The Morse complex for gradient flows on compact manifolds

Transcription

1 The Morse complex for gradient flows on compact manifolds 1 Preliminaries Let M be a compact smooth manifold of dimension n, without boundary, and let f : M R be a smooth function. A point x M such that df(x) = 0 is said a critical point of f. The set of critical points of f is closed, and it will be denoted by crit(f). If 0 is a critical point of g : R n R, and ϕ : R n R n is a diffeomorphism such that ϕ(0) = 0, in the identity ij (g ϕ) = i [ h g(ϕ) j ϕ h ] = kh g(ϕ) i ϕ k j ϕ h + h f(ϕ) ij ϕ h the last term vanishes in 0, so denoting by Hess g the hessian matrix of g, Hess (g ϕ)(0) = dϕ(0)hess g(ϕ(0))dϕ(0) T. This formula shows that a change of coordinates makes the the Hessian matrix at a critical point change within its similarity class. So the following definitions do not depend on the choice of a coordinate system near x: Definition 1.1 A critical point x crit(f) is said non-degenerate if the Hessian of f at x in local coordinates is invertible. The Morse index of x is the number of negative eigenvalues of such Hessian, and it is denoted by ind (x). The inverse mapping theorem implies that non-degenerate critical points are isolated points of crit(f). When all the critical points are non-degenerate, f is said a Morse function. Since M is compact, a Morse function on M has finitely many critical points. Theorem 1.1 The set of Morse functions on M is open and dense in the space of all smooth functions, endowed with the C 2 topology. The fact that the Morse condition is open in C 2 is an easy consequence of the inverse mapping theorem. The density comes from an argument involving Sard s theorem. A proof can be found in [Hir70], Chapter 6, Theorem 1.2. From now on, f will be a Morse function. 2 The Morse relations If A is a finitely generated Abelian group, the elements of finite order in A form the torsion subgroup T, and the quotient group A/T is free Abelian (see [Lan64], pag. 45). The minimal number of generators of A/T is called the rank of A. The rank of the singular homology group H k (X; Z) is called the k-th Betti number of the topological space X. Denote by β k = β k (M; Z) the k-th Betti number of M. The classical version of the Morse relations is the following: Theorem 2.1 Let c k be the number of critical points of index k. Then for every k = 0, 1,..., n we have the inequality and the equality holds when k = n. c k c k 1 + c k 2 ± c 0 β k β k 1 + β k 2 ± β 0, 1

2 In particular, c k β k for every k = 0, 1,..., n. The above result may be restated as follows: there exists a polinomyal Q with integer non-negative coefficients such that λ ind (x) = P (M) + (1 + λ)q(λ), x crit(f) where P (M) denotes the Poincaré polinomyal of M: P (M) = n β k (M; Z)λ k. k=0 A direct proof of Theorem 2.1 can be found in [Mil63], Part I 5, or in [Hir70], Chapter 6, Theorem 3.4. Here we will use a different approach, building the so called Morse complex. 3 The gradient flow Let us fix a Riemannian metric, on M, so that we can introduce the gradient vector field of f, i.e. the vector field grad f such that df(p)[ξ] = grad f(p), ξ, (p, ξ) T M. The gradient flow of f is the map φ : R M M solving { t φ(t, p) = grad f(φ(t, p)), φ(0, p) = p, t R, p M. The compactness of M ensures that φ is defined over all R M. It is a smooth map, the map φ t ( ) := φ(t, ) is a diffeomorphism of M, and φ is a flow, meaning that the group properties φ 0 = id M and φ s+t = φ s φ t, s, t R, hold. The function f is strictly decreasing on the non-constant orbits of φ: = t s f(φ(t, p)) f(φ(s, p)) = df(φ(σ))[ t φ(σ, p)] dσ = t s t s d f(φ(σ, p)) dσ dσ grad f(φ(σ, p)) 2 dσ. The critical points of f are exactly the fixed points of the flow φ, and x crit(f) is nondegenerate if and only if it is a hyperbolic fixed point for φ. This means that the linear mapping dφ t (x) : T x M T x M does not have eigenvalues of modulus 1, for t 0. In other words, denoting by H : T x M T x M the infinitesimal generator of dφ t (x), i.e. dφ t (x) = e th, t R, we have that H has no purely imaginary eigenvalues. Indeed, in our case H is conjugated to a local representation of the Hessian of f at x, so its spectrum is real, and the critical point is non-degenerate if and only if 0 is not an eigenvalue of H (actually, H is the Riemannian Hessian of f at x). Let T x M = E u x E s x, be the spectral decomposition of T x M into the generalized eigenspace Ex u corresponding to the positive eigenvalues of H, and the generalized eigenspace Ex s corresponding to the negative eigenvalues of H. Since H is conjugated to a local representation of the Hessian of f at x, we have (1) dim E u x = ind (x), dim E s x = n ind (x). 2

