Morse Theory and Thom-Smale Homology Qingyuan Bai Last update: May - - PDF document

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Morse Theory and Thom-Smale Homology Qingyuan Bai Last update: May 16, 2019 Abstract This survey focuses on the Morse theory of finite dimensional compact manifolds. We introduce Morse function and discuss its first applications. Then we turn

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Morse Theory and Thom-Smale Homology

Qingyuan Bai Last update: May 16, 2019

Abstract This survey focuses on the Morse theory of finite dimensional compact

manifolds. We introduce Morse function and discuss its first applications.

Then we turn to study the pseudo gradient field X adapted to some Morse function f. There is a complex (Cf

∗ , ∂X) associated to such pair (f, X)

if X satisfies the so called ‘Smale property’. We prove that for different choice of (f, X) , the associated complexes are naturally chain homotopic. So the ‘Morse Homology’ is actually a smooth invariant of manifolds.

1 Introduction 1 2 Basic Morse Theory 3 2.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . 3 2.2 Genericness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Homotopy type and critical values . . . . . . . . . . . . . . . . . 6 3 Thom-Smale Complex and Morse Homology 9 3.1 Pseudo gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Smale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Thom-Smale complex . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Manifold with boundary (or corners) . . . . . . . . . . . . . . . . 22 4 A priori: the invariance of Thom-Smale complex 23 References 27

1 Introduction

In this section, we briefly talk about the history of Morse theory. 1

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In the original work of Marston Morse, he defined Morse function f on a manifold M and its index λp at each critical point p (the exact definition of these objects will be given in the next section). His idea was that the ana- lytic behaviour of functions on a manifold M should reflect its topology. This connection was clarified by the following Morse’s inequality: Theorem 1.1 (Morse’s Inequality). Let Mt(f) :=

p tλp be the polynomial in

t whose n-th coefficient is the number of critical points whose index is n. Let Pt(M; K) be the Poincar´ e polynomial of M taken over K. Then we have Mt(f) − Pt(M; K) = (1 + t)Q(t) (1) where Q(t) is a polynomial in t whose coefficients are nonnegative. The original proof works for functions on finite dimensional manifolds, but the same strategy can be applied to functionals on infinite dimensional spaces (like loop spaces). With this in mind, Morse proved his famous theorem assert- ing that for any Riemannian metric on Sn, there are infinitely many geodesics connecting two arbitrarily chosen points on it. Moreover, Bott used the same strategy to prove his periodicity theorem concerning the homotopy groups of O(n) and U(n). We will not touch upon the infinite dimensional set up in our treatment, however. The curious reader might read [5]. It was Smale’s idea to focus on the (pseudo-)gradient flows on the manifold. By counting flow lines, we can define a complex (Cf

∗ , ∂X) freely generated by

critical points of f. It was proved that the homology of (Cf

∗ , ∂X) is isomorphic to

the singular homology of M, so the classical inequality follows as an algebraic

fact. This construction, as Bott put it, is ‘the most beautiful formulation of

nondegenerate Morse theory’. The Thom-Smale complex was famous again in 80s after Witten’s famous

paper. In his work, Witten considered a Riemannian manifold with a Morse
function. He used this model to describe physic phenomenon that were first

proposed in inifnite dimensional settings of Yang-Mills theory. It turns out that this complex was exactly the Thom-Smale complex that we defined above. So the theory can serve as a toy model to help us understand the general theory. Similarly, it works as a toy model of Floer theory, in which we try to define ‘Thom-Smale complex’ in the infinite-dimensional spaces. The later theory has become a central tool in modern symplectic geometry. The organization of this survey is as follows. In section 2, we define Morse function and show its genericness. Then we develop basic Morse theory, culmi- nating in a proof of the Morse’s Inequality. In section 3, we introduce pseudo- gradient flow that satisfies Smale property and show its genericness. The rest

f the section is devoted to the definition and basic properties of Thom-Smale

complex. 2

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2 Basic Morse Theory

In this section, we develop Morse theory on finite dimensional manifolds. Note that in our treatment, we always assume M to be a compact manifold of dimension m (with or without boundary).

2.1 Definition and first examples

Given a smooth function f : M → R, x ∈ M is called a critical point of f if d f = 0 at x. In such case, there is a symmetric bilinear form Hxf (so called Hessian form) defined on TxM given by Hxf(v1, v2) := V1d f(V2)|x (2) where vi ∈ TxM and Vi is arbitrary locally defined smooth vector field that coincides with vi at x. Proposition 2.1. The bilinear form Hxf is well defined and symmetric.

