfies the following constraint: if f ∆n′ ∆n is the map of simplices associated to a nondecreasing function↦0 1 ::: n 0 1 n , then m !σ f !σ ∆n′ . We let Ωpoly X denote the set∶ of all→ singular′ m-forms on X. + { < < < } → { < < ⋯ < } If○ ! is a singular m-form on X, then we can define a singular m 1 -form d! on X=byS the formula d! σ( )d !σ . Using this differential, we can regard 0 Ω0 X Ω1 X Ω2 X ( + ) ( poly) = ( ) poly poly as a chain complex of rational vector spaces, which we will refer to as the poly- → ( ) → ( ) → ( ) → ⋯ nomial de Rham complex of X and denote by Ωpoly X . We regard Ωpoly X as a commutative differential graded algebra over Q∗ , with multiplication∗ given by ! ! σ !σ !σ. ( ) ( ) The′ construction′ X Ω X determines a contravariant functor ( ∧ ) = ∧ poly Topological Spaces Commutative∗ Differential Graded Algebras over Q : ↦ ( ) It is closely related to the singular cochain functor X C X; Q : one can { } → { } show that integration of differential forms defines a quasi-isomorphism∗ of chain complexes: Ωpoly X C X; Q . In particular, the↦ rational( cohomology) ∗ ∗ H X; Q can be computed by the chain complex Ωpoly X . We can therefore ask∗ another∫ variant∶ of( Question) → ( 2: ) ∗ ( ) ( ) Question 5. Let X and Y be simply connected topological spaces, and suppose that Ωpoly X and Ωpoly Y are quasi-isomorphic (as commutative differential graded∗ algebras). Does∗ it follow that X and Y are rationally homotopy equiva- lent? ( ) ( ) Perhaps surprisingly, the answer to Question 5 turns out to be \yes", at least if we impose some mild finiteness assumptions. To formulate this precisely, we need to introduce a bit of terminology. Notation 6. Let Top denote the category of topological spaces and let Topsc Top be the full subcategory spanned by the simply connected topological spaces. For every field k, we let Topft k Topsc denote the full subcategory spanned by⊆ n those simply connected spaces( X) for which the cohomology group H X; k is a finite-dimensional vector space over⊆ k, for each n 0 (note that this condition depends only on the characteristic of k). ( ) ≥ 4 LECTURE 1: OVERVIEW We let h Q denote the category obtained from Topsc by formally inverting all rational homotopy equivalences, and we let h ft Q denote the category obtained from Topft Sby formally inverting all rational homotopy( ) equivalences (one can show that this is a full subcategory of h ft Q . S dg ( ) Let CAlgQ denote the category of commutative differential graded algebras dg S dg over Q, and let hCAlgQ denote the category obtained from CAlgQ by formally inverting all quasi-isomorphisms. Since the construction X Ωpoly X carries rational homotopy equivalences to quasi-isomorphisms, it induces∗ a functor of homotopy categories ↦ (Q )op dg Ωpoly h hCAlgQ : We then have the following: ∗ ∶ ( S ) → Theorem 7 (Sullivan). The functor Ωpoly determines a fully faithful embedding ft Q op dg ∗ h hCAlgQ . Moreover, the essential image of this embedding consists of those( ) commutative differential graded algebras A which satisfy the following conditions:( S ) ↪ ∗ i The unit map Q H0 A is an isomorphism. n ii The cohomology groups H∗ A vanish for n 1 and n 0. n iii( ) The cohomology→ groups( H) A∗ are finite-dimensional (as vector spaces ( ) over Q) for n 2. ( ∗) = < (It follows) from Theorem 7 that( the) polynomial de Rham complex Ω X ≥ poly does an excellent job of satisfying requirements a and b above. ∗ a By virtue of Theorem 7, we can say exactly what information about( a) topological space X is captured by the( polynomial) ( ) de Rham complex ( ) Ωpoly X : namely, the rational homotopy type of X (at least if X is rationally∗ of finite type). ( ) b The polynomial de Rham complex Ωpoly X is a commutative differential graded algebra over Q, and therefore∗ belongs to a class of objects which ( ) can be studied by purely algebraic techniques.( ) Remark 8. One might argue that the polynomial de Rham functor Ωpoly fails to satisfy b for one of the following reasons: ∗ 1 Even when the topological space X is relatively simple, the commutative ( ) differential graded algebra Ωpoly X tends to be enormous (and hence not ( ) really suitable for calculations).∗ ( ) 2 The isomorphism class of the algebra Ωpoly X is not a rational homotopy invariant: only its quasi-isomorphism class∗ is.
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