3 See Appendix A for more details. Consider the φ-invariant sets W u (x) = {p M φ(t, p) x for t }, W s (x) = {p M φ(t, p) x for t + }. The sets W u (x) and W s (x) are called the unstable manifold and the stable manifold of x. This terminology is justified by the fact that they are actually smooth objects: Theorem 3.1 The unstable and the stable manifolds W u (x) and W s (x) are smooth submanifolds of M, diffeomorphic to E u x and E s x, respectively, and such that T x W u (x) = E u x, T x W s (x) = E s x. See [Shu87], Chapter 5, for a proof. In general, if x is a hyperbolic fixed point of an arbitrary flow, its unstable and stable manifold are just images of injective smooth immersions, and they need not be locally closed. The fact that we are dealing with a gradient flow guarantees that these immersions are actually embeddings. The fact that f strictly decreases along the non-constant orbits of φ easily implies that for every p M the limit of φ(t, p) for t + (resp. t ) exists, and that it is a critical point. Therefore {W s (x) x crit(f)} and {W u (x) x crit(f)} are two partitions of M. Remark 3.1 One may actually work with gradient-like flows, i.e. flows which are generated by the gradient of a Morse function near the critical points, and such that there exists a Lyapunov function, i.e. a smooth g : M R such that d/dtg(φ t (p)) < 0 for every regular point p M. This situation is only apparently more general, because every gradient-like flow is a gradient flow of some Morse function with respect to some metric, see [Sma61], Theorem B. 4 The Morse-Smale condition The pair (f,, ) satisfies the Morse-Smale condition if for every pair of critical points x, y crit(f), the intersection W u (x) W s (y) is transverse. This means that T p W u (x) + T p W s (y) = T p M, p W u (x) W s (y). In this case, the implicit function theorem implies that W u (x) W s (y) is a submanifold of dimension dim W u (x) W s (y) = dim W u (x) + dim W s (y) n = ind x ind y. The intersection W u (x) W s (y) is φ-invariant, so when it is non-empty and x y, it contains a non-constant orbit, so its dimension is at least 1. This shows that the Morse-Smale property has the following important consequence: W u (x) W s (y) = whenever x y are critical points such that ind x ind y. The implicit function theorem and Sard s theorem again imply a genericity result for Morse- Smale pairs: Theorem 4.1 Given a Morse function f, the set of metrics, such that (f,, ) satisfies the Morse-Smale condition is open and dense in the set of all smooth metrics, endowed with the C 2 topology. The proof of a weaker result can be found in [Sma61], Theorem A. For a different approach, see [Sch93], section 2.3. From now on,, will be a metric such that (f,, ) satisfies the Morse-Smale condition. 3

4 5 Orientations The unstable manifolds are diffeomorphic to Euclidean spaces, so we can fix an orientation for every W u (x), x crit(f). These orientations induce an orientation on every non-empty intersection W u (x) W s (y), as we are going to show. Let p W u (x) W s (y) and choose a linear subspace V of T p W u (x) such that T p W u (x) = T p (W u (x) W s (y)) V. (2) Then V T p W s (y) = (0), and by the Morse-Smale condition V T p W s (y) = T p W u (x) + T p W s (y) = T p M. It is a general fact about flows with a hyperbolic fixed point that, if p W s (y) and V is a direct summand of T p W s (y) in T p M, the image of V by the differential of the flow converges to the tangent space of the unstable manifold of y: dφ t (p)v T y W u (y) for t +. Here the definition of convergence of linear subspaces of the tangent bundle of M is reduced via local charts to the usual notion of convergence in the Grassmannian of R n. See Appendix A. Therefore, when t is large the orientation of T y W u (y) induces an orientation of dφ t (p)v, and thus of V. We will orient T p (W u (x) W s (y)) in such a way that (2) is an oriented sum. Since the set of direct summands of T p W s (y) is connected, a simple homotopy argument shows that the orientation we have defined does not depend on the choice of V. 6 Compactness of the set of gradient flow lines Recall that the class of all compact subsets of M (indeed of any metric space) is a metric space with the Hausdorff distance dist (K 1, K 2 ) := sup inf dist (p, q) + sup inf dist (p, q). q K 2 q K 2 p K 1 Then dist (K n, K) n 0 if and only if for any ɛ > 0 K n is contained in an ɛ-neighborhood of K and K is contained in an ɛ-neighborhood of K n, for n large enough. Consider a sequence of gradient flow lines connecting two critical points x, y, and the sequence of their closures: S n = φ(r {p n }) {x, y}, (p n ) W u (x) W s (y). Since M is compact, up to a subsequence we may assume that p n p, and the continuity of φ would give us the convergence φ(, p n ) φ(, p) uniformly on compact subsets of R. However, it may happen that p / W u (x), or p / W s (y), so φ(, p) could be a gradient flow line connecting two other critical points, and the convergence would not be uniform on R. We will show that in this case a subsequence of (S n ) converges to a broken gradient flow line in the Hausdorff topology. p K 1 Definition 6.1 Let x, y crit(f). A broken gradient flow line from x to a y is a set S = {z 0 } φ(r {p 1 }) {z 2 } φ(r {p 2 }) {z k 1 } φ(r {p k }) {z k }, where k 1, x = z 0 z 1 = z k 1 z k = y are critical points, and p i W u (z i 1 ) W s (z i ). 4

