aa r X i v : . [ m a t h . A T ] S e p GEOMETRIC HOMOLOGY
MAX LIPYANSKIY Introduction
The purpose of this paper is to introduce a version of singular homology basedon smooth mappings of manifolds with corners. Although variants of such a theoryexists in the literature (see [3], [1], [16]), we felt that certain points were not ade-quately addressed. In particular, our goal is to construct a chain level theory basedon smooth mappings of manifolds with corners. In addition, we will avoid using thefact that smooth manifolds with can be triangulated. As we shall see, transversalityand intersections play a major role in setting up this theory. From a pedagogicalviewpoint, having intersection theory arguments available from the start facilitatessimple and intuitive computations.The motivation for our construction comes from several sources. There has been arenewed interest in defining invariants based on geometric chains mapping to configu-ration space. Such chains appear as solutions of various nonlinear partial differentialequations in Gromov-Witten theory as well as gauge theory. In addition, geometricchains also appear in the work of Sullivan and Chas on String topology [2]. Thegeometric construction of various ∞ -structures is closely related to a chain level con-struction of intersection theory.Ultimately, however, we are interested in the infinite dimensional variant of this the-ory. In [5], we introduced a new approach to Floer theory based on a notion of asemi-infinite cycle. This approach provides an alternative (but equivalent) approachto the Morse theory construction of Floer homology. The fundamental objects in thistheory are semi-infinite chains which are mappings of Hilbert manifolds σ : P → B meeting certain topological axioms. The theory we discuss in this paper is a finite di-mensional analogue of this geometric construction. As it turns out, many argumentsin Floer theory (for instance, the proof of the Morse Homology theorem) are nearlyidentical to their finite dimensional counterparts once certain topological assumptionsare satisfied. For these reasons, we thought it would be useful to expose our approachto geometric homology in the finite dimensional case to help motivate the analogousarguments in Floer theory. cknowledgement. We wish to thank Tom Mrowka and Dennis Sullivan foruseful conversations. In addition, we would like to thank the Simons Center ForGeometry and Physics for their hospitality while this work was being completed.2.
Geometric Preliminaries
Manifolds With Corners.
Here we review the concept of a manifold withcorners and prove various geometric lemmas that are used in the paper.
Definition 1.
A map R k × [0 , j → R l × [0 , k is smooth if locally it is a restrictionof a smooth map on R k + j → R l + k . Definition 2.
Let P be a Hausdorff, 2nd countable topological space. P is a smoothmanifold with corners if it is equipped with an open cover { U i } together withhomeomorphisms φ i : R k × [0 , n − k → U i We demand that φ − j ◦ φ i are smooth where defined. The collection ( U i , φ i ) is an atlas for P . As usual, picking a maximal compatible atlas specifies a smooth structure on P . P has a natural structure of a stratified space. Indeed, Let P i be the set of points p with a local chart of the form R n − i × [0 , i where p = { } . We have P = P ∪ P ∪ P ∪ P . . . One may check, using the inverse function theorem, that definition of strata is in-dependent of the choice of chart. Note that P i is a manifold without corners ofdimension n − i . We will assume that, for a given P , each connected component ofthe top stratum has the same dimension n . Let M be a smooth manifold withoutboundary and f : P → M be a smooth map. Definition 3.
A map f : P → M is transverse to a submanifold Y ⊂ M if itstransverse on each open stratum P i of P . Lemma 1.
Consider a manifold M with closed submanifold Y . Given a smooth map σ : P → M such that σ is transverse to Y , σ − ( Y ) is a manifold with corners.Proof. Locally, we may write σ as σ : R l × [0 , i → R n with Y = R k × ⊂ R n . Let p ∈ R l . We will produce a chart for σ − ( Y ) near p × π : R n → R n − k be the projection to the last n − k factors. The map f : R l × [0 , i → R n − k × [0 , i iven by f ( v, t ) = ( π ( σ ( v, t )) , t ) is a surjection in view of our transversality assump-tion. The implicit function theorem assures that we can pick coordinates U × [0 , i for around p such that π ( σ ( u, t )) = φ ( u ) for some smooth function φ . Therefore,another application of the implicit function theorem yields that φ − (0) × [0 , i is achart for σ − ( Y ). (cid:3) Boundary Operator.
Our goal is to define a geometric boundary operator onmanifolds with corners. For [0 , k let ∂ i ([0 , k ) = [0 , k − with the natural face inclusion s i : ∂ i ([0 , k ) → [0 , k that omits the i th factor. For [0 , k , let ∂ [0 , k = G i ∂ i [0 , k Given a P , we construct ∂P as follows. We replace the chart R n − k × [0 , k by R n − k × ∂ [0 , k . The transition functions for P induce transition functions for ∂P .Note that locally, we get one copy of P , two copies of P , etc.2.3. Orientations.Definition 4. P is said to be orientable if P is orientable. An orientation of P is an orientation of P . The following definition is important for the construction of homology groups:
Definition 5.
Given a manifold with corners P , a mapping σ : P → M is said tobe trivial , if there exists an orientation reversing diffeomorphism f : P → P with σ ◦ f = σ . Lemma 2.
An orientation of P induces an orientation of ∂P .Proof. Since ( ∂P ) = P , this is a consequence of the fact that an orientation of P induces an orientation on P using the “outward normal first” convention. (cid:3) Since diffeomorphism of P induces one of ∂P , we have: Lemma 3. If σ : P → M is trivial, so is ∂σ : ∂P → M . The following lemma will imply that ∂ = 0 in the homology theory we define: Lemma 4.
Given any σ , we have ∂ σ trivial. roof. Locally, ∂ ( V × [0 , k ) = V × ∂ [0 , k . Now, ∂ [0 , k = a i = j ∂ i ◦ ∂ j [0 , k Thus, ∂ [0 , k is naturally a disjoint union of two spaces corresponding to when i < j and i > j . This provides the orientation reversing involution of ∂ σ . (cid:3) We now recall the notion of orientable map between finite dimensional manifolds.For the general discussion, see [4]. Given a smooth map σ : P → M of finite dimensional manifolds, recall the construction of determinant bundle det ( σ ) → P Intuitively, the line over a point p ∈ P corresponds to the spaceΛ max KerDσ p ⊗ Λ max ( CokerDσ p ) ∗ Since in general the dimension of kerDσ may jump, to construct a locally trivialbundle one proceeds as follows. Locally, one may pick a trivialized bundle R n → P and a map of bundles L : R n → σ ∗ ( T M )such that the map Dσ ⊕ L : T P ⊕ R n → σ ∗ ( T M ) is surjective. One then defines det ( σ ) as Λ max ( Ker ( Dσ ⊕ L )) ⊗ (Λ n R n ) ∗ One can check that does not depend on the local choices and defines a line bundleover P . A useful fact in dealing with orientations is that any exact sequence:0 → V → V → · · · V n → ⊗ even Λ max V i ∼ = ⊗ odd Λ max V i Definition 6.
An orientation of a map σ is an orientation of det( σ ) . We will need several basic results about oriented maps. Given transverse maps σ : P → N and f : M → N , we may form the pullback f ∗ ( σ ) : f − ( P ) → M This is defined as the subset of M × P such that ( m, p ) ∈ f − ( P ) if f ( m ) = σ ( p ). Lemma 5.
We have f ∗ ( det ( σ )) ∼ = det ( f ∗ ( σ )) roof. The tangent space to f − ( P ) is given by ( v, w ) ∈ T M × T P with Df ( v ) = Dσ ( w ). The kernel of the map f ∗ σ : f − ( P ) → M is given by points ( v, w ) with v = 0 and Dσ ( w ) = 0. Thus, ker ( f − σ ) is naturally identified with ker ( σ ). In viewof the isomorphism T M/im ( f − σ ) → T N/im ( σ )the cokernels are also naturally identified. (cid:3) This implies that an orientation of the map σ induces one on f ∗ ( σ ). Lemma 6. If f is oriented and P is oriented, so is f − ( P ) .Proof. The tangent space to f − ( P ) is given by ( v, w ) ∈ T M ⊕ T P with Df ( v ) = Dσ ( w ). The exact sequence0 → KerDf → T M → T N → CokerDf → max T M ⊗ Λ max ( T N ) ∗ while the sequence0 → T f − ( P ) → T M ⊕ T P → T N → max T f − ( P ) is equivalent to that of Λ max T M ⊗ Λ max ( T N ) ∗ ⊗ Λ max T P . (cid:3) Note that given an oriented map σ : P → M , we get an induced oriented map ∂σ : ∂P → M . The following is straighforward to verify using a local chart: Lemma 7.
Given an oriented map σ : P → M , there exists a diffeomorphism φ : ∂ P → ∂ P commuting with σ such that φ ∗ ( det ( σ )) ∼ = − det ( σ ) as oriented line bundles. Cutting by a Hypersurface.
We will need to cut a manifold by a hypersurfaceobtaining a decomposition of the manifold into two parts. Let σ : P → M be as above. Consider a smooth map f : M → R with p a regular value of f ◦ σ . Let σ + = σ | ( f ◦ σ ) − [ p, ∞ ) σ − = σ | ( f ◦ σ ) − ( −∞ ,p ] and σ = σ | ( f ◦ σ ) − ( p ) et P ′ = ( f ◦ σ ) − ( p ) Theorem 1. σ ± are manifolds with corners. Furthermore, ∂ ( σ ± ) = ( ∂σ ) ± ⊔ ± σ Proof.
Locally we have σ : V × [0 , k → M where V is a manifold without boundary. Applying the inverse function theorem, wecan represent σ ± as ( σ | V ) ± × [0 , k . The formula for the boundary follows since ∂ ( σ ±| V × [0 , k ) = ∂σ ±| V × [0 , k ⊔ σ ±| V × ∂ [0 , k and ∂σ ±| V = ± σ | V (cid:3) We also need to prove a result that will allows us to cut cycles into smaller pieces.We introduce a manifold
Cre ( σ ) that interpolates between σ and ”creasing” σ along σ . Let D be subset of R given by { ( x, y ) ∈ R | ≤ y ≤ − | x | , | x | < } D inherits the structure of a manifold with corners as a subset of the plane. We definea homeomorphism φ : D → ( − , × [0 , φ ( x, y ) = ( x, y − | x | )Note that φ is a diffeomorphism outside x = 0.Let Cre ( σ ) be obtained as follows. As a topological space, Cre ( σ ) = P × [0 , P ′ × [0 , P − P ′ ) × [0 , P ′ , we can locallywrite P as P = V × ( − ǫ, ǫ ) with f ( σ ( v, t )) = t and V is a manifold with corners. Wetake the manifold structure to be V × D ǫ where D ǫ = { ( x, y ) ∈ D | = − ǫ < x < ǫ } Here we identify P = V × ( − ǫ, ǫ ) ⊂ V × D ǫ as points with y = 0. The overlap chartmap V × D ǫ → V × ( − ǫ, ǫ ) × [0 , φ to φ ǫ : D ǫ → ( − ǫ, ǫ ) × [0 , ince σ induces a map Cre ( σ ) by composing σ with the smooth projection V × D ǫ → V × ( − ǫ, ǫ )we have Lemma 8. σ induces smooth map on Cre ( σ ) . Furthermore, if σ is trivial, so is Cre ( σ ) . Lemma 9. ∂ ( Cre ( σ )) = − σ ⊔ σ + ⊔ σ − ⊔ Cre ( ∂σ ) Proof.