Proof. For any local vector field V1, V2, we have:

V1(V2f) − V2(V1f) = [V1, V2]f (3) where [V1, V2] is the Lie bracket of two vector fields. Since d f = 0 at p, we see that V1(V2f)|p = V2(V1f)|p (4) Now that this bilinear form will be symmetric if it is well-defined. Note that V1(V2f)|p = V1|p(V2f) = v1|p(V2f) (5) So H(v1, v2) doesn’t depend on the choice of V1. Same argument shows that it doesn’t depend on the choice of V2. So it is well defined. Under local charts (x1, ..., xm), we have a basis for TxM = span{∂x1, ...∂xn}. The matrix of Hx(f) with respect to this basis is just the ‘Hessian’ matrix: (

∂2f ∂xi∂xj ). If at each critical point of f, the associated bilinear form is nondegen-

erate, we say f is a Morse function. In such case, the index of f at x is defined to be the number of negative eigenvalues (counted with multiplicities) of Hxf. The following lemma tells us Morse functions behave well locally. Lemma 2.1. Let f be a smooth function on M. If d f = 0 at x and Hxf is

nondegnerate. Then we can find some neighbourhood x ∈ U and φ : U → Rm

such that in the chart f takes the form f ◦ φ−1 = c −

x2

i + n

j=k+1

x2

(6) 3

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Proof. This is a local statement, so we might assume p = 0 ∈ Rm and ∂xif = 0

at 0. Then locally f − c can be written as f − c =

xixjgij (7) where gij = gji are smooth functions. The symmetric matrix (gij)|0 is nonde- generate by Morse condition. Since the diagonalizing process is C∞, we can find a diffeomorphism defined near 0 such that f takes the required form un- der this chart. More precisely, there is a GL(R, n) valued smooth function A defined near 0 such that At(gij)A is of the form diag{1, 1, ..., −1, ... − 1}. Then x → A−1x is a diffeomorphism near 0. Name the new coordinate y, then f−c = xt(gij(x))x = ytAt(gij(x(y))Ay = k

i=1 y2 i −n j=k+1 y2 j as promised.

For further use, we will call the chart given by Lemma 2.1 a Morse chart. Corollary 2.1. A Morse function’s critical points are discrete. If M is compact, then f has only finitely many critical points. Example 2.1. The first example is sphere in R3: S2 := {(x, y, z) : x2+y2+z2 = 1} equipped with the ‘height function’ f = z. Then f has a critical point of index 0 at its minimum and a critical point of index 2 at its maximum. Example 2.2. The (probably) most famous example of Morse theory is the embedded Torus T 2 in R3 equipped with the ‘height function’ f = z. f has a critical point of index 0 at its minimum and a critical point of index 2 at its

maximum. Moreover, it has 2 critical points of index 1 corresponding to two

‘saddle points’ of T 2. Example 2.3. A less trivial Morse function on T 2 is given by f = sin 2πx + sin 2πy. This function can be viewed as obtained from (S1, sin 2πx) via product

construction. Likewise, this function has 1 critical point of index 0, 2 critical

points of index 1 and 1 point of index 2.

2.2 Genericness

However, so far we haven’t been assured that there exists Morse function on arbitrary manifold. We intend to prove the following theorem. Theorem 2.1. Let M be a compact manifold. The set of Morse function is C0 dense in C∞(M). In other words, any smooth function on M can be uniformly approximated by Morse function. By the following lemma, we reduce problem to embedded manifolds in Rn. Lemma 2.2. Any compact manifold M (with or without boundary) admits an embedding into some Euclidean space RN. 4

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Proof. Let M be of dimension m. Since it is compact, it admits a finite coordi-

nate chart cover {B1, ..., Bn} such that each Bi is a coordinate ball or half-ball and that for each i there exists a coordinate domain B′

i ⊃ Bi. We require the

coordinate map φi : B′

i → Rm restricts to a diffeomorphism from Bi to a com-

pact subset of Rm. For each i we choose ρi : M → R a smooth bump function that is equal to 1 on Bi and supported in B′

i. Then we have a smooth map

F : M → Rmn+n F(p) = (ρ1(p)φ1(p), ... , ρn(p)φn(p), ρ1(p), ... , ρn(p)) this is well defined on M by zero-extending ρiφi out of the support of ρi. We claim that F is an injective smooth immersion. Since M is compact, this implies F is an embedding. First we show F is injective. If F(p) = F(q), then we choose some Bi contains p, then ρi(p) = ρi(q) = 0. So q lies in the support of ρi. But coordinate map φi(p) = φi(q), so p = q. Then we show F is an immersion. For any p ∈ M, choose some Bi contains p. Since ρi ≡ 1 on Bi, we have dρiφi|p = ρi(p)dφi(p) is an injection. Now we assume M ⊂ Rn and N ⊂ R2n to be the total space of the normal bundle of M taken with respect to the standard metric on RN, i.e. N = {(q, v) : q ∈ M, v ⊥ TqM} Define E : N → Rn, (q, v) → q + v. Following Milnor’s notion, we will call the points e = q + v a focal point of (M, q) if the Jacobian of the map E at (q, v)

degenerates. Note that e is a focal point for some (M, q) if and only it is a

critical value of E. Applying Sard’s theorem, we know that focal points are of measure 0 in Rn. Conversely, fix e ∈ Rn and define Le : M → R, q → ||q − e||2. Then we have: Proposition 2.2. Let q ∈ M be a critical point of Le. Then it is a degenerate critical point of Le if and only if e is a focal point of (M, q).