5 When k = 1, a broken gradient flow line is just the closure of a genuine gradient flow line. If S is a broken gradient flow line as in the above definition, the following inequalities must hold: f(x) > f(z 1 ) > > f(z k 1 ) > f(y). (3) It is easy to check that a compact set S M is a broken gradient flow line from x to y if and only if (i) x, y S, (ii) S is φ-invariant, (iii) the intersection S {p M f(p) = c} consists of a single point if c [f(y), f(x)], and it is empty otherwise. The Morse-Smale condition implies that, for a broken gradient flow line as in Definition 6.1, Now we can state the compactness result. ind (x) > ind (z 1 ) > > ind (z k 1 ) > ind (y). (4) Proposition 6.1 Let (p n ) W u (x) W s (y), and set S n := φ(r {p n }) {x, y}. Then (S n ) has a subsequence which converges to a broken gradient flow line from x to y, in the Hausdorff topology. Proof. The space of compact subsets of a compact metric space is compact with respect to the Hausdorff topology, so (S n ) has a subsequence (S n) which converges to a compact set S M. Then x, y S, and since S n is φ-invariant, so is S. Since S n f 1 ([f(y), f(x)]), we obtain that the set S {p M f(p) = c} (5) is empty for every c / [f(y), f(x)]. Let c [f(y), f(x)]. Then (S n) has a subsequence (S n) such that S n {p M f(p) = c} converges to some point in (5), which is then non-empty. If the set (5) contains two points p, q, the fact that (S n ) is a sequence of flow lines allows us to find a sequence (p n ) converging to p, and numbers t n R such that φ(t n, p n ) q. By reversing if necessary the role of p and q, we may assume that t n 0 for every n, and by (1) we deduce the convergence: tn 0 grad f(φ(t, p n )) 2 dt = f(p n ) f(φ(t n, p n )) f(p) f(q) = 0. Then, either t n 0 or the sequence of sets φ([0, t n ] {p n }) converges to a critical point. In both cases, we obtain that p = q. This shows that the set (5) consists of a single point. Hence S is a broken gradient flow line from x to y. 7 The Morse complex Let x, y crit(f) be critical points with ind (x) ind (y) = 1. (6) If W u (x) W s (y) is non-empty, it is a one-dimensional submanifold of M, so each connected component is a gradient flow line from x to y. We claim that W u (x) W s (y) consists of finitely many connected components. Indeed, if the points p n belong to different connected components, Proposition 6.1 implies that, up to a subsequence, the sequence S n := φ(r {p n }) {x, y} converges to a broken gradient flow line S from x to y. Such a broken gradient flow line cannot 5