We check the statement in a neighborhood of P ′ . We have the chart V × D ǫ .We have ∂ ( V × D ǫ ) = ∂V × D ǫ ⊔ V × ∂D ǫ while ∂D ǫ = ( − ǫ, ǫ ) ⊔ ( − ǫ, ⊔ [0 , ǫ ) ∂V × D ǫ corresponds to Cre ( ∂P ) V × ( − ǫ, ǫ ) corresponds to PV × [0 , ǫ ) corresponds to σ + and V × ( − ǫ,
0] to σ − . (cid:3) Definition of the Homology Groups
Fix some countable infinite dimensional Hilbert space H once and for all. To avoidset-theoretic complications, all our chains will be subsets of H . Let M be a smoothparacompact manifold without boundary. We wish to construct a homology theoryfor M based on mappings of manifolds with corners into M . Although we focus onthe case M is finite dimensional, most of the methods of this paper carry over ratherdirectly to the case when M is an infinite dimensional paracompact Banach manifold. Definition 7. A chain is a smooth map σ : P → M where P is a compact oriented manifold with corners embedded in H . Two chains σ : P → M and τ : Q → M are said to be isomorphic if there exists an orientationpreserving diffeomorphism f : P → Q such that τ ◦ f = σ . We will assume that all the components of P have the same dimension. Definition 8.
A chain σ : P → M is said to be trivial if there exists an orientationreversing diffeomorphism f : P → P with σ ◦ f = σ . Definition 9.
A chain σ : P → M is said to have small image if σ ( P ) ⊂ g ( N ) where g : N → M is a smooth map from a manifold with corners (not necessarilycompact) with dim ( N ) < dim ( P ) . efinition 10. A chain σ is said to be degenerate if σ has small image and ∂σ isisomorphic to a disjoint union of a trivial chain and a chain with small image. As the simplest example, note that the constant map [0 , → pt is degenerate butthe boundary does not have small image. Lemma 10.
Let τ be a trivial chain. If σ ⊔ τ is trivial, then σ is trivial.Proof. We decompose σ into mutually isomorphic components as σ = σ ⊔ σ . . . Here, each σ i consists of the disjoint union of isomorphic connected components of(up to orientation) of σ and no σ i , σ j share isomorphic components when i = j . Anautomorphism of σ preserved these components. Therefore, σ is trivial exactly whenfor each i, either the number of components of σ i is zero when counted with orientationor each such component admits an orientation reversing isomorphism. Since τ istrivial, σ ⊔ τ either adds components which are trivial or adds zero componentswhen counted with sign. Therefore, the only way σ ⊔ τ is trivial is if σ was alreadytrivial. (cid:3) Definition 11.
Let Q ( M ) be the set of chains isomorphic to α ⊔ β where α is trivialand β is degenerate. We allow α and β to be empty. Lemma 11. If σ is in Q ( M ) , so is ∂σ .Proof. Since the boundary of a trivial chain is trivial, we can focus on when σ isdegenerate. We have ∂σ = α ⊔ β with α trivial and β with small image. We needto show that β is degenerate. Since ∂ σ is trivial, we have ∂α ⊔ ∂β trivial. By theprevious lemma ∂β is trivial, hence β is degenerate as desired. (cid:3) The following is the key step to defining an equivalence relation based on chains in Q ( M ): Lemma 12. If σ ⊔ τ ∈ Q ( M ) for some τ ∈ Q ( M ) , we have σ ∈ Q ( M ) .Proof. We decompose σ as σ ⊔ σ . . . as in the lemma above. The hypothesis impliesthat each σ i is either trivial or has small image. Thus, we can write σ = α ⊔ β with α trivial and β having small image. We show that β is degenerate. We have ∂σ ⊔ ∂τ ∈ Q ( M ). By repeating the argument, this implies that ∂β is a union of asmall chain and a trivial chain. Therefore, β is degenerate. (cid:3) We are ready to define our chain complex:
Definition 12.
Define an equivalence on chains as follows: σ ∼ τ if σ ⊔ − τ is in Q ( M ) . We denote the resulting set by C ∗ ( M ) . Lemma 13. ∼ is an equivalence relation. The geometric boundary operator ∂ inducesthe structure of a chain complex on C ∗ ( M ) with addition given by disjoint union. roof. Reflexivity and symmetry are clear. To check transitivity, note that σ ⊔ − τ ∈ Q ( M ) and τ ⊔ − ρ ∈ Q ( M ) imply σ ⊔ − τ ⊔ τ ⊔ − ρ ∈ Q ( M ). By the previous lemma, σ ⊔ − ρ is in Q ( M ) as desired. The additive structure is induced by disjoint union:Given σ : P → M and τ : Q → M we must choose an embedding of P ⊔ Q in ourHilbert space H . However, any two such choices will be equivalent under our relationsince the difference will be isomorphic to a trivial chain. C ∗ ( M ) has inverses since σ ⊔ − σ is trivial for any σ . Finally, C ∗ ( M ) forms a complex since ∂Q ( M ) ⊂ Q ( M )and ∂ σ is always trivial. (cid:3) Note that C ∗ ( M ) has a natural grading given by the dimension of the chains. Definition 13.
Let H ∗ ( M ) denote the homology groups associated to the complex C ∗ ( M ) . Let us observe that cycles with small image are 0 in C ∗ ( M ): Lemma 14. If σ has small image and ∂σ ∼ , then σ ∼ .Proof. We have that ∂σ ∈ Q ( M ). Therefore, σ is degenerate and hence also in Q ( M ). (cid:3) As will be proved in subsequent sections, the homology groups introduced aboveare isomorphic to the singular homology of M .4. Cohomology
There is a variant of the groups that is based on proper rather than compact chains.These will give a geometric version of cohomology.
Definition 14. A cochain is a smooth oriented map σ : P → M where P is asmooth manifold with corners and σ is a proper map. Not to confuse the two distinct notions of orientation we will say that cochains are cooriented . Given σ : P → M and τ : P ′ → M and a diffeomorphism f : P → P ′ such that σ ◦ f = τ , we get an induced isomorphism det ( σ ) → f ∗ ( det ( τ )). Therefore,we can ask whether this map preserves orientations: Definition 15.
A cochain σ : P → M is said to be trivial if there exists a diffeo-morphism f : P → P with σ ◦ f = σ which changes the orientation of the map. The rest of the definitions go through as before and give rise to a chain complexwith the boundary operator given by geometric boundary.
Definition 16.
Let C ∗ ( M ) be the chain complex of proper maps. We denote itshomology groups by H ∗ ( M ) . ∗ ( M ) is graded by the codimension of the map σ : P → M . With this conven-tion, ∂ has degree 1. As we will show later, H ∗ ( M ) is a geometric realization of thesingular cohomology of M . Remark.
In the case that M is an infinite dimensional manifold, cochains are givenby proper Fredholm maps σ : P → M , where P is necessarily infinite dimensional.The rest of the construction proceeds with little change.5. Transversality and Pairing
In this section we use standard results on transversality to obtain a pairing: ∩ : H a ( M ) ⊗ H b ( M ) → H a − b ( M )The arguments are quite standard, but we need pay attention to the fact that per-turbations should preserve trivial and degenerate chains. Definition 17.
Maps σ : P → M , τ : Q → M are in general position if theysatisfy the following conditions:1. They are transverse on each open stratum.2. If P i a component of an open stratum of P with small image, then there existsa smooth map g : T → M of smaller dimension covering the image of P i which istransverse to all the strata of τ .3. If Q i a component of an open stratum of Q with small image, then there exists asmooth map g ′ : T ′ → M of smaller dimension covering the image of Q i , which istransverse to all the strata of σ . Now we exhibit enough perturbations to ensure transversality while preservingdegenerate chains. To this end, we have the following basic result:
Lemma 15.
There exists a smooth connected manifold P and a smooth map F : M × P → M such that D F is surjective at all points. We assume F ( · , p ) = Id on M for some p ∈ P . Using standard transversality arguments we have the following:
Lemma 16.
Given σ : P → M and τ : P → M there exists a ∈ P such that F ( · , a ) ◦ σ and τ are in general position. Given any two such a, b ∈ P , there exists apath γ : [0 , → P such that γ (0) = a , γ (1) = b and the map Σ : P × [0 , → M given by Σ( p, t ) = F ( σ ( p ) , γ ( t )) is transverse to τ . roof. This is a standard transversality argument. We need to ensure transversalityfor each open stratum as well as for some choice of maps g i : T i → M coveringcomponents with small image. Since each chain has at most a countable numberof components in each stratum, we can ensure simultaneous transversality by anapplication of Sard’s theorem. (cid:3) Definition 18.
Given σ and τ in general position, let σ ∩ τ = ( σ × τ ) − (∆) where ∆ is the diagonal in M × M . Note that σ ∩ τ is oriented and comes with a map to M . Also, since σ has compactdomain and τ is proper, σ ∩ τ is compact. The need for such specific perturbationsis to ensure taking intersections preserves degenerate and trivial chains: Lemma 17.
Assume σ and τ are in general position. If σ is in Q ( M ) , so is σ ∩ τ .Proof. If σ has a orientation reversing self-diffeomorphism f such that σ ◦ f = σ , then F ( · , a ) ◦ σ ◦ f = F ( · , a ) ◦ σ Therefore, σ ∩ τ has an induced orientation reversing diffeomorphism as well. If σ has small image contained in g : T → M , then F ( · , s ) ◦ σ has image contained in F ( · , s ) ◦ g as long as g is transverse to τ . Thus, σ ∩ τ has small image contained in g ∩ τ . Furthermore, if we assume σ is degenerate, we have σ ∩ τ has small image and ∂ ( σ ∩ τ ) = ∂σ ∩ τ ⊔ ± σ ∩ ∂τ is a union of s trivial chain and a chain with small image. (cid:3) Theorem 2.