Proof. To help us get down to computation, we will make use of the local

coordinates. Let ui be a chart arond q and q(u1, ..., um) be corresponding parametrization of M ⊂ Rn. Choose another n − m normal vector field around q : nm+1, ...,

nn. If q is a critical point of Le, we have:

∂uiLe = ∂ui(q − e) · (q − e) = 2∂uiq · (q − e) = 0 , ∀i (8) ∂2

uiujLe = 2[∂uiq · ∂ujq − ∂2 uiujq · (q − e)]

(9) Then (q−e) is perpendicular to the tangent space of M at q. Thus e is in the im- age of E : N → Rn. Use (u1, ..., um, tm+1, ..., tn) → (q(u1, ..., um), tm+1 nm+1 + ... + tn nn as a chart on N, we compute: ∂uiE = ∂uiq +

k=m+1

tk∂ui nk (10) 5

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∂tjE = nj (11) To see the rank of E at (q, e − q), we note that {∂uiq, nk} is a basis of TeRn. So we take the inner product of these two sets of vectors to get a matrix whose rank is equal to the rank of Jacobian of E:

∂uiq · ∂ujq + ∂uiq · tk∂uj

nk ∗ Id

Note that the first block of the above matrix ∂uiq · ∂ujq + ∂uiq · tk∂uj

nk = ∂uiq · ∂ujq − ∂2

uiujq · (q − e) at (q, e − q) ∈ N. We conclude that E is singular if

and only if HqLe is singular. The theorem follows. For any f ∈ C∞(M), we choose an embedding of M into Rn written as (f, f1, f2, ..., fn). Fix constant c sufficiently large and let p = (−c + ǫ1, ǫ2, ..., ǫn) close to (−c, 0, ..., 0) satisfying Lp : M → R is nondegenerate. Then g(x) :=

Lp−c2 2c

is non-degenerate and approximates f as required when c are large and ǫi are small.

2.3 Homotopy type and critical values

Let f be a Morse function on M. By perturbing locally with a bump funciton, we might assume that f takes different values at its critical points. Define M a := {x ∈ M : f(x) ≤ a}. For a that’s not a critical value of f, M a is a manifold with boundary f −1(a). As a runs from −∞ to +∞, M a grows from empty to the whole of M. The following two theorems tells us how the manifold changes its homotopy type at a critical point of index k. These results follow the spirit of our previous slogan:“the analytic behaviour of functions on a manifold M should reflect its topology”. Theorem 2.2. If [a, b] contains no critical value, then M a ∼ = M b.

Proof. Choose an Riemannian metric g on M, then we have the (negative)

gradient vector field defined by: ∇f := −g∗(d f) (12) On f −1[a, b], we have ∇f = 0. Choose a smooth function h defined on M such that coincides with g(∇f, ∇f)−1 on f −1[a, b] and vanishes outside a neighbour- hood of it (which contains no critical points). Then X := h · ∇f is well defined throughout M. Let Φt be the corresponding flow of X. Then for q ∈ f −1[a, b]: d dtf(Φt(q))|0 = X · f|q = g(X, g∗(d f))|q = −g(∇f, ∇f) g(∇f, ∇f) = −1 (13) So Φb−a sends M b diffeomorphically onto M a. The theorem follows. Theorem 2.3. If p is an critical point of index k and c = f(p), then for sufficiently small ǫ > 0, we have M c+ǫ ≃ M c−ǫ ∪φ ek, where φ is a continuous map from Sk−1 to M. In other words, when we cross a critical point of index k, M a changes its homotopy type by attaching a k-cell. 6

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Figure 1: An easy example for Theorem 2.3

Proof. See Figure 1 for an illustration. We introduce another function F which

coincides with the Morse function f except that F < f near p. Then M c+ǫ = F −1(−∞, c + ǫ] is diffeomorphic to F −1(−∞, c − ǫ] by Theorem 2.2. We choose a suitable cell ek ⊂ F −1(−∞, c − ǫ] and show that M c−e ∪ ek is a deformation retract of F −1(−∞, c − ǫ]. Then the theorem follows. First we set up some notations. Let (U, {ui}) be a Morse chart centered at p, then locally we have f = c − u2

1 − ... − u2 k + u2 k+1 + ... + u2 m

(14) Mimicing the case of dimension 2, we define ξ := k

i=1 u2 i and η := n i=k+1 u2 i .