6 be a genuine flow line, because otherwise W u (x) W s (y) would not be a submanifold. Therefore S must contain another critical point, but then (4) and (6) are incompatible. Every connected component of W u (x) W s (y) has the orientation defined in section 5. Moreover in the case of index difference 1, it has also the orientation defined by declaring the tangent vector grad f(p) to be a positive basis of T p (W u (x) W s (y)). We can associate a coefficient +1 or 1 to every connected component of W u (x) W s (y), depending on whether these orientations coincide or not, and we can define c(x, y) Z to be the sum of all these coefficients. Let C k be the free Z-module spanned by the critical points of index k: C k = Z x. x crit(f) ind (x)=k We define the homomorphism δ k : C k C k 1 by setting δ k x := c(x, y) y, x crit(f), ind (x) = k. y crit(f) ind (y)=k 1 The following theorem is the main result of this chapter. Theorem 7.1 The following statements hold: (i) δ k 1 δ k = 0, for every k = 1,..., n. So {C, δ } is a complex of finitely generated free Abelian groups. (ii) The homology of the complex {C, δ } is isomorphic to the singular homology of M: H k ({C, δ }) := ker(δ k : C k C k 1 ) ran (δ k+1 : C k+1 C k ) = H k (M; Z). The complex {C, δ } is said the Morse complex of the pair (f,, ). In general, it depends on the Morse-Smale metric chosen. Theorem 7.1 implies the Morse relations of Theorem 2.1. Our strategy to prove Theorem 7.1 will be to build a suitable cellular filtration of M such that the associated complex is isomorphic to the Morse complex of (f,, ). 8 Cellular filtrations Let X be a topological space. Definition 8.1 A cellular filtration of X is a sequence of subspaces X k X, k Z, such that (i) H i (X k, X k 1 ) = 0 if i k; (ii) every simplex in X is contained in X k, for some k. In particular, X = k Z Xk. If the last identity holds, (ii) is automatically fulfilled when the sets X k are open. In our case, the filtrations will be actually finite. Given a cellular filtration of X, we set W k X := H k (X k, X k 1 ; Z), and we define the homomorphism k : W k X W k 1 X as the composition H k (X k, X k 1 ) H k 1 (X k 1 ) H k 1 (X k 1, X k 2 ). Then k 1 k = 0, because already the composition H k 1 (X k 1 ) H k 1 (X k 1, X k 2 ) H k 2 (X k 2 ) vanishes. Therefore {W X, } is a complex of Abelian groups, and it is said the cellular complex of X, induced by the filtration {X k }. Cellular filtrations are a useful tool to compute homology, because of the following result (see [Dol72], Chapter V, Proposition 1.3). 6

7 Theorem 8.1 There exists an isomorphism H k ({W X, }) = H k (X, X 1 ; Z), for every k N. In our case, we will take X = M and X 1 =. 9 Construction of the cellular filtration Denote by B r (p) the open ball of radius r centered in p M. Proposition 9.1 If r > 0 is small enough, then φ(r + B r (x)) B r (y) =, whenever x y are critical points such that ind (y) ind (x). Proof. Arguing by contradiction and using the fact that crit(f) is finite, we can find critical points x y such that ind (y) ind (x), and sequences (p n ) M, p n x, (t n ) R +, such that φ(t n, p n ) y. Then the fact that f decreases strictly on non-constant orbits implies that f(y) < f(x). Up to a subsequence, we may assume that the gradient flow lines through the points p n connect the same two critical points. Then Proposition 6.1 implies that there exists a broken gradient flow line passing through x and y. Since f(y) < f(x), the inequalities (3) and (4) imply that ind (y) < ind (x), against our assumption. Let us fix an r > 0 so small that the conclusion of Proposition 9.1 holds, and let us define the sets X k := φ(r + B r (x)). x crit(f) ind (x) k Each X k is open and positively invariant: φ t (X k ) X k for every t > 0. Moreover X k X k+1 and, since every p M belongs to the unstable manifold of some critical point, X n = M. We will prove that {X k } is a cellular filtration. Remark 9.1 When the Morse function is self-indexing, i.e. f(x) = ind (x) for every critical point x, a natural cellular filtration is given by { X k = p M f(p) < k + 1 }. 2 Furthermore, it could be shown that if (f,, ) has the Morse-Smale property, the gradient flow of f is also the gradient flow of a self-indexing function, see [Sma61] and [Mey68]. 10 Local description near a critical point Let x crit(f) and choose a chart ψ x : T x M = R n M such that ψ x (0) = x and dψ x (0) = id TxM. Then 0 is a hyperbolic fixed point for the local flow on T x M conjugated to φ by ψ x, and the corresponding splitting T x M = E u x E s x is given by E u x = T x W u (x), E s x = T x W s (x). Denoting by E(ɛ) the closed ball of radius ɛ in the linear space E, we set Q ɛ (x) = ψ x (E u x (ɛ) E s x(ɛ)), Q ɛ (x) = ψ x ( E u x (ɛ) E s x(ɛ)). The following results are proved in Appendix A. They are consequences of the fact that x is a hyperbolic fixed point, and that we are dealing with a gradient flow. 7