Transverse intersections induce a well defined map: ∩ : H a ( M ) ⊗ H b ( M ) → H a − b ( M ) Proof.
Given a cycle σ : P → M and a cocycle τ : Q → M , we will use a generichomotopy F : M × [0 , → M to obtain a representative F ( · , ◦ σ of the homology class of σ that is transverse to τ . Suppose now that σ ∼ ∂σ where σ is transverse to τ . If σ is also transverse to τ , then ∂ ( σ ∩ τ ) ∼ σ ∩ τ as desired. If not, pick a generic homotopy F as above andlet σ ( p, t ) = F ( σ ( p ) , t )Since ∂ ( σ + ∂σ ) ∼ σ , we need only check that σ + ∂σ is equivalent to a transversechain. Let σ ( p, t ) = F ( σ ( p ) , t )By construction, σ + ∂σ ∼ σ ⊔ σ ( · , F , σ and σ ( · ,
1) willbe transverse to τ as desired. (cid:3) y a similar technique applied to proper maps , we obtain a geometric version ofthe cup product: H a ( M ) ⊗ H b ( M ) → H a + b ( M )6. Eilenberg-Steenrod Axioms
In this section we verify that our theory satisfies the Eilenberg-Steenrod axioms.We begin by computing the homology of a point:
Theorem 3. H k ( pt ) = 0 when n = 0 and H ( pt ) ∼ = Z .Proof. Since any map σ : P → pt is constant any such chain has small image when dim ( P ) >
0. If σ is a cycle, we have ∂σ a union of trivial and degenerate chains.Therefore, σ is degenerate and hence 0 in C ∗ ( pt ). To verify the claim in dimensionzero note that 0 dimensional cycles are finite collections of (oriented) points, while1-dimensional cycles are compact intervals. (cid:3) Next, we verify a version of functoriality:
Theorem 4.
Given a smooth map f : M → N between smooth manifolds, we have aninduced chain map f ∗ : C ∗ ( M ) → C ∗ ( N ) . f ∗ extends to a functor from the categoryof smooth finite dimensional manifolds and smooth maps to the category of chaincomplexes over Z .Proof. It suffices to note that the image of a trivial chain is trivial (with the sameorientation reversing diffeomorphism) and given a map σ with small image containedin g : T → M , we have f ◦ σ has image contained in f ◦ g : T → N . (cid:3) Theorem 5.
Given a smooth homotopy F : M × [0 , → N , we have that F ∗ ( · , ischain homotopic to F ∗ ( · , .Proof. The chain homotopy H is defined as follows. Given a chain σ : P → M let H ( σ ) : P × [0 , → M be the map with H ( σ )( p, t ) = F ( σ ( p ) , t ). We have ∂H ( σ ) = ( − | P | ( F ( σ, − F ( σ, H ( ∂σ ). (cid:3) In the case f is not smooth one can still define a map on the level of homology.Indeed, one can approximate any continuous f by a smooth f ′ . Such a choice isunique up to smooth homotopy. Therefore, the map on homology is still well definedas follows from the proof of the homotopy axiom. Remark.
The restriction that M is smooth is not significant. Since any finiteCW complex X has the homotopy type of a finite dimensional manifold (possiblynoncompact), we can define our homology groups for such X by choose a homotopyequivalent smooth model for X . iven an open set U ⊂ M we have the exact sequence:0 → C ∗ ( U ) → C ∗ ( M ) → C ∗ ( M ) /C ∗ ( U ) → C ∗ ( M, U ) as C ∗ ( M ) /C ∗ ( U ) and obtain the long exact sequence · → H ∗ ( U ) → H ∗ ( M ) → H ∗ ( M, U ) → · · · Finally, we verify a version of the excision axiom. Let U ⊂ M be an open subset and K ⊂ U be compact set. For excision, as well as later applications, we will need to cutcycles into pieces without changing the homology class. Consider a smooth function f : M → R . Let σ + = σ | ( f ◦ σ ) − [ p, ∞ ) σ − = σ | ( f ◦ σ ) − ( −∞ ,p ] and σ = σ | ( f ◦ σ ) − ( p ) Lemma 18.
Given σ : P → M with σ ∈ Q ( M ) , we have σ ± , σ ∈ Q ( M ) forgeneric p . Let σ : P → M with ∂σ ∼ τ where τ ∈ U ⊂ M . For generic p , we have [ σ ] = [ σ + ⊔ σ − ] in H ∗ ( M, U ) .Proof. If σ is trivial so are σ ± and σ since the involution descends to them. If σ isdegenerate, we have that im ( σ ) ⊂ im ( g ) for g : T → M . In addition, we have that ∂σ = α ⊔ β with β trivial and α covered by g ′ of smaller dimension. Therefore, bychoosing a generic p , we can assume its a regular value of g ′ , g and σ . Now, σ ± hassmall image since σ has small image. σ has image contained in g and hence hassmall image as well. We have ∂σ ± = ( ∂σ ) ± ⊔ ± σ while ∂σ = ( ∂σ ) . Since α ± has small image and β ± is we see that σ ± and σ aredegenerate as well.We apply the crease construction to conclude that ∂Cre ( σ ) = σ + ⊔ σ − ⊔ − σ ⊔ Cre ( ∂σ )Since ∂σ ∼ τ we have Cre ( ∂σ ) ∼ Cre ( τ ). Indeed, given a chain ρ ∈ Q ( M ), we have Cre ( ρ ) ∈ Q ( M ). This is clear for trivial chains. For degenerate chains, observe that Cre ( ρ ) has same image as ρ and hence is small. Also, ∂Cre ( ρ ) = Cre ( ∂ρ ) ⊔ − ρ ⊔ ρ + ⊔ ρ − This shows that ∂Cre ( ρ ) has small image as well. Since Cre ( τ ) ∈ C ∗ ( U ), we concludethat [ σ ] = [ σ + ⊔ σ − ] in H ∗ ( M, U ) as desired. (cid:3)
We can now prove the excision axiom:
Theorem 6.
The natural inclusion i ∗ : H ∗ ( M − K, U − K ) → H ∗ ( M, U ) is anisomorphism. e break the proof into two lemmas: Lemma 19. i ∗ is a surjection.Proof. Given a cycle σ in H ∗ ( M, U ), we show that it is homologous to σ ′ with im ( σ ′ ) ⊂ M − K Take a smooth function f : M → R with f = 0 on K and f > U . Let p ∈ (0 ,
1) be a regular value of f ◦ σ . Cutting by the preimage of p decomposes σ as σ + + σ − with im ( σ − ) ⊂ U . Since, [ σ ] = [ σ + ⊔ σ − ] in H ∗ ( M, U ) and [ σ − ] = 0 in C ∗ ( M, U ), we have [ σ ] = [ σ + ] in H ∗ ( M, U ). (cid:3) Lemma 20. i ∗ is an injection.Proof. Consider a cycle in H ∗ ( M − K, U − K ) represented by a chain σ . Suppose[ σ ] = 0 in H ∗ ( M, U ). Thus, there exists τ and η ∈ C ∗ ( U ) with ∂τ ⊔ − σ ∼ η . Sincethe image of σ avoids K , we can cut our chains by a generic hyperplane as above suchthat σ + = σ and σ − = ∅ . Since cutting by a generic plane preserves Q ( M ), we get( ∂τ ) + ⊔ − σ ∼ η + We have ∂ ( τ + ) = ( ∂τ ) + ⊔ τ . Therefore, ∂ ( τ + ) ⊔ − σ ⊔ − τ ∼ η + Since each chain in this union lies in M − K , [ σ ] = 0 in H ∗ ( M − K, U − K ) as well. (cid:3) Much of the previous discussion applies to the cohomology groups. For instance,given a smooth map f : M → N of manifolds without boundary, we get an inducedmap: f ∗ : H ∗ ( N ) → H ∗ ( M )defined by pullback. Note that it is only well defined on the homology level since wemay need to perturb a cocycle representative to ensure that it is in general position.In the special case of an inclusion of an open set, i : U ⊂ M , the map i ∗ is welldefined on the chain level since transversality is automatic. We define H ∗ ( M, U ) tobe the homology of the kernel of i ∗ .7. Thom Isomorphism
In this section we prove the Thom isomorphism theorem. As we will show, thisresult is a direct consequence of the definitions and does not require an inductiveMayer-Vietoris argument. Let M be a smooth connected m -manifold without bound-ary and V → M a real vector bundle over M of dimension n . Let π : V → M be theprojection and i : M → V be the zero section. Theorem 7. π ∗ : H ∗ ( M ) → H ∗ ( V ) is an isomorphism with inverse i ∗ . Similarly,the map i ∗ : H ∗ ( M ) → H ∗ ( V ) is an isomorphism with inverse π ∗ . roof. Follows immediately from the fact that both π ◦ i and i ◦ π are homotopic toidentity. (cid:3) Since we defined cohomology groups independently of homology, the fact that i ∗ isan isomorphism does not immediately imply the same for i ∗ . Let us restate the tworesults in geometric terms. The isomorphism for i ∗ simply asserts that any homologyclass in V may by pushed down to lie on the zero section. On the other hand, i ∗ identifies cocycles of different dimension. In essence, it asserts that a cohomologyclass is completely determined by its restriction to the zero section.Assume that M is compact and that V is oriented. Note, that this implies thatboth π and i are oriented maps. Since i is proper, we get a map induced by inclusion: i ∗ : H ∗ ( M ) → H ∗ + n ( V, V − i ( M ))Recall that H ∗ + n ( V, V − i ( M )) is defined by considering the subcomplex of C ∗ ( V )of chains that vanish on V − i ( M ). We have the following version of the ThomIsomorphism: Theorem 8. i ∗ : H ∗ ( M ) ∼ = H ∗ + n ( V, V − i ( M )) with inverse induced by i ∗ .Proof. The proof is very similar to the proof of excision. To show i ∗ is surjectiveconsider [ σ ] ∈ H ∗ ( V, V − i ( M )). Introduce a metric on V . We may cut σ by the unitsphere bundle to get [ σ ] = [ σ + ⊔ σ − ] with σ + supported away from the zero section.By hypothesis, σ + ∼
0. Therefore, we may replace σ by a chain contained in the unitdisk bundle. Projecting σ to the zero section shows that σ is homologous to a chainin im ( i ∗ ). Now, we establish surjectivity. Suppose i ∗ ( σ ) ∼ ∂τ in C ∗ ( V, V − i ( V )).Again, we may cut τ by the unit sphere bundle to produce τ − in the unit disk bundleof V with ∂τ − ∼ σ . Projecting to the zero section yields τ ′ with ∂τ ′ ∼ σ . (cid:3) We end this section by pointing out the relationship between our proof and themore standard proofs based on the notion of a Thom class. When the bundle isoriented, i : M → V may be viewed as a cohomology class [ U ] ∈ H n ( V, V − i ( V )). The restriction of [ U ]is any fiber V p is the origin and hence the generator of H n ( V p , V p − V ∗ p ). This is thecharacterization of the Thom class. The isomorphism i ∗ : H ∗ ( M ) → H ∗ + n ( V, V − i ( M ))may be viewed as the composition of pulling back by π ∗ : H ∗ ( M ) → H ∗ ( V )followed by intersecting (the geometric analogue of the cup product) with the class[ U ]. Note that the pullback, i ∗ ([ U ]) ∈ H n ( M ) is the Euler class associated to thebundle V . This corresponds the intersection of the zero section with a generic section. ne can also recast the Thom isomorphism is a slightly different form. Define H ∗ c ( M )to be the group generated by oriented cochains which have compact domain. If V isoriented, we have i ∗ : H ∗ ( M ) ∼ = H ∗ + nc ( V )induced by inclusion. 8. Morse Homology Theorem
In this section we prove the Morse homology theorem identifying the homologygroups of the complex we constructed with those of the Morse complex. There aremany versions of this theorem in the literature. We decided to include a proof sinceit has a direct generalization to the infinite dimensional theory.8.1.