Figure 2: Local situation Choose ǫ > 0 sufficiently small so that [c − ǫ, c − ǫ] contains no other critical

value. Define:

ek = {x ∈ U : ξ(x) ≤ ǫ; η(x) = 0} (15) Note that ek ∩M c−ǫ = Sk−1, so we have ek attached to M c−ǫ as expected. Now we give the definition of F. Let µ be sa smooth function on R satisfying: µ(0) > ǫ (16) 7

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µ(0) = ǫ for r ≤ 2ǫ (17) − 1 < µ′(r) ≤ 0 (18) Let F := f − µ(ξ + 2η), then F extends to a smooth function on M. We claim the following: (1) F −1(−∞, c + ǫ] = f −1(−∞, c + ǫ] (2) F has the same critical points as f does (3) M c−ǫ ∪ ek is a deformation retract of F −1(−∞, c − ǫ] The first two properties follow directly from definition. Note that F has no critical value in [c − ǫ, c + ǫ], so M c+ǫ = F −1(−∞, c + ǫ] is diffeomorphic to F −1(−∞, c − ǫ] by Theorem 2.2. We give the explicit construction for (3) to conclude the proof. Keep in mind that we need to construct a family of map rt such that r1 = id and r0 is retraction from F −1(−∞, c − ǫ] to M c−ǫ ∪ ek. The definition is as follows:

For p such that ξ ≤ ǫ, define rt(u1, ..., un) = (u1, ..., uk, tuk+1, ..., tun).
For p such that ǫ ≤ ξ ≤ η+ǫ, define rt(u1, ..., un) = (u1, ..., uk, stuk+1, ..., stun),

where st = t + (1 − t)(ξ − ǫ)1/2(η)−1/2.

For p ∈ M c−ǫ, define rt =Id.

As an application of our works, we will prove Morse’s inequality (1.1). Theorem 2.4. Let f be a Morse function on a compact manifold M. Then M has the homotopy type of a finite CW complex with one k-cell attached for each critical point of index k.

Proof. As we have assumed that f has distinct critical values, let α0 < α1 <

... < αl be the critical values of f. Choose ǫ0 > 0 sufficiently small, M α0+ǫ0 is diffeomorphic to a k-dimensional closed disc. Choose again ǫ1 > 0 sufficiently small, by Theorem 2.2 and 2.3, we have M α1+ǫ1 has the homotopy type of M α0+ǫ0 with a indα1-cell attached. Then we can conlude the proof by induction. With theorem 2.4 at hand, the Morse’s Inequality follows directly from the following algebraic fact. Proposition 2.3. Let C∗ be a finite complex of vector spaces and H∗ its ho-

mology. Then for P(t) := dim Citi and Q(t) := dim Hiti. The ”Morse’s

Inequality” (1) holds.

Proof. We might assume that Ci = 0 for i < 0 and i > m. Let di : Ci → Ci+1

be the i-th differention. For simpicity, let pi and qi be the i-th coefficient of 8

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corresponding polynomial. Consider the n-th coefficient of (1 + t)−1Q(t): ((1 + t)−1Q(t))n = qn − qn−1 + qn−2 + ... + (−1)nq0 = dim ker dn − dim im dn−1 − dim ker dn−1 + ... + (−1)n dim ker d0 = dim ker dn − pn−1 + pn−2 + ...(−1)np0 ≤ pn − pn−1 + pn−2 + ...(−1)np0 (19) which is the n-th coefficient of (1 + t)−1P(t). The proposition follows. Remark 2.1. Note that the corresponding cellular chain complex can be viewed as ‘freely generated by critical points of f’, though in this way we cannot read

ff the information of differentiation ∂. In next section we will see how one can

define the complex directly.

3 Thom-Smale Complex and Morse Homology

In this section, we define Thom-Smale complex and prove some of its fun- damental properties. Throughout this section, M will be a compact manifold without boundary. Although at times we need to deal with manifold with boundary to help us develop the theory. To clarify the notion, we define the Ck topology that we will use in this section. First consider the local (Euclidean) case: fix a map f : U → V between open subsets of Rn. A sequence of maps converges to f in Ck topology means that up to k-th derivatives, it converges to f uniformly on each compact subset of

U. Now consider a map φ : M → N, fix a collection of chart Ui covering M

such that for each Ui, f(Ui) is contained in a chart Vi ⊂ N. A sequence of maps converges in Ck topology to φ means for each i, the restricted sequence converges to f|Ui : Ui → Vi as maps bewteen open subsets of Euclidean space.

3.1 Pseudo gradients

From the proof of Theorem 2.2, we see that the gradient flow of a Morse funci- ton is of fundamental importance in understanding the topology of a manifold. We give the following definition to describe a slightly general family of vector fields. Definition 3.1. A vector field X on M is called a pseudo-gradient field adapted to Morse function f if the following holds:1

Xf ≤ 0, with the equality holds only at the critical points of f.
For each critical point x of f, there exists a Morse chart centered at x

such that in this chart X is the (negative) gradient of f with respect to the stardard metric of Rm. 9

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Figure 3: How pseudo gradient behaves near a critical point Proposition 3.1. For any Morse function f on M, there exists pseudo gradient field adapted to f.