8 Lemma 10.1 If ɛ > 0 is small enough, Q ɛ (x) is an eventual exit set from Q ɛ (x): (i) if p Q ɛ (x) and φ(t, p) / Q ɛ (x) for some t > 0, then there exists s [0, t) such that φ(s, p) Q ɛ (x); (ii) if p Q ɛ (x), then φ(t, p) / Q ɛ (x) for every t > 0. Lemma 10.2 If ɛ > 0 is small enough, then: (i) W u (x) Q ɛ (x) = ψ x (graph ζ x ), where is a smooth map. ζ x : E u (ɛ) E s (ɛ) (ii) Let p W s (x), and let N M be a submanifold such that N W s (x) = {p} and this intersection is transverse. Then there exists ρ > 0 and t 0 0 such that for every t t 0, there holds φ t (N B ρ (p)) Q ɛ (x) = ψ x (graph η t ), for a smooth map η t : E u (ɛ) E s (ɛ). We will choose ɛ so small that the conclusions of the above lemmas holds, and Q ɛ (x) B r (x), x crit(f), where r has been chosen in section Deformation argument If K M is compact, A M is open and positively invariant, and lim φ(t, p) A, p K, t + then there exists t 0 0 such that φ t (K) A, for every t t 0. Indeed, for every p K we can find t(p) such that φ(t(p), p) A, so there exists an open neighborhood U(p) of p such that φ(t(p), q) A for every q U(p). Let {U(p j ) j = 1,..., k} be a finite covering of K, and set t 0 := max{t(p 1 ),..., t(p k )}. The fact that A is positively invariant allows to conclude. Lemma 11.1 If t 0 is large enough, then for every critical point x such that ind (x) = k. φ t (Q ɛ (x)) X k 1, Proof. Let x be a critical point such that ind (x) = k. If p Q ɛ (x), the limit of φ(t, (p) for t + cannot be x, by Lemma 10.1 (ii), so by Proposition 9.1, this limit is a critical point y with ind y < ind x. In particular, it lies in the positively invariant open set X k 1, and the compactness of the union of the Q ɛ (x), for ind (x) = k, allows to conclude. Let us choose t 0 large enough so that the conclusion of the above lemma holds, for every k = 0, 1,..., n, and set Q(x) := φ t (Q ɛ (x)), Q (x) := φ t (Q ɛ (x)). Then Q (x) is an eventual exit set from Q(x), in the sense of Lemma Together with Lemma 11.1, this implies that the set Y k := X k 1 Q(x) is positively invariant. x crit(f) ind (x)=k 8

9 Lemma 11.2 If t > 0 is large enough, then φ t (X k ) Y k. Proof. Let p X k : there exist x crit(f) with ind (x) k, (p n ) B r (x), and (t n ) R + such that φ(t n, p n ) p. Let y crit(f) be the limit of φ(t, p) for t +. Then there exists s 0 such that φ(s, p) B r (y), and by the continuity of φ s, φ(s + t n, p n ) B r (y) for n large. By Proposition 9.1, either y = x or ind (y) < ind (x) k. In any case y belongs to the interior part of Y k, and the conclusion follows from the compactness of X k. 12 The homology of the filtration Lemma 11.2 implies that the inclusion (Y k, X k 1 ) i (X k, X k 1 ) is a homotopic equivalence: there exists a continuous map h : (X k, X k 1 ) (Y k, X k 1 ), such that i h and h i are homotopic to the identity mappings of the respective pairs. Indeed, it is enough to choose h = φ t, for t large enough. Therefore and by excision H i (X k, X k 1 ) = H i (Y k, X k 1 ), H i (Y k, X k 1 ) = H i ( Q(x), Q (x) ), where the unions are taken over all critical points of index k. Proposition 9.1 implies that the compact sets Q(x) are pairwise disjoint, so H i ( Q(x), Q (x)) = H i (Q(x), Q (x)). By construction, (Q(x), Q (x)) is diffeomorphic to so since E s x(ɛ) is contractible, we obtain x crit(f) ind (x)=k (E u x (ɛ) E s x(ɛ), E u x (ɛ) E s x(ɛ)), H i (Q(x), Q (x)) = H i (E u x (ɛ), E u x (ɛ)) = { Z if i = k, 0 if i k. Therefore H i (X k, X k 1 ) = { Z k = C k if i = k, 0 if i k., and {X k } is a cellular filtration. In order to compare the boundary homomorphism of the cellular complex related to {X k } to the homomorphism δ k : C k C k 1, we need to be more precise about the choice of an isomorphism H k (X k, X k 1 ) = C k. By construction, we have the diffeomorphisms (Ex u (ɛ) Ex(ɛ), s Ex u (ɛ) Ex(ɛ)) s ψx (Q ɛ (x), Q ɛ (x)) φ t (Q(x), Q (x)), 9