Passing a Critical Point.
Let g be a Riemannian metric and f : M → R be a proper Morse function, i.e. all critical points of f are nondegenerate. We assumethat f is bounded below. Choose a metric on M such that ∇ f is complete. Wealso assume that there is at most one critical point at each level, although this isnot essential for the arguments that follow. Given a flow line γ : [ a, b ] → M with γ ′ ( t ) = −∇ f we define the energy of γ as Z ba | γ ′ ( t ) | dt = f ( a ) − f ( b )Let M c = f − (( −∞ , c )). We let F t ( M ) be the time t downward gradient flow. Thus, F t ◦ F τ = F t + τ The flow acts on chains by F t ( σ ) = F t ◦ σ .First we show that only the presence of critical points can change the topology: Lemma 21.
Given a < b . Assume there are no critical values of f in [ a, b ] . Let i : M a → M b be the inclusion. We have the isomorphism i ∗ : H ∗ ( M a ) → H ∗ ( M b ) Proof.
Given a cycle σ ∈ H ∗ ( M b ), F t ( σ ) lies in M a for sufficiently large t . Since F t ( σ )is homologous to σ , we have that i ∗ is surjective. Now, assume i ∗ ( σ ) = 0. Thus, ∂τ ∼ σ with τ ∈ H ∗ ( M b ). For large t , we have F t ( τ ) ∈ C ∗ ( M a ) and ∂ F t ( τ ) ∼ F t ( σ ).Since the gradient flow preserves the homology class, we have [ σ ] = 0 in H ∗ ( M a ) aswell. (cid:3) An easy compactness argument shows that: emma 22. Assume the f has a unique isolated critical point y at level c . Choose c ′ > c so close that M c ′ has no new critical points. Let U be a small neighborhood of y . There exists ǫ > such that any chain σ : P → M c ′ may be pushed by the flowdown to σ with im ( σ ) ⊂ U ∪ M c − ǫ . Let V be a finite dimensional vector space with dim ( V ) = dim ( M ). The Morselemma allows us to identify a neighborhood U of y with a neighborhood of the originin V = V + ⊕ V − and f : V → R with f ( v ) = −| v − | + | v + | + f ( y )Let ind ( y ) = dim ( V − )By taking ǫ to be sufficiently small in the previous lemma, we can assume that U contains the 2 √ ǫ ball in V . Theorem 9.
Let k = ind ( y ) . For ǫ > small, H k ( M c + ǫ , M c − ǫ ) = Z and H j ( M c + ǫ , M c − ǫ ) =0 for j = k .Proof. Let D √ ǫ denote the closed disk centered at 0 of radius 2 √ ǫ in V and let D ± (2 √ ǫ ) denote the closed disk in V ± . By flowing σ using the downward gradientflow, we may assume that the image of σ is contained in M c − ǫ ∪ D √ ǫ . By a smallperturbation, we may also assume that σ is transverse to D + (2 √ ǫ ).Our next task is to flatten out σ . Pick a smooth function φ : V → [0 , φ be 1 on D √ ǫ and 0 outside D √ ǫ . Consider the map H t : V → V with H t ( v − , v + ) = ( v − , (1 − tφ ( v )) v + )and t ∈ [0 , H = Id while H maps V + to 0 on D √ ǫ . H t ◦ σ provides the desiredhomotopy that flattens out σ near the origin. Let σ = H ◦ σ . Note that f ( σ ( p )) ≤ f ( σ ( p ))Thus, we can assume that points in P that don’t land in M c − ǫ map to V − in the localcoordinates. Also note that 0 is a regular value of this map since σ was transverse to D + (2 √ ǫ ).We may cut our cycle σ into two pieces σ − ⊔ σ +0 by taking the preimage of a suffi-ciently small sphere S − ( δ ) of radius δ < √ ǫ around the origin in V − . Here, we take σ +0 to be the piece contained inside the ball D − ( δ ) bounded by the sphere. By using he crease construction, we have [ σ ] = [ σ − ⊔ σ +0 ] in H j ( M c + ǫ , M c − ǫ ). Now argumentdivides into cases depending on the dimension of the cycle.Case j < k : We have that σ +0 is empty and thus σ = σ − is contained in theregion where f < c . It may be pushed to lie in M c − ǫ by the flow. Therefore, H j ( M c + ǫ , M c − ǫ ) = 0.Case j = k : Here σ +0 is a finite collection of disks mapping diffeomorphically to D − ( δ ).Note that for j = k , D − (2 √ ǫ ) defines an element in H k ( M c + ǫ , M c − ǫ ). By replacing σ with σ ⊔ m [ D − ǫ ], we can assume that σ +0 ∼ C ∗ ( M c + ǫ ). Therefore, [ σ ] + m [ D − ǫ ]is equivalent to a cycle strictly below c and hence 0 in homology. Thus, D − (2 √ ǫ )generates the homology in this dimension. It’s a free generator in H k ( M c + ǫ , M c − ǫ )since its intersection with D + ( √ ǫ ) is 1.Case j > k : In this final case, we note that since σ +0 maps to D − ( δ ) it has smallimage as it is contained in a manifold of smaller dimension. We have ∂σ +0 = ( ∂σ ) + ⊔ σ Since ∂σ ∼ ∂σ ) + ∼
0. Also, σ maps to S − ( δ ) and hence has smallimage. Therefore, σ +0 is degenerate. Thus, σ is homologous to a cycle with imagebelow the critical point and is therefore nullhomologous in H j ( M c + ǫ , M c − ǫ ). (cid:3) More generally, we may assume that a critical level has k critical points all of thesame index. In that case, H ∗ ( M c + ǫ , M c − ǫ ) ∼ = Z k This proof is along the same lines as the one above.8.2.
Self-Indexing Case.
In the previous section, we saw that if there is a uniquenondegenerate critical point y at f ( y ) = c then H j ( M c + ǫ , M c − ǫ ) = Z when j = ind ( y )and 0 else. Suppose that the next critical point z above y is at f ( z ) = d with ind ( z ) = ind ( y ) + 1Consider the composite map: H j +1 ( M d + ǫ , M d − ǫ ) ∂ ∗ −→ H j ) ( M d − ǫ ) i ∗ −→ H j ( M d − ǫ , M c − ǫ )Since we assumed there are no other critical points between y and z , we have: H ∗ ( M d − ǫ , M c − ǫ ) ∼ = H ∗ ( M c + ǫ , M c − ǫ ) ∼ = Z We wish to compute i ∗ ◦ ∂ ∗ . Since H ( M d + ǫ , M d − ǫ ) is generated by the unstable disk[ D − z ( ǫ )], i ∗ ◦ ∂ ∗ ([ D − z ( ǫ )]) = [ ∂D − z ( ǫ )] iewed as an element of H ∗ ( M d − ǫ , M c − ǫ ). Using the gradient flow, we may push thiscycle down to M c + ǫ . As discussed above, the image in H ∗ ( M c + ǫ , M c − ǫ ) is computedby counting intersections of the cycle with the disk D + y ( ǫ ) in a neighborhood of d .This a version of the familiar statement that boundary operator may be computedby counting the intersections of attaching/belt spheres.Now, assume that M is compact and f is self indexing. Thus, the critical pointsof index i are on level i of f . We have a filtration ∅ = M − ǫ ⊂ M ǫ ⊂ · · · M n − ǫ ⊂ M n + ǫ = M with H ( M j + ǫ , M j − ǫ ) = Z k , where k is the number of critical points of index j . Definition 19.
Let C f ∗ ( M ) = ( ⊕ j C fj , ∂ f ) be the free chain complex with C fj = H ∗ ( M j + ǫ , M j − ǫ ) The differential ∂ f : C f ∗ → C f ∗− , ∂ f = i ∗ ◦ ∂ ∗ arises from composing ∂ ∗ : H j ( M j + ǫ , M j − ǫ ) → H j − ( M k +1 ) with i ∗ : H j − ( M j − ǫ ) → H j − ( M j − ǫ , M j − ǫ ) Let H f ∗ ( M ) be the homology of the chain complex C f ∗ ( M ) . Theorem 10.
Given a self-indexing Morse function f as above, we have H f ∗ ( M ) ∼ = H ∗ ( M ) Proof.
Given the local computation, the remaining algebraic argument is identical tothe proof of the cellular homology theorem (see for example [10]). (cid:3)
Let us summarize what we have accomplished. Our only assumption on f is thatit is Morse and not necessarily Morse-Smale. From this we deduced the existence ofa chain complex generated by the critical points whose homology is isomorphic tothe geometric singular homology of M . The arguments, as well as those in the nextsection, make no use of gluing theory for gradient flow trajectories that is crucial inthe Floer construction of the Morse complex. Furthermore, when applied to a suitableinfinite dimensional variant of the theory, they give an alternative construction of theFloer chain complex.8.3. General Case.