Proof. This follows from the abundance of Riemannian metrics on M. Let us

fix a collection of Morse charts centered at each critical points and add some more charts to make it a collection of charts covering M. Note that each chart is equipped with a metric induced by the standard metric on Euclidean space. Now we have a Riemannian metric g on M pieced together by partition of unity. This metric still coincides with the standard metric on the (possibly smaller) Morse charts. So the gradient field of f taken with respect to g would give us a pseudo gradient field. For a pseudo-gradient field X, we can integrate it to get a family of diffeo- morphisms ΦX

for t ∈ R. The fixed points of this flow are exactly the critical points of f. Moreover, every point on M flows to (and from) some critical point

f f. This is guaranteed by the following result.

Proposition 3.2. Let M be a compact manifold without boundary, f a Morse function on M and X a pseudo-gradient field adapted to f. For any point x ∈ M, there exist critical points of f, p and q such that: lim

t→−∞ ΦX t (x) = p

(20) lim

t→+∞ ΦX t (x) = q

(21)

Proof. We might assume that x ∈ M is not a critical point of f. Let ΦX

t (x)

be the flow line of x. Note that f(ΦX

t (x)) is strictly decreasing in t. So ΦX t (x)

1If M has boundary, X is required to point strictly inwards along the boundary.

10

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never visits the same Morse chart twice. If ΦX

t (x) doesn’t converges to some

critical points q as t goes to +∞, then it will stay strictly away from all the Morse charts when t is sufficiently large. By the definition of pseudo gradient field, Xf ≤ 0 and equality holds only at critical points of f. So there exists ǫ > 0 such that outside these Morse charts we have Xf ≤ −ǫ (22) By definition of the flow, we have: d dtf(ΦX

t (x)) = X(ΦX t (x))f ≤ −ǫ

(23) Integrating this equation, we see that f(ΦX

t (x)) decreases to −∞ as t goes to

+∞, which is absurd since f is bounded below. The case where t tends to +∞ is similar. Remark 3.1. This result is certainly not true for manifold with boundary, where the flow line might end at a boundary point. Following the idea of above proposition, we can define a decomposition of M: Corollary 3.1. Given a gradient flow ΦX

t , for each critical point x of f, we

consider: M u

X(x) := {p ∈ M :

lim

t→−∞ ΦX t (p) = x}

(24) M s

X(x) := {p ∈ M :

lim

t→+∞ ΦX t (p) = x}

(25) where the superscript ‘u’ and ‘s’ stand for ‘unstable’ and ‘stable’. Then M =

x∈Crit(f)

M u

X(x) =

x∈Crit(f)

M s

X(x)

(26) are both decompositions of X as disjoint unions. Proposition 3.3. For a critical point x of index k, M u

X(x) is a submanifold of

M diffeomorphic to an open k-disk. Likewise, M s

X(x) is an embedded subman-

ifold of M diffeomorphic to an open (m − k)-disk.

Proof. Inspecting the local behaviour might be of help to understand how the

stable/unstable manifolds behave. Under a Morse chart U centered at some critical point p, we assume Morse function looks like f = c +

m−k

x2

i − k

y2

(27) Then the unstable manifold of x in this chart is exactly those points where xi all vanish: U ∩ M u

X(x) = {q ∈ U : ∀i, xi(q) = 0}

(28) 11

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For sufficiently small ǫ, we have an embedding: φ−1 : Sk−1 = {(xi, yi) ∈ Rm : xi = 0;

i = ǫ} → M

(29) where φ = (xi, yi) : U → Rm is the coordinate map. This map can be extended to F : Sk−1 × [−∞, +∞)/Sk−1 × {−∞} → M F(x, t) := ΦX

t (φ−1(x))

F(x, −∞) := p (30) Here Sk−1 × [−∞, +∞)/Sk−1 × {−∞} is homeomorphic to Rk by the obvious map (x, t) → exp(t)x. And the smooth structure on it is induced from Rk by this map. It is obvious that F is an embedding away from origin. However, by definition of the pseudo gradient field, we find that under the Morse chart, exp(t)x is exactly the flow ΦX

t (x). So φ−1 ◦F is identity map near origin. Thus

F is an embedding of Rk(Dk) into M and its image is exactly M u

X(x). The case

f stable manfold is proved likewise.

Definition 3.2. We call M u

X(x) and M s X(x) stable manifold and unstable man-

ifold of x. Let’s revisit the examples we considered above. Example 3.1. First consider the embedded sphere S2 in R3 equipped with height function. With the induced metric we have a gradient field X on S2, and its flow ‘flows from high to low’ as expected. Let’s name the north pole p and south pole q. Then M u

X(p) = S2 − {q} and M s X(p) = {p}. Likewise,

M s

X(q) = S2 − {p} and M u X(q) = {q}.