10 and we recall that dψ x (0) = id TxM, and that the linear space E u x = T x W u (x) is oriented. Denote by D k the closed unit disk in R k, with its standard orientation, and let ω k be the corresponding generator of H k (D k, D k ). Choose an orientation preserving diffeomorphism α x : (D k, D k ) (E u x (ɛ), E u x (ɛ)). By Lemma 10.2 (i), W u (x) Q ɛ (x) is the image by ψ x of the graph of the smooth map Let γ x : D k M be given by the composition D k Then γ x induces an isomorphism ζ x : E u x (ɛ) E s x(ɛ). α x E u x (ɛ) id ζx E u x (ɛ) E s x(ɛ) ψx Q ɛ (x) φ t Q(x) M. H k (D k, D k ) H k (Q(x), Q (x)), and H k (X k, X k 1 ) is generated by the images of ω k by the injective homomorphisms γ x : H k (D k, D k ) H k (X k, X k 1 ) for all critical points x of index k. Therefore we can consider the isomorphism θ k : C k H k (X k, X k 1 ), θ k x = γ x (ω k ). By construction, the maps γ x are smooth embeddings, and they have the further properties: γ x (D k ) W u (x), γ x ( D k ) = W u (x) Q (x). We claim that the diagram C k δ k θ k Hk (X k, X k 1 ) k (7) C k 1 θ k 1 Hk 1 (X k 1 X k 2 ) commutes. Therefore {C, δ } is a complex, proving assertion (i) of our main Theorem 7.1, and it is isomorphic to the cellular complex {H (X, X 1 ), }. Hence by Theorem 8.1 also statement (ii) holds. The commutativity of the diagram above will be proved in the next section. Remark 12.1 The proof that {X k } is a cellular filtration uses only the Morse-Smale condition up to order 0: it is required that W u (x) W s (y) = whenever x y and ind (x) ind (y) 0 (see the proof of Proposition 9.1). To define δ k, one needs the Morse-Smale condition up to order 1: the intersection W u (x) W s (y) should be transverse whenever ind (x) ind (y) 1. This condition is actually sufficient also to prove the isomorphism between the cellular complex and the Morse complex, as the argument of the next section will show. 13 Commutativity of diagram (7) Let x be a critical point of index k. We must prove that k θ k x = θ k 1 δ k x. (8) 10

11 The naturality of the boundary in singular homology yields the commutative diagram H k (D k, D k ) H k 1 ( D k ) γ x Hk (X k, X k 1 ) τ Hk 1 (X k 1 ) (9) where τ : D k X k 1 is the restriction of γ x. Let σ k 1 be the image of ω k by the left vertical arrow: σ k 1 generates H k 1 ( D k ) = H k 1 (S k 1 ) = Z, and it determines an orientation of D k, that is the one for which R k = Rv T v D k, v D k is an equality between oriented spaces, when Rv is oriented by v (the outward orientation). The map τ is a diffeomorphism onto the (k 1)-sphere W u (x) Q (x), which is then oriented by τ (σ k 1 ). So the hyperplane V (v) := T τ(v) (W u (x) Q (x)) T τ(v) W u (x), v D k, is oriented. Since the vector grad f(τ(v)) points inward γ x (D k ), we deduce that T τ(v) W u (x) = R( grad f(τ(v))) V (v), v D k, is an equality between oriented spaces, where R( grad f(τ(v))) is oriented by grad f(τ(v)). Every gradient flow line converging to x for t, and to a critical point of index k 1 for t +, meets W u (x) Q (x) in a single point. Let y 1,..., y r be the critical points of index k 1 whose stable manifolds have non-empty intersection with W u (x), and let vi 1,..., vmi i, i {1,..., r}, be the distinct points in D k such that { } τ(v j i ) j = 1,..., m i = W u (x) W s (y i ) Q (x). Then by definition where the coefficient c j i r m i δ k x = c j i y i, (10) i=1 j=1 is +1 or 1, depending on whether in the limit dφ t (τ(v j i ))V (vj i ) T y i W u (y i ), for t +, the orientation is preserved or reversed. Denote by N ρ (v), v D k, the open ball in D k, with radius ρ and center v. By Lemma 10.2 (ii), we can find ρ > 0 and t 0 0 such that for every i {1,..., r}, j {1,..., m i }, t t 0, the set φ t (τ(n ρ (v j i ))) Q ɛ(y i ) is the image by the chart ψ yi of the graph of a smooth map η i,j t : E u y i (ɛ) E s y i (ɛ). We may assume that ρ is so small that the closed balls N ρ (v j i ) are pairwise disjoint. If v D k \ i,j N ρ(v j i ), the limit of φ(t, τ(v)) for t + is a critical point of index not exceeding k 2, so we can find an s t 0 such that ( φ s D k \ ) i,j N ρ(v j i ) X k 2. 11