The results of the previous section are not sufficient for ap-plications in Floer theory. The main issue is that the assumption that the Morsefunction is self-indexing is too restrictive. Indeed, the perturbations in Floer theoryare usually taken to be small and it is not clear to us how to perturb the relevantfunctional to obtain a self-indexing one. In this section we explain how given that ∇ f is Morse-Smale, we can define the Morse chain complex and show that its homologycoincides with the geometric cycle homology. e will make use of the following basic compactness result (see [13]): Theorem 11.
Let x and y be critical points of f .Given a sequence of possibly broken trajectories γ i : [ a i , b i ] → M with γ i ( a i ) → x and γ i ( b i ) → y , some subsequence converges to a possibly brokentrajectory between x and y . Assume we are given a proper Morse function f which is Morse-Smale with respectto some metric on M . The Morse-Smale condition implies that given critical points x and y with ind ( x ) ≤ ind ( y ), there are no gradient flow lines from x to y . In fact,there exist open neighborhoods x ∈ U x and y ∈ U y such that there are no gradientflow lines from U x to U y . Indeed, arguing by contradiction we can shrink the sets U x , U y to obtain in the limit a possibly broken gradient flow line from x to y . Definition 20.
A chain σ is said to be k − small if each critical point of index j ≥ k has a neighborhood U such that U ∩ F t ( σ ) = ∅ for all t ≥ . Definition 21.
Let C k ∗ ⊂ C ∗ ( M ) be the subcomplex generated by k -small chains. Wehave the filtration: C ∗ ⊂ C ∗ ⊂ . . . C dim ( M +1) ∗ = C ∗ ( M ) Lemma 23. H j ( C k ∗ ) = 0 when j > k .Proof. Given a cycle σ : P → M in C kj let us see what happens as we try to push it down by the flow. As we attempt topush it past a critical point x , two things can occur. If ind ( x ) > k , F t ( σ ) stays awayfrom an entire neighborhood of the critical point. Therefore, we focus on flowing σ past a critical point with ind ( x ) ≤ k < j . We will perturb σ to σ ′ in a smallneighborhood of x to be transverse to the stable manifold of x . Here, we use the localstable manifold V + of the local model V + provided by the Morse lemma. A crucialpoint is that this perturbation does not take us out of C k ∗ . Indeed, since the flow isMorse-Smale, a small neighborhood of x contains no points flowing arbitrary closethe critical points of index ≥ k . Therefore the homotopyΣ : P × [0 , → M perturbing σ has image different from σ only in a small neighborhood of x . We have j > k , therefore σ ′ intersects the stable manifold in a manifold of dimension at least1. By the arguments in the section on passing a critical points, we can modify σ ′ to σ ′′ locally around x so that σ ′′ lies below x and [ σ ′′ ] = [ σ ] in H ∗ ( C k ). By repeatingthis a finite number of steps, we can modify σ to be empty and thus 0 in H j ( C k ∗ ). (cid:3) emma 24. H j ( C k ∗ , C k − ∗ ) = 0 for j = k .Proof. The argument is quite similar to the proof of the previous lemma. We aregiven a chain σ ∈ C kj and ∂σ ∈ C k − j − . As before, we are trying to push our chainpast a critical point x . We need to check that all the constructions used in the lemmaregarding passing a critical point carry over to this setting. First we need perturb thechain in a neighborhood of x to be transverse to the stable manifold. The perturbationgives a chain Σ : P × [0 , → M with Σ | P × = σ and ∂ Σ = Σ | P × ⊔ Σ P × ⊔ Σ ∂ ( P ) × [0 , Note that Σ ∂ ( P ) × [0 , ∈ C k +1 ∗ since the perturbation is supported in a neighborhood of x and thus is zero in C k ∗ /C k +1 ∗ . Therefore, we can perturb the cycle to be transversewithout changing its homology class. Next, we project the cycle to manifold V − . Thisdoes not change our class. If j < ind ( x ) the intersection with V + is empty hence wecan flow past it. Otherwise we cut our σ to produce σ + ⊔ σ − with σ − strictly belowthe critical point. Since j > ind ( x ), σ + is degenerate and thus can be discarded.Therefore we can flow past any critical point x without altering the homology class.Eventually, the class is empty. (cid:3) Take D ± ( x m ) to be a disk in V ± ( x m ) around each critical point x m of index k smallenough to be in U x m . Note that D − ( x m ) is a cycle in C kk /C k − k . Indeed, the boundaryof such a disk can flow only to a critical point of index ≤ k − C kk − . Our next task is to verify that these disks freely generate H k ( C k ∗ , C k − ∗ ) ∼ = Z n .We say that chain σ lies below S ⊂ M if F t ( σ ) is disjoint from S for all t ≥ Lemma 25.
Any chain in C k ∗ can be pushed down by the flow to lie below ∪ m ∂D + ( x m ) Proof.
Consider F t ( σ ) for t very large. We claim that im ( F t ( σ )) ∩ ( ∪ m ∂D + ( x m )) = ∅ for all t sufficiently large. By contradiction, assume there are large times t i andtrajectories starting at points of σ and ending on some ∂D m . By compactness suchtrajectories must approach some critical point arbitrarily closely somewhere along theway. Let y be such a critical point. Note that ind ( y ) ≤ k since σ ∈ C k ∗ ( M ). Weclaim that ind ( y ) < k . First, if y = x m , the value of f along the trajectory would liestrictly below ∂D + m . If y = x m , part of the trajectory represents a flow line betweensmall neighborhoods of critical points of the same index. This is not possible since D + ( x m ) is chosen to lie in U x m . (cid:3) Lemma 26. H k ( C k ∗ , C k − ∗ ) ∼ = Z n , where n is the number of critical points of index k . roof. We define a map Ev : H k ( C k ∗ , C k +1 ∗ ) → Z n which will turn out to be be an isomorphism. By the previous lemma, we can pushany σ ∈ H k ( C k ∗ , C k +1 ∗ ) to lie below ∪ m ∂D + ( x m ). We may perturb such σ to betransverse to each D + ( x m ) by altering it near x m . Let Ev ( σ ) = ( a x , a x , · · · )be the intersections of σ with D ( x i ). Note that this does not depend on the smallperturbation since ∂σ lies in C k − ∗ and thus avoids U x m . We now assert that thisgives rise to a well defined intersection with the D + ( x m ). To see this, take σ below ∪ m ∂D + ( x m ) and assume σ = ∂τ with τ ∈ C k ∗ . Taking T sufficiently large F T ( τ )lies below ∪ m ∂D + ( x m ) and ∂ F T ( τ ) = F T ( σ ). However, σ and F T ( σ ) are homotopicon the complement of ∪ m ∂D + ( x m ). Therefore, they have the same intersection with ∪ m D + ( x m ). Thus, Ev vanishes on boundaries and is therefore well defined. Ev is surjective since D − ( x a ) ∩ D + ( x b ) = δ ab . Ev is also injective since if σ has zeroalgebraic intersection with all D + ( x m ) we can modify it to lie below all critical pointsof index k , as in the previous lemmas. (cid:3) With these results in place, we can proceed to define the Morse complex as C fi ( M ) = H i ( C i ∗ , C i − ∗ )The differential ∂ f : C f ∗ → C f ∗− , ∂ f = ∂ ∗ ◦ i ∗ arises from the connecting homomor-phism: ∂ ∗ : H k ( C k ∗ , C k − ∗ ) → H k − ( C k − ∗ )and the one induced by inclusion: i ∗ : H k − ( C k − ∗ ) → H k − ( C k − ∗ , C k − ∗ )Geometrically, the differential may be interpreted as follows. The generators for H k ( C k ∗ , C k − ∗ ) can be taken to be small oriented disks D − x m in the unstable manifoldsaround the critical points as above. Let D + y l be the stable cooriented disks aroundcritical points of index k −
1. Take the intersections F T ( ∂D − ( x m )) ∩ D + ( y l ) for T large. These give you the coefficients of the incidence matrix. By the same argumentsas in [10], we have the following Morse Homology Theorem: Theorem 12.
There exists an isomorphism H ∗ ( M ) ∼ = H f ∗ ( M )Finally, we would like to remark that there is a version of the Morse homologytheorem for cohomology. In this case, one considers the upward gradient flow whichexists as long as ∇ f is complete. We can define H ∗ f ( M ) much like before. We have: Theorem 13.
There exists an isomorphism H ∗ ( M ) ∼ = H ∗ f ( M ) . Equivariant Theory
Basic Construction.
The theory of geometric cycles has an equivariant gen-eralization. Below, we describe a construction which can be viewed as the geometricanalogue of the Cartan construction for deRham theory. A similar theory for singularhomology appears in [14]. For this section, assume M has a smooth action S × M → M Definition 22.
Let J be the map taking σ : P → M to J ( σ ) : S × P → M with J ( σ )( e iθ , p ) = e iθ σ ( p )We define our complex as follows. Let C + ∗ = ( C ∗ ( M ) ⊗ Z [ u ] , ∂ J )with u a formal variable of degree − ∂ J ( σu k ) = ( ∂σ ) u k + J ( σ ) u k +1 The grading of σu k is dim ( σ ) − k . Lemma 27. ∂ J = 0 .Proof. We compute ∂ J σ = ∂ J ( ∂ ( σ ) + J ( σ ) u ) = ∂ ( σ ) + J ( ∂σ ) u + ∂ ( J σ ) u + J ( σ ) u Observe that since J ( P ) is already S invariantim J σ = im( J σ )and im( ∂J σ ) = J ( ∂J ( σ ))By definition, this implies that J ( P ) is degenerate and hence zero in C ∗ ( M ). J ( ∂σ ) : S × ∂P → M while ∂ ( J σ ) : ∂ ( S × P ) → M . Therefore, J ( ∂σ ) = − ∂ ( J σ ). (cid:3) Definition 23.
Let H + ∗ ( M ) be the homology associated with the complex C + ∗ ( M ) . We have the following analogue of the Gysin sequence:
Theorem 14.
There is a long exact sequence: · · · → H + k +2 ( M ) → H + k ( M ) → H k ( M ) → . . . roof. We have an exact sequence of complexes:0 → C + ∗ +2 ( M ) u −→ C + ∗ ( M ) → C ∗ ( M ) → (cid:3) Theorem 15.