Example 3.2. For the embedded torus T 2 equipped with height function and induced metric, the corresponding unstable manifold for each point is indicated as below. The open 2-cell is contained in Mu

X(d). (To get the stable manifolds,

turn the figure upside down.) Figure 4: Decomposing T 2 into unstable manifolds. Example 3.3. For (T 2, f = sin 2πx + sin 2πy), we equip T 2 with flat metric and thus get a gradient field. The corresponding unstable manifold for each point is indicated as below. 12

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Figure 5: Decomposing T 2 into unstable manifolds (again).

3.2 Smale condition

Inspecting the example of T 2 equipped with standard metric induced from R3 above, we see that the unstable manifold and stable manifold of two in- dex 1 critical points doesn’t intersect transversely. To avoid such pathological behaviour of submanifolds, we introduce the Smale condition: Definition 3.3. We say a pseudo gradient flow X (adapted to some Morse function f) satisfies Smale condition if for all x, y ∈ Crit(f) M u

X(x) ⋔ M s X(y)

(31) To be more precise, recall that two submanifold L, N ⊂ M intersects trans- versely if for each point x ∈ L ∩ N, we have: TxL + TxN = TxM (32) The following is an easy consequence of transversal property and implicit func- tion theorem. Proposition 3.4. If L ⋔ N = ∅, then L ∩ N is a submanifold of M, with codim L ∩ N = codim L + codim N (33) Also note that dim M u

X(x) = m−ind(x) and dim M s X(y) = ind(y), so Smale

condition is equivalent to saying that:

If ind(x) < ind(y), then M u

X(x) ∩ M s X(y) = ∅.

If ind(x) ≥ ind(y), then M u

X(x)∩M s X(y) is either empty set or a subman-

ifold of dimension ind(x) − ind(y). The other examples above all satisfy the Smale condition. The following theorem shows the genericness of Smale condition. Theorem 3.1. Any pseudo gradient field X can be approximated arbitrarily in C1 topology by another pseudo gradient field X′ that satisfies Smale condition. 13

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Remark 3.2. To clarify the notion, ‘C1 approximation’ means that we can find pseudo gradient field X′ arbitrarily C1 close to X as maps M → TM which satisfies Smale condition. In other words, for every ǫ > 0, for every cover of M by coordinate charts Ui and for every collection of compact subsets Ki, there is a psuedo gradient field X′ such that under the chart Ui ||X′ − X|| ≤ ǫ (34) for C1 norm on Ki.

Proof. Without loss of generality, we may assume that f takes different value

at each critical point.2 Assuming the critical points are {ci} satisfying f(c1) > f(c2) > ... > f(cq), we prove the theorem by inductively modifying the given gradient field X. The induction is based on the following fact: For each ci, there is a pseudo gradient flow X′ C1 approximatex X as required and a poisitive number ǫ such that:

X′ coinsides with X outside f −1[f(ci) + ǫ, f(ci) + 2ǫ].
The stable manifold of cj (for X′) is transversal to the unstable manifolds
f all critical points (for X′).

With this fact at hand, we can apply it inductively on c1, ..., cq to get X′ as

required. Now we focus on proving this fact.

Given X and ci. We choose ǫ so small that f(ci) + 2ǫ < 2f(ci−1) (35) and that Q := M s

X(ci) ∩ f −1(f(ci) + 2ǫ) lies in the Morse chart centered at ci.

Then Q is diffeomorphic to Sn−k−1. Choose some tubular neighborhood of Q inside f −1(f(ci) + 2ǫ): Q × Dk → f −1(f(ci) + 2ǫ) (36) We extend it into an embedding: Φ : Dk × Q × [0, m] → f −1[f(ci) + ǫ, f(ci) + 2ǫ] (37) such that

Φ(0, ∗, 0) embeds Q into f −1(f(ci) + ǫ)
Φ(0, ∗, m) embeds Q into f −1(f(ci) + 2ǫ)

2As we’ve seen above, this can be achieved by perturbing f locally with a function that is

constant on Morse chart. Then X will still be a pseudo gradient field adapted to the perturbed function.

14

SLIDE 15

If z is the coordinate in [0, m], then

Φ∗(−∂z) = X (38) This extension can be obtained directly by integrating the vector field X. In fact, suppose under Morse chart U we have f = x2

i − y2 i , then pseudo

gradient X takes the form X = −2(

xi∂xi −
yi∂yi)

(39) Then along the flow we have d dtf(Φt

X(q)) = Xf(Φt X(q)) = 4||X(Φt X(q))||2

(40) Solving this ODE, we see that Φt

X satisfies: for q1, q2 ∈ M u(cj) ∩ U such that

f(q1) = f(q2), we have f(Φt

X(q1) = f(Φt X(q2). Thus we may choose m such

that Φm

X({0} × Q) ⊂ f −1(f(ci) + 2ǫ). Any unstable manifold M u X is transversal

to the level sets, so it intersects with Dk × Q × {m} along a manifold P ′. If M s

X(ci) ⋔ P ′, then we necesarrily have M s X(ci) ⋔ M u

X. So we need to modify

X to X′ inside Im(Φ) ⊂ f −1[f(ci) + ǫ, f(ci) + 2ǫ] so that M s

X′(ci) ⋔ P ′.