12 So, considering the homotopy φ t τ, t [0, s], we obtain the commutative diagram H k 1 ( D k ) τ Hk 1 (X k 1 ) (11) H k 1 ( D k, D k \ i,j N ρ(v j i )) By excision, the vertical arrow in the commutative diagram (φ s τ) Hk 1 (X k 1, X k 2 ) H k 1 ( D k, D k \ i,j N ρ(v j i )) (φs τ) H k 1 (X k 1, X k 2 ) i (φ s τ) H k 1 ( i,j N ρ(v j i ), i,j N ρ(v j i )) (12) is an isomorphism. Moreover, if we fix orientation preserving diffeomorphisms it is easy to see that the image of σ k 1 by H k 1 ( D k ) H k 1 ( D k, D k \ i,j β j i : (D k 1, D k 1 ) (N ρ (v j i ), N ρ(v j i )), N ρ (v j i i 1 )) H k 1 ( N ρ (v j i ), N ρ (v j i )) i,j i,j = i,j H k 1 (N ρ (v j i ), N ρ(v j i )) is the element i,j βj i (ω k 1). Hence the commutativity of diagrams (9), (11), (12), implies that We claim that k γ x (ω k ) = r m i (φ s τ β j i ) (ω k 1 ). (13) i=1 j=1 (φ s τ β j i ) (ω k 1 ) = c j i γ y i (ω k 1 ). (14) Then (8) follows from (10), (13), and (14). There remains to prove (14). Let D N ρ (v j i ) be the closed neighborhood of vj i in D k such that φ s (τ(d)) = ψ yi (graph ηs i,j ), φ s (τ( D)) = ψ yi (graph ηs i,j ) Q ɛ (y i ), Then D is diffeomorphic to the (k 1)-disk, so we can fix an orientation preserving diffeomorphism µ from (D k 1, D k 1 ) onto (D, D). Then the diagram β j i H k 1 (D k 1, D k 1 ) H k 1 (N ρ (v j i ), N ρ(v j H i ))(φs τ) k 1 (X k 1, X k 2 ) (15) µ H k 1 (D, D) φ t (φ s τ) H k 1 (Q ɛ (y i ), Q ɛ (y i )) commutes. By construction, the map φ t γ yi : (D k 1, D k 1 ) (Q ɛ (y i ), Q ɛ (y i )) maps D k 1 diffeomorphically onto W u (y i ) Q ɛ (y i ) = ψ yi (graph ζ yi ), 12