Given a smooth S -equivariant map f : M → N , we have an inducedmap f ∗ : H + ∗ ( M ) → H + ∗ ( N ) Proof.
We define f ∗ ( σu k ) as ( f ◦ σ ) u k . If σ : P → M , we have J ( f ◦ σ ) : S × P → N with J ( f ◦ σ )( e iθ , p ) = e iθ f ( σ ( p ))Equivariance of f implies that f ∗ ( J ( σ ))( e iθ , p ) = f ( e iθ σ ( p )) = e iθ f ( σ ( p ))Therefore, J ( f ∗ ) = f ∗ ( J ) as desired. (cid:3) The homotopy axiom is also easy to verify:
Theorem 16.
Given an equivariant map H : M × [0 , → N , we have that H ∗ ( · ,
0) = H ∗ ( · , . We now introduce two variants of the equivariant construction:
Definition 24.
Let H ∞∗ ( M ) be the homology associated with the complex C ∞∗ ( M ) = ( C ∗ ( M ) ⊗ Z [ u, u − ] , ∂ J ) and H −∗ ( M ) be the homology of the complex C −∗ ( M ) = C ∗ ( M ) ⊗ Z [ u, u − ] /u · ( C ∗ ( M ) ⊗ Z [ u ])Since localization is exact, we have: Theorem 17. H ∞∗ ( M ) ∼ = u − H + ∗ ( M ) Theorem 18.
There is a long exact sequence: · · · → H + k +2 ( M ) → H ∞ k ( M ) → H − k ( M ) → . . . Proof.
This follows from the short exact sequence of complexes:0 → C + ∗ +2 ( M ) u −→ C ∞∗ ( M ) → C −∗ ( M ) → (cid:3) emark. One can show (see [15] for a discussion) that H ∞∗ ( M ) is the contributionto the homology coming from the fixed points of the action. In particular, one hasthe localization formula: H ∞∗ ( M ) ∼ = H ∗ ( M fix ) ⊗ Z [ u, u − ]In the case M is infinite dimensional, one must consider a completion of the groupswith respect to u for the localization formula to hold.9.2. Cohomology and Pairing.
We can define cohomological versions of thesegroups that will turn out to be dual to the ones in the previous section.
Definition 25.
Let v be a formal variable of degree 2. Let H ∗ + ( M ) (resp. H ∗− ( M ) , H ∗∞ ( M ) ) be the group associated to the complex ( C ∗ ( M ) ⊗ Z [ v ] , ∂ J ) (resp. ( C ∗ ( M ) ⊗ Z [ v, v − ] , ∂ J ) , ( C ∗ ( M ) ⊗ Z [ v, v − ] /v ( C ∗ ( M ) ⊗ Z [ v ]) , ∂ J )) . Now, we discuss the cohomology pullback map which will use transversality theory.We state the results for the “+” version, but they have straightforward generalizationsto all the versions.
Theorem 19.
Given an equivariant map f : M → N , we have an induced map f ∗ : H ∗ + ( N ) → H ∗ + ( M ) . This construction is functorial.Proof. Given a cocycle σ + σ v · · · our first task is to perturb it to be transverse to f .We must not change the equivariant cohomology class during the perturbation. Usingusual transversality arguments, let ∂τ = σ − σ ′ with σ ′ transverse to f . Therefore,we may alter our class to be σ ′ + ( σ − J τ ) v + · · · Now, perturb σ − J τ to be transverse to f . This time the correction lies in the v term. We may continue in this fashion until the terms become degenerate. Therefore,we may assume that σ = σ + σ v · · · is transverse to f . We set f ∗ ( σ ) = σ ∩ f + σ ∩ f v · · · We need to check that f ∗ commutes with ∂ J on transverse chains. It is clear that ∂ ( f ∗ ( σ )) = f ∗ ( ∂σ )If σ i : P → N we have f ∗ ( σ i ) : Q → M where Q = ( p, m ) with f ( m ) = σ ( p ). Therefore, J ( f ∗ ( σ i )) : S × Q → M Meanwhile, f ∗ ( J σ i ) : Q ′ → M here Q ′ = ( e iθ , p, m ) with f ( m ) = e iθ σ i ( p ). The diffeomorphism φ : Q → Q ′ sending ( e iθ , p, m ) to ( e iθ , p, e iθ m ) provides the isomorphism of J ( f ∗ ( σ i )) and f ∗ ( J ( σ i )).Therefore, ∂ J ( f ∗ ( σ )) = f ∗ ( ∂ J σ ) as desired. (cid:3) By a similar pullback argument, we have the homotopy axiom:
Theorem 20.
Given an equivariant map H : M × [0 , → N , we have that H ∗ ( · ,
0) = H ∗ ( · , Lemma 28.
There is a pairing H a − ( M ) ⊗ H + a ( M ) → Z Proof.
Let us say that σ is strongly transverse to τ τ ⋔ σ and τ ⋔ J ( σ ). Givenchains Σ ni =0 σ i v i ∈ C + ∗ ( M ) and Σ i = − n τ i u i ∈ C −∗ ( M ) with σ i strongly transverse to τ − i we define the intersection as Σ ni =0 σ i ∩ τ − i ) ∈ Z The calculation below is based on the following observation. Given σ : P → M and τ : Q → M where σ has dimension i and τ has codimension i −
1, we have J ( σ ) ∩ τ ) = − | σ | +1 σ ∩ J ( τ ))Indeed, we have the bijection ( e iθ , p, q ) = ( p, e − iθ , q )for triples with e iθ σ ( p ) = τ ( q ).We have ∂ J ( σ ) ∩ τ + ( − | σ | σ ∩ ∂ J τ )= Σ ni =0 ∂σ i + J σ i − ) ∩ τ − i + ( − | σ | σ i ∩ ( ∂τ − i + J τ − i − ))= Σ ni =0 ∂σ i ) ∩ τ − i ) + ( − | σ | σ i ∩ ∂τ − i ) = Σ ni =0 ∂ ( σ i ∩ τ − i )) = 0This calculation implies that the intersection count is well defined. To complete theconstruction we must check that the chains may be perturbed to be transverse withoutchanging the homology class. Assume inductively that σ through σ k − are stronglytransverse to all τ j . Let ρ : P k × [0 , → M e a generic homotopy of σ k such that ρ ( · ,
0) = σ k and ρ ( · ,
1) = σ ′ k is stronglytransverse to τ − j . Taking ∂ J ρ we may replace σ k by σ ′ k + ρ ′ where ρ ′ is a generichomotopy of ∂σ k . However, ∂σ k ∼ J ( σ k − ) since σ is a cycle. Since σ k − is stronglytransverse to all τ j , we may assume by a generic choice of ρ that ρ ′ equivalent on thelevel of chains to a strongly transverse representative. After finitely many steps, weobtain a chain where the only non transverse chains have coefficient u l so large that τ − l is empty. (cid:3) Remark.
It is not immediate how to define a cup product on H ∗ + ( M ) directly.Indeed, on M × M with the diagonal S -action, the operator J does not decomposeas an operator on the factors. Thus, given cohomology classes a, b ∈ H ∗ + ( M ), a × b does not define an equivariant class in M × M . In fact, the difference { a, b } := J ( a ∩ b ) − J ( a ) ∩ b − ( − | a | a ∩ J ( b )is a finite dimensional analogue of the string bracket of Chas-Sullivan (see [2]).9.3. Some Calculations.
Here are a few simple calculations:
Lemma 29. H + ∗ ( pt ) ∼ = Z [ u ] ; H ∞∗ ( pt ) ∼ = Z [ u, u − ] ; H −∗ ( pt ) ∼ = Z [ u, u − ] / Z [ u ] Proof.
These follow directly from the fact that for a point, both J and ∂ are zero. (cid:3) Lemma 30. H + ∗ ( S ) ∼ = Z ; H ∞∗ ( S ) ∼ = 0 ; H −∗ ( S ) ∼ = Z Proof. H + ( S ): Let σ = σ + σ u + σ u · · · be a cycle and let id : S → S be the identity map.Suppose σ is a cycle of degree >
1. Therefore, σ i is automatically small. We have ∂σ ∼
0, and thus σ is degenerate while σ i is degenerate for i > dim ( σ i ) > σ has degree 1. Since dim ( σ i ) > i > σ i ∼ i >
0. Furthermore, ∂ J τ = ∂τ . Therefore, in degree 1, we are reduced to the ordinary homology of S which has id as a free generator.Suppose σ has degree 0. σ i ∼ i > dim > σ = σ + σ u with J ( σ ) ∼ − ∂σ . σ is a finitecollection of oriented points. Therefore, J σ is a finite number of copies of i . By thecomputation of the ordinary homology of S , J ( σ ) ∼ − ∂σ implies that the algebraiccount of σ is trivial. Thus, there exists τ with ∂τ = σ . Therefore, we can assumethat σ = σ u . In this case, σ is degenerate since it has small image and ∂σ ∼ σ has degree <
0. We can factor σ = u k ( σ ′ ) with ∂ J σ ′ = 0 and degree σ ′ either 0 or 1. If degree σ ′ is 0, we have σ ′ = ∂ J τ hence σ = ∂ ( u k τ ). Now, assumethat the degree of σ ′ is 1. If k > σ = u k − ( uσ ′ ). Since uσ ′ is a cycle f degree >
1, we have ∂ J τ = uσ ′ and thus ∂ J σ = ∂ J ( uτ ). Finally, we deal withthe case σ = uσ ′ where σ ′ has degree 1. We have [ σ ′ ] = n [ id ] in H + ∗ ( M ). However, ∂ J ( pt ) = id [ u ] which implies that σ is a boundary. H ∞ ( S ): By localization, H ∞ ( S ) = u − H + ( S ) = 0Of course, a direct argument is also possible. H − ( S ): This follows from the long exact sequence connecting the groups. Notethat in this case [ pt ] is a cycle since ∂ J [ pt ] = id [ u ] = 0and u = 0. id : S → S on the other hand is a boundary since [ S ] = ∂ J [ pt ] u − . (cid:3) Relation to the Borel Construction.
Let us first recall the Borel construc-tion of equivariant homology. Let S ∞ be the unit sphere in an infinite complex Hilbertspace. S ∞ is contractible and has a smooth S action. We let C P ∞ be the quotient S ∞ /S . Let M be a smooth manifold with a smooth S -action. The Borel model of M is defined as M S = ( S ∞ × M ) /S Let H B ∗ ( M ) = H ∗ ( M S )In this section we demonstrate a natural isomorphism H −∗ ( M ) ∼ = H B ∗ ( M ) Remark.