Let P := Φ−1P ′ ⊂ Dk × Q × [0, m]. The desired transersality condition in the model reads M s

X′ ⋔ P where

M s

X′ = {Φt X′(0, q, 0) : t < 0, q ∈ Q}

(41) When X′ = X = −∂z, P ∩ M s

X = P ∩ {(0, q, m) : q ∈ Q} = p−1(0)

(42) where p : P → Dk is the projection to Dk. By Sard’s theorem, we can find ω ∈ Dk arbitrarily close to 0 such that for any unstable manifold M u and P corresponds to it, ω is a regular value for the projection p. Then we can choose a perturbation X′ such that W s

X′ ∩ {z = m} = Φ−m X′ (0, q, 0) = (ω, q, m).

Now W s

X′ ∩ P = p−1(ω) is a submanifold of codimension k in P. Counting

the codimension would imply transversality between W s

X′ and P. Thus we can

conclude the proof by the following construction: Claim: There is vector field X′ arbtrarily C1 close to X such that

X′ = −∂z near ∂Dk × Q × [0, m]
Φt

X′(0, q, 0) = (ω, q, m)

Let ω = (v1, ..., vk) ∈ Dk, define X′ := −∂z −

βi(z)γ(x)∂xi (43) where 15

SLIDE 16

γ is a bump function on Dk centered at origin and |∂γ/∂xi| ≤ 2.
βi is 0 near the boundary of [0, m] and satisfies |βi(s)| < η and |β′

i(s) < η|,

where η is a small positive number that’s chosen together with ω. Then X′ satisfies the claimed property. Definition 3.4. Suppose X is a pseudo gradient field adapted to some Morse function f, then the union of flow lines connecting two critical points a, b M u

X(a) ∩ M s X(b) = {x :

lim

t→−∞ Φt X(x) = a; lim t→+∞ Φt X(x) = b}

(44) is either empty or a submanifold of dimension ind(a) − ind(b). If a = b, there is a free R-action on M u

X(a) ∩ M s X(b) by (t, x) → ΦX t (x). We define the ‘moduli

space of trajectories (flow lines) connnecting two critical points a, b’ to be Ma,b := M u

X(a) ∩ M s X(b)/R

(45) Remark 3.3. An another approach to understand this moduli space Ma,b is to choose a regular value x of f lying inside (f(b), f(a)). Then f −1(x) is a submanifold of M that intersects transversely with M u

X(a) ∩ M s X(b). Ma,b can

then be identified with f −1(x) ∩ M u

X(a) ∩ M s X(b) since each flow from a to b

intersects with f −1(x) exactly once.

3.3 Thom-Smale complex

As we have promised, we will construct a complex free generated by critical points of f over Z/2.

We have a differential graded Z/2-vector space Cf

∗

whose degree k part is Cf

k :=

x∈Crit(f);ind(x)=k

xZ/2 (46) And we have a differential ∂X : Cf

∗ → Cf ∗ whose action on each x ∈ Crit(f) is

defined by ∂Xx :=

y∈Crit(f)

N(x, y)y (47) where N(x, y) = #Mx,y if ind(y) = ind(x) − 1

therwise

(48) We recall the definition of Mx,y = M u

X(x) ∩ M s X(y)/R. So it is an manifold

f dimension ind(x) − ind(y) − 1. If ind(y) = ind(x) − 1, Mx,y is a discrete

space, we will show that it is also compact. Then ‘counting number’ map is well

3This construction can surely be carried out over field of arbitrary character. It suffices to

consider orientation in each step below.

16

SLIDE 17

defined. To do so, we first define the compactification of moduli space for more

general cases: M

x,y :=

z1,..,zi

Mx,z1 × Mz1,z2 × ... × Mzi,y (49) where i ≥ 0 is arbitrary, and ind(x) > ind(z1) > ... > ind(zi) > ind(y). The elements of M

x,y are called ‘broken trajectories’ connecting x and y, for we

can view it as flows consecutively visting x, z1, z2, ..., zi, y. We equip it with the following topology. To clarify the notions, we have the following definition: Definition 3.5. For a Morse function f on M, we fix Morse charts U(x) for each critical point x and choose ǫ(x) > 0 sufficiently small. Then we have Ω±(x) := U(x) ∩ f −1(f(x) ± ǫ(x)). When a trajectory (or a broken trajectory) meets Ω(x)±, we say that it enters (leaves) the corresponding Morse chart. Let γ = (γ1, γ2, ..., γi) ∈ Mx,z1 × Mz1,z2 × ... × Mzi−1,y, a neighbourhood U ǫ of γ in M

x,y contains τ such that:

There exist 1 ≤ i1 < ... < ij−1 ≤ i − 1 such that τ = (τ1, ..., τj) ∈

Mx,zi1 × ...Mzij−1,y.