13 preserving the orientation. On the other hand, the map maps D k 1 diffeomorphically onto Consider the diffeomorphism φ s τ µ : (D k 1, D k 1 ) (Q ɛ (y i ), Q ɛ (y i )) φ s (τ(d)) = ψ yi (graph η i,j s ). π : ψ yi (graph η i,j s ) ψ yi (graph ζ yi ), π(ψ yi (ξ u, η i,j s (ξ u ))) = ψ yi (ξ u, ζ yi (ξ u )). Since the tangent space of φ s (τ(d)) at φ s (τ(v j i )) is it is easy to see that dφ s (τ(v j i ))V (vj i ), π φ s τ µ : D k 1 W u (y i ) Q ɛ (y i ) preserves the orientation if c j i = +1, and it reverses it if c j i = 1. Two diffeomorphisms of (D k 1, D k 1 ) onto itself induce the same isomorphism in homology if and only if they induce the same orientation, otherwise they induce opposite isomorphisms. Then, since π is the restriction of a map of (Q ɛ (y i ), Q ɛ (y i )) into itself which is homotopic to the identity, we deduce that (φ s τ µ) (ω k 1 ) = c j i (φ t γ y i ) (ω k 1 ). Hence the commutativity of (15) implies (14). 14 Some bibliography and further remarks Theorem 2.1 was proved by Morse in his [Mor25], see also [Mor34, Mor47]. A classical reference for Morse theory is Milnor s book [Mil63], and more recently the review papers by Bott [Bot82, Bot88]. The dynamical system point of view arose after the seminal work of Smale, see [Sma60, Sma61] and the beautiful foundational paper [Sma67], and it immediately had influences in topology, see Milnor s book on the h-cobordism theorem [Mil65]. In this framework, one can consider Morse- Smale flows, which are dynamical systems more general than gradient flows because they may have periodic orbits. The connection between Morse theory for Morse-Smale flows and the homotopy of a manifold has been further clarified by Franks [Fra79], see also his lecture notes [Fra80]. Interpreting the boundary homomorphism of a cellular filtration in terms of an algebraic count of the gradient flow lines connecting critical points of index difference 1, was already implicit in a paper by Thom [Tho49], who however did not clarify the conditions required on the gradient flow. This interpretation was pointed out by Witten in his [Wit82], where it is deduced quite indirectly from a relationship between Morse theory and certain deformations of the Laplace- Beltrami operator. The first explicit proof of Theorem 7.1 is due to Floer [Flo89], see also [Sal90]. Floer s proof makes use of Conley index theory, a general and powerful method to decompose a dynamical system into simpler invariant sets, see [Con78, CZ84], and Salamon s exposition [Sal85]. A systematic study of the Morse complex of a function as a tool to build a homology theory which satisfies the Eilenberg-Steenrod axioms can be found in Schwarz s book [Sch93]. In this book, the methods are closer to the ones used in Floer homology for symplectic fixed points: instead of working with the intersection between the unstable manifold of x and the stable manifold of y, one deals with the moduli spaces of curves solving the gradient flow equation and converging to two critical points x and y, for t and t +, respectively. Seeing these moduli spaces as the inverse images of the zero-section by suitable sections of infinite dimensional Banach bundles allows to prove their regularity and to compute their dimension. The boundary property δ k 1 δ k = 0 is proved by the so called gluing method, using a suitable version of the implicit 13

14 function theorem. The isomorphism with the singular homology is deduced by the fact that all the homology theories which satisfy the Eilenberg-Steenrod axioms are equivalent on compact CW -complexes. A more direct proof of this isomorphism, still in this spirit, can be obtained by interpreting singular homology theory in terms of pseudo-cycles, see [Sch99]. Replacing the compactness of M by suitable assumptions on the behavior of the function f, Morse theory can be extended to infinite dimensional Hilbert manifolds, at least in the case of critical points with finite Morse index, a situation which arises in many variational problems. See the foundational paper by Palais [Pal63], and the book of Chang [Cha93] for many applications. References [Bot82] R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N.S.) 7 (1982), [Bot88], Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68 (1988), [Cha93] K. C. Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser, Boston, Mass., [Con78] C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, Amer. Math. Soc., Providence, R.I., [CZ84] C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), [Dol72] A. Dold, Lectures on algebraic topology, Grundlehren der Mathematischen Wissenschaften, vol. 200, Springer-Verlag, New York-Berlin, [Flo89] A. Floer, Witten s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), [Fra79] J. Franks, Morse-Smale flows and homotopy theory, Topology 18 (1979), [Fra80], Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Amer. Math. Soc., Providence, R.I., [Hir70] M. W. Hirsch, Differential topology, Springer, New York-London, [Lan64] S. Lang, Algebra, Addison-Wesley, Reading, MA, [Mey68] R. Meyer, Energy functions for Morse-Smale systems, Amer. Jour. Math. 90 (1968), [Mil63] [Mil65] J. Milnor, Morse theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, Princeton, N.J., 1963., Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville, VA, [Mor25] M. Morse, Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27 (1925), [Mor34], The calculus of variations in the large, Amer. Math. Soc. Colloq., vol. 18, Amer. Math. Soc., Providence, [Mor47], Introduction to analysis in the large, Princeton Univerity Press, Princeton,

15 [Pal63] R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), [Sal85] [Sal90] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1 41., Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22 (1990), [Sch93] M. Schwarz, Morse homology, Birkhäuser, Basel, [Sch99], Equivalence for Morse homology, Geometry and topology in dynamics (Winston- Salem, NC, 1998/San Antonio, TX, 1999 (Providence, RI), Contemp. Math., vol. 246, Amer. Math. Soc., 1999, pp [Shu87] M. Shub, Global stability of dynamical systems, Springer, New York, [Sma60] S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), [Sma61], On gradient dynamical systems, Ann. of Math. 74 (1961), [Sma67], Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), [Tho49] R. Thom, Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris 228 (1949), [Wit82] E. Witten, Supersymmetry and Morse theory, J. Differential Geometry 17 (1982),