Instead of introducing infinite dimensional manifolds one may take a finitedimensional approximation S k to S ∞ and define H B ∗ ( M ) as the limitlim k →∞ H ∗ (( M × S k ) /S )As a preliminary step, we have the following lemma: Lemma 31.
Assume that M has a free S action. We have H ∗ ( M/S ) ∼ = H −∗ ( M ) with the map induced by the projection π : M → M/S roof. Given a chain σ = P ni =0 σ i u − i ∈ H −∗ ( M ), let F ∗ ( σ ) = σ ∈ C ∗ ( M/S ). Tosee that F ∗ is a chain map note that F ∗ ( J ( σ )) is always degenerate and hence 0 in C ∗ ( M/S ). The rest of the argument is a standard application of the Mayer-Vietorissequence. Let U ⊂ M/S be a small ball. The restriction of π to U is equivalent tothe projection S × U → U . Since H −∗ ( S × U ) ∼ = Z with the generator given by p × q for any ( p, q ) ∈ U , we have the isomorphism H −∗ ( S × U ) ∼ = H ∗ ( U ) induced by F ∗ . We may cover M/S by convex balls andproceed by induction to deduce the general case. (cid:3) Corollary 1.
The projection π : M × S ∞ → M S induces an isomorphism π ∗ : H −∗ ( M × S ∞ ) ∼ = H ∗ ( M S )The isomorphism H −∗ ( M ) ∼ = H B ∗ ( M ) now follows from the following lemma: Lemma 32.
The equivariant projection π : M × S ∞ → M induces an isomorphismon homology.Proof. As S ∞ is contractible, the result is a consequence of the following general fact.Suppose we have an equivariant map f : M → N . If f ∗ : H ∗ ( M ) → H ∗ ( N )is an isomorphism, then so is f ∗ : H −∗ ( M ) → H −∗ ( N )The proof of this fact is the following filtration argument. Let C −∗ be the chaincomplex of either M or N . Let F k C −∗ ⊂ C −∗ be the subcomplex generated by sums P ki =0 σ i u − i . We have F k C −∗ /F k − C −∗ = C ∗− k Therefore, since C a = 0 for a <
0, in each degree l the filtration F k C − l is stable for k sufficiently large. The conclusion follows from the 5-lemma by comparing the longexact sequences for the pair ( F k C −∗ , F k − C −∗ ). (cid:3) Chern Classes and Equivariant Cohomology.
Assume π : V → M is a complex vector bundle of dimension n . In this section we describe a constructionof Chern classes using equivariant cohomology. In particular, transversality theoryplays a major role in the construction. Observe that V has a natural S -actioninduced by the restricting the action of C ∗ to S on each fiber. We think of M ashaving a trivial S -action. Let i : M → V be the zero section. heorem 21. We have i ∗ : H ∗ + ( V ) ∼ = H ∗ + ( M ) ∼ = H ∗ ( M ) ⊗ Z [ v ] Proof.
Follows from the fact that π ◦ i = Id , while i ◦ π is equivariantly homotopic tothe identity. (cid:3) Recall that the Thom class U is given by i : M → V . In this section, we view it asequivariant cohomology class in H ∗ + ( V ). Definition 26.
Let Ch ( V ) = i ∗ ( U ) ∈ H ∗ + ( M ) be the total Chern class. Note that Ch ( V ) decomposes as[ ch n ( V )] + [ ch n − ( V )] v + [ ch n − ( V )] v · · · with ch n − i ( V ) of degree 2( n − i ) Let us unravel this definition. We need to modify U to produce an equivariant class transverse to i : M → V . Take τ with ∂τ ∼ U − U ′ where U ′ is transverse to i . We have U − ∂ J τ = U ′ − ( J τ ) v Now, pick τ such that J τ − ∂τ is transverse to i . This time the correction termis ( J τ ) v . Repeat until the correction term is degenerate and hence can be ignored.We have produced a chain U ′ + a v + a v + · · · such that each term is transverse to i . We can pull this back to obtain Ch ( V ) asdesired. In particular, this implies that ch n ( V ) is the Euler class of V . Example.
Here is the simplest version of this construction. Let us consider C → pt as a line bundle over the origin. We define the homotopy H : [0 , → C with H ( t ) = t ⊂ C . The resulting section is 1 ⊂ C that is now transverse to 0 andits pullback is the empty set. Thus, the first Chern class is 0. According to ourconstruction, we must examine the map σ : S × [0 , → C given by mapping ( e iθ , t ) to e iθ t . σ does not have zero as a regular value. However,any other point in the open unit ball is a regular value of σ and has intersection 1with the chain σ . A small perturbation of σ produces a chain σ ′ that has intersection1 with the origin. Therefore, the pullback of the zero section is 1 ∈ H ∗ ( pt ). A versionof this argument applies generally to a complex vector bundle V → M to concludethat ch ( V ) = 1. Lemma 33.
We have Ch ( V ⊕ V ) = Ch ( V ) ∪ Ch ( V ) . roof. Let U i be the equivariant cochain in C ∗ + ( V i ) given by the zero section s i : M → V i Consider the vector bundle V ⊕ V over M × M . The cochain U × U ∈ C ∗ + ( V ⊕ V )given by the zero section i : M × M → V ⊕ V is the Thom class of V ⊕ V . Since, π ∗ i : H ∗ + ( M ) → H ∗ + ( V i )is an isomorphism, we can take π ∗ i ( Ch ( V i )) = U ′ i as a cocyle that also represents theThom class U i . In other words, U i − U ′ i ∼ ∂ J τ i We claim that U ′ × U ′ represents the Thom class of V ⊕ V . This follows from theLeibniz rule for ∂ J . Indeed, one must check that J ( σ × τ ) = J ( σ ) × τ in the specialcase where τ has an S -action. If σ : P → V and τ : Q → V , the map sending( e iθ , p, q ) to ( e iθ , p, e iθ q ) identifies J ( σ × τ ) with J ( σ ) × τ . As an application of this,we have that ∂ J ( τ × U ) ∼ U × U − U ′ × U and thus U × U is homologous to U ′ × U . Similarly, U ′ × U is homologous to U ′ × U ′ . The desired conclusion now follows from pulling back Ch ( V ⊕ V ) to M viathe diagonal. (cid:3) Theorem 22. Ch ( V ) agrees with the usual definition of the total Chern class.Proof. By the splitting principle [8], it suffices to treat the case of a line bundle. (cid:3)
Steenrod Operations and Equivariant Cohomology.
Let V → M be areal vector bundle. V has a natural fiberwise Z -action given by( m, v ) ( m, − v )As in the complex case, the Thom class U is given by i : M → V where i is theinclusion of M as the zero section. We view U as a Z -equivariant cohomology classin H ∗ + ( V ; Z ). In direct analogy with our construction of Chern classes, one maydefine the total Stiefel-Whitney class: Definition 27.
Let SW ( V ) = i ∗ ( U ) ∈ H ∗ + ( M ; Z ) be the total Stiefel-Whitney class. The verification of the product formula is identical to the case of Chern classes asdiscussed above.We may also use equivariant cohomology to define Steenrod operations. Considerthe action of Z on M × M that sends ( x, y ) to ( y, x ). Let∆ : M → M × M e the inclusion of M as the diagonal. Given a cohomology class τ : Q → M , notethat τ × τ defines an equivariant class in H ∗ + ( M × M ; Z ). Definition 28.
The total Steenrod square class is defined as Sq ( τ ) = ∆ ∗ ( τ × τ ) ∈ H ∗ + ( M ; Z )The top degree part of Sq ( τ ) is just the cup product τ ∪ τ . The verification of thefundamental properties of the Steenrod operations, including the Cartan formula,is nearly identical to the case of Chern classes. In addition, a neighborhood of thediagonal in M × M may be identified with the tangent bundle of M in a manner thatrespects the Z -action. It follows that the total Stiefel-Whitney class of the tangentbundle coincides with the Steenrod square of the diagonal.10. Relation To Other Theories
In this section we construct natural maps between H ∗ and various forms of (co)homologyfound in the literature. The fact that the natural maps induce isomorphisms followfrom the standard Mayer-Vietoris arguments and thus we mostly restrict our atten-tion to describing the relevant maps.The standard construction of singular homology H sing ∗ ( M ) is based on mappings σ : ∆ n → M where ∆ n is an n -simplex. If M is smooth, one may restrict to smooth mappings([7]). By definition, each smooth σ is a well defined chain in M . Since the definitionof the boundary maps are compatible, we have a well defined map H sing ∗ ( M ) → H ∗ ( M )There is a variant of singular theory that is based in mappings of cubes (see [11]) σ : [0 , n → M To obtain the correct chain complex one must quotient out degenerate cubes. Bydefinition, these are mappings σ that are independent of one of the coordinates. Notethat such cubes are also degenerate from our point of view. Indeed, lets say that σ : [0 , n = [0 , n − × [0 , → M is independent of the last coordinate. It follows that σ has small image. In addition, ∂σ : ∂ ([0 , n − ) × [0 , ⊔ [0 , n − × ∂ [0 , → M is a union of a chain with small image and a trivial chain. If H cube ∗ ( M ) denotes cubicalhomology, we have defined a natural map H cube ∗ ( M ) → H ∗ ( M ) et Ω ∗ ( M ) be the chain complex of smooth differential forms on M . Integrationdefines a map Ω ∗ deRham ( M ) → C ∗ ( M ; R ) ∗ To see that this map is well defined, we must check that a form ω integrates to zeroon trivial chains and chains with small image. Indeed, if σ : P → M has small image, σ ∗ ( ω ) vanishes since the rank of σ is less than the dimension of P atevery point in the domain. If σ is trivial, let φ : P → P be an orientation reversingdiffeomorphism that commutes with σ . We have σ ∗ ( ω ) = φ ∗ ◦ σ ∗ ( ω ). On the other hand, since φ reverses orientation, Z P φ ∗ ◦ σ ∗ ( ω ) = − Z P σ ∗ ( ω )Therefore, R P σ ∗ ( ω ) = 0.Let us now discuss the equivariant situation. Assume that S acts smoothly on M . Let ξ denote the vector field induced by the infinitesimal action. The Cartanconstruction gives us a deRham version of equivariant cohomology as follows. LetΩ ∗ S ( M ) be the complex of S -invariant forms. Form a new complex(Ω ∗ ( M ) ⊗ R [ v ] , d S )where v has degree 2 and the differential acts as d S ( ω ) = dω + i ξ ωv The resulting cohomology groups H S ,deRham ∗ ( M ) are the Cartan model for the equi-variant cohomology of M [15]. We define a chain mapΩ ∗ deRham ( M ) ⊗ R [ v ] → C −∗ ( M ; R ) ∗ as follows. Given a chain P Ni =0 σ i u − i ∈ C − k ( M ) with σ i of dimension k − i and acochain N X i =0 ω i v i ∈ Ω ∗ deRham ( M ) ⊗ R [ v ]we have the evaluation map N X i =0 Z P i σ ∗ i ( ω i ) ∈ R To check that this defines a chain map, we use Stoke’s theorem and the fact that R S × P σ ∗ ( ω ) = R P σ ∗ ( i ξ ω ) as ω is S -invariant. Universal Coefficients
The proof of the universal coefficient theorem for homology has a subtlety in ourcase. As defined, it is not obvious that C ∗ ( M ) is free. However, we have: Lemma 34.
For any M , C ∗ ( M ) is torsion-free.Proof. Suppose there is σ ∈ C ∗ ( M ) with kσ ∼ k ∈ Z + . As in proof oflemma 10, we decompose σ as σ = σ ⊔ σ . . . where each σ i consists of the union of isomorphic connected components (up to ori-entation) of σ and no σ i , σ j share isomorphic components when i = j . Since kσ ∼ kσ i is trivial or has small image for each i . This implies thateach σ i is either trivial or has small image. We can ignore all the trivial componentsand assume that each σ i has small image and thus σ has small image. By hypothesis, k∂σ ∼
0. By repeating the argument, it follows that ∂σ is a union of a trivial chainand a chain with small image. This implies that σ ∼ (cid:3) By standard homological arguments (see [6]), C ∗ ( M ) is thus a flat Z -module. Sup-pose A is a Z -module. Let H ∗ ( M ; A ) denote the homology of C ∗ ( M ) ⊗ Z A . Wemay now use the standard homological arguments (see [6]) to prove the universalcoefficients theorem: Theorem 23.
For any Z module A , we have the exact sequence → H ∗ ( M ; Z ) ⊗ A → H ∗ ( M ; A ) → T or Z ( H ∗− ( M ; Z ) , A ) → Duality
Our definition of cohomology groups is geometric and does not follow the standardroute of dualizing the homology chain complex. Recall that we have defined a capproduct H a ( M ; Z ) ⊗ H b ( M ; Z ) → H a − b ( M ; Z )by taking intersections of generic transverse representatives for our cycles. Letting b = a and taking coefficients in a field K , we get a map: D : H a ( M ; K ) → Hom ( H a ( M ) , K ) Theorem 24.
The map D is an isomorphism.Proof. This is a variant of the standard Mayer-Vietoris argument (see [8]). One covers M by convex balls and inducts on the number of balls in a cover. The inductive stepconsists of recognizing that for a open ball in R n , we have a perfect paring between thecycle represented by a point and the cocyle represented by the entire open ball. (cid:3) emark. Our version of geometric duality does not assume M is compact. Now,assume that M is a compact oriented manifold of dimension n . We have an identifi-cation H ∗ ( M ; Z ) ∼ = H n −∗ ( M ; Z )sending a cycle σ of dimension k to the cocyle represented by σ of codimension n − k .Note that an oriented chain may be viewed as a cooriented cochain using the orien-tation of M . From our point of view, this last isomorphism is basically a tautologysince we do not define cohomology groups by dualizing homology. Therefore, the realcontent of duality lies in the nondegeneracy of the intersection pairing.13. Spaces with a Functional
Basic Definitions.
So far, we have worked exclusively with smooth manifoldsand smooth maps. In a sense, this is not a great loss of generality since any finiteCW complex has the homotopy type of a smooth manifold (possibly noncompact).However, there are certain natural geometric constructions that are not possible ifone restricts to smooth spaces. For example, given a subset S ⊂ M one may form thequotient M/S . However, even if S is a smooth submanifold, M/S does not in generalinherit a smooth structure. For instance, the construction of cones and suspensionsthat is central to homotopy theory leads to singular spaces. One of the goals of thissection is to provide a language to model these nonsmooth phenomena in the smoothcategory. This leads us to the notion of a space with functional and morphismsbetween such spaces. This formalism plays a crucial role in the construction of Floertheory in [5] and we hope it is also illuminating in finite dimensions.
Definition 29. A space with a functional ( M, f ) , is a pair where M is a smoothfinite dimensional manifold without boundary and f : M → R is a continuous function. The purpose of introducing f is that it provides a way of characterizing the pointsin M that approach ±∞ . We come to the definition of homology groups of ( M, f ): Definition 30.
Let P be a smooth oriented manifold with corners. A smooth map σ : P → M is a chain if given p i ∈ P , either p i has a converging subsequence or lim i →∞ f ( σ ( p i )) = −∞ Intuitively, the only source of noncompactness of σ comes from points going to −∞ . We may now define trivial and degenerate chains as in our definition of geo-metric homology in section 3. We let H ∗ ( M, f ) denote the resulting groups. Theyare graded by the dimension of the chains. xample 1. Consider ( R , f ) for several different choices of f . If f = 0, we re-cover the usual homology of R . If f ( t ) = t then in fact H ∗ ( R , t ) = 0. Indeed, given apoint m ∈ R , we may form the homotopy( −∞ , → R mapping t ∈ ( −∞ ,
0] to m + t . The same homotopy shows that all 1-cycles are trivialas well. If we take f ( t ) = − t , we have H ( R , − t ) = 0 by a similar argument. On theother hand, this time H ( R , − t ) = Z . It is generated by the identity map R → R .This cycle is nontrivial since it pairs with a point to give 1. Example 2.
Given a vector bundle V → M , fix a metric on V . Let f ( m, v ) = −| v | .One may view ( V, f ) as the smooth analogue of the Thom space construction. Notethat any fibre defines a homology class in H ∗ ( V, f ), provided V is oriented. Example 3.
Let w be a closed 1-form on M and let f M be a cover. Suppose that thepullback of w is an exact 1-form df . Consider ( f M , f ). H ∗ ( f M , f ) is the topologicalanalogue of Novikov homology (see [9]). Indeed, in addition to compact cycles, weallow for countable sums of compact cycles as long as the images tend to −∞ . Example 4.
We say that (
M, f ) is pointed if each component of M contains asequence that tends to −∞ . For such spaces, H ( M, f ) vanishes since a point in M may be moved to −∞ by a path γ : ( −∞ , → M . Given any manifold M , we mayform a pointed space by removing a point p i from each component M i and taking f i : M i − p i → R to be a function that approaches −∞ as points approach p i . For instance, startingfrom S we obtain ( R , − t ) by this procedure. Given two pointed spaces ( M, f ) and(
N, g ) we have the product ( M × N, f + g ). This is actually the smash product forpointed spaces. For example, ( R n , −| x | ) is the n th iterated product of ( R , − t ). Itis the pointed version of S n . Example 5.
Our formalism makes its possible to construct cones in the smoothcategory. Let S ⊂ M be a closed submanifold. Take a function f : M → [0 , V of S and vanishes outside a larger tubularneighborhood U . Let I = ( −∞ , − M × I, f · t ) where t is the coordinateon R . We decompose our space M × I as ( M − U ) × I and U × I . On ( M − U ) × I , f · t = 0 and thus, up to homotopy, we get M − U . On U × I , on the other hand,we claim that every cycle is nulhomotopic. Indeed, given a cycle P , first deform itto lie in V × I , Now, observe that by moving the t coordinate to −∞ we provide a omotopy from our cycle to the empty set.We also have the notion of a cochain: Definition 31.
Let Q be a smooth manifold with corners. A cochain τ : Q → M is asmooth cooriented map such that given q i ∈ Q , either q i has a converging subsequenceor lim f ( τ ( q i )) = ∞ . We let H ∗ ( M, f ) denote the resulting cohomology groups which are graded bycodimension. Note that we have pairing H ∗ ( M, f ) ⊗ H ∗ ( M, f ) → H ∗ ( M )since the intersections of a cycle and a cocycle must be compact.The notion of cohomology of a space with functional is a generalization of usualnotion of cohomology. Namely, given any smooth manifold M , let f be any exhaust-ing function on M that is bounded below. Thus, m i ∈ M either has a convergentsubsequence or lim i →∞ f ( m i ) = ∞ . We associate to M the space ( M, f ). In thiscase, we have:
Lemma 35.
We have H ∗ ( M, f ) ∼ = H ∗ ( M ) and H ∗ ( M, f ) ∼ = H ∗ ( M ) Remark.
Note that if we take f = 0 in the previous construction, we get coho-mology with compact supports.13.2. Morphisms.Definition 32.
Let U be open in M . A morphism φ : ( M, f ) → ( N, g ) is a smoothmap φ : U → N such that if m i ∈ U does not have a convergent subsequence in U , then g ( φ ( u i )) →−∞ . Lemma 36.
Let φ be a morphism. Given a chain σ : P → M , φ ◦ σ | σ − ( U ) : σ − ( U ) → N is a chain.Proof. Suppose p i ∈ σ − ( U ) is a sequence without a convergent subsequence. Weclaim that σ ( p i ) does not have a convergent subsequence in U . Indeed, if somesubsequence σ ( p j ) had a limit y ∈ U , then p j must have a convergent subsequence p k ∈ P . But this implies that lim p k ∈ σ − ( U ), contradicting the assumption on p i .Therefore, since σ ( p i ) does not have a convergent subsequence in U , g ( φ ( σ ( p i ))) →−∞ as desired. (cid:3) xample 1. Let S ⊂ M be a compact submanifold. Let f : M − S → R be a function that is bounded outside a neighborhood of S and approaches −∞ as apoint approaches S . We define the collapsed space to be ( M − S, f ). The morphism φ : M → M − S defined by the identity map on M − S is our smooth analogue of the collapsing map.An important special case that is central to homotopy theory is the collapsing map S n → S n ∨ S n that is obtained by collapsing an equatorial S n − . Here we view S n ∨ S n as a disjointunion of two open hemispheres of the original S n . References [1]
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Simons Center for Geometry and Physics, Stony Brook University, Stony Brook,NY 11794
E-mail address : [email protected]@math.columbia.edu