For each x, z1, z2, ..., y, and its corresponding Morse chart, we require that

τ enters (exits) the chart ǫ-close to γ. Remark 3.4. The topology defined above is obviously Hausdorff and second

countable. The inclusion i : Mx,y → M

x,y is an embedding.

Theorem 3.2. The space M

x,y is compact.

Proof. It suffices to show sequential compactness since the space is second count-
able. As the definition above, we fix Morse charts U(x) for each critical point

x, and fix Ω±(x) to determine when a trajectory leaves or enter a Morse chart. Let ln be a sequence of trajectories in M

x,y. First assume that all ln lies in

Mx,y, let l∓

n be the points where ln leaves U(x) and enters U(y). Note that

Ω−(x) ∩ M u

X(x) is diffeomorphic to Sind(x)−1 which is compact. So we may

assume l−

n converges to x− and likewise, l+ n converges to y+. Let τ1 ∈ Mx,z be

the trajectory through x− and τ +

1 be the point where τ1 enters U(z). By the

dependence of solution on initial data, ln will enter U(z) for n >> 0. Let mn be the points where ln enters U(z), we claim that m+

n would converge to τ + 1 .

This will follow from the lemma 3.1 below. If z = y, then by above claim, we have ln → τ1. If this were not the case, then m+

n is not contained in M s X(z) for n large. So ln will leave U(z) through

m−

n . Again, we assume m− n converges to z− and claim that z− lies on M u X(z).

If this were not the case, then z− would be on the flow line of some point z+ with f(z+) = f(m+

n ) such that z+ is not in M s X(z). But by lemma 3.1 again,

m+

n converges to z+ and τ + 1 at the same time. So z+ = τ + 1 ∈ M s X(z) which is

absurd. 17

SLIDE 18

Now we are at the situation as the beginning of the proof, so we might repeat the argument above until we stop at some zi = b and get a sequence converges to (τ1, ..., τi). For the general case where ln ∈ M

x,y, we may pass to a subsequence such

that for all n ln = (τ 1

n, ..., τ i n) ∈ Mx,z1 × ... × Mzi−1,y

(50) and apply the above result to τ 1, ...τ i consecutively. Lemma 3.1. Let xn ∈ M be a sequence converging to x which is not a critical point of f and f(x) = f(xn). If y lies on the flow line of x and yn lies on the flow line of xn such that f(y) = f(yn), then we have: lim

n→∞ yn = y

(51)

Proof. As we’ve done before, choose some neighbourhood U of Crit(f) that

doesn’t touch x, y, xn, yn. We have a flow Y defined through M which satisfies Y = − 1 d f(X)X on M − U (52) Then there is some t0 such that Φt0

Y (x) = y

(53) and Φt0

Y (xn) = yn

(54) Then lim

n→∞ yn = lim n→∞ Φt0 Y (xn) = Φt0 Y (x) = y

(55) Corollary 3.2. If ind(y) = ind(x) − 1, then Mx,y is a finite set. Lemma 3.2. Mx,y is dense in M

x,y. To be more precise, for every element in

M

x,y, each of its neighbourhood contains an element in Mx,y.

Clearly ∂ is of degree [−1]. It remains to show that it squares to 0. To do so, we study the structure of M

x,y for ind(y) = ind(x) − 2.

Theorem 3.3. For such x, y, the compactified space M

x,y is an compact 1-

manifold with boundary. Moreover, its boundary corresponds exactly to those points in

z Mx,z × Mz,y. To be more precise, let λ1 ∈ M(x, z) and λ2 ∈

M(z, y), there is an embedding ψ from [0, δ) onto a neighbourhood of (λ1, λ2) in M

x,y which is smooth on (0, δ) and satisfies

ψ(0) = (λ1, λ2)
ψ(s) ∈ Mx,y for s = 0

18

SLIDE 19

Moreover, if ln ∈ Mx,y converges to (λ1, λ2), then ln lies in the image of ψ for n sufficiently large.

Proof. We start by clarifying the notions. Assume the index of x, z, y is

k+1, k, k−1, repectively. As the definition above, we’ve chosen U(x), Ω±(x), ǫ(x) for all critical points. Let α := f(z) and S+(z) := M s

X ∩ f −1(α + ǫ(z))

(56) S−(z) := M u

X ∩ f −1(α − ǫ(z))

(57) Let a1 = S+(z)∩λ1. The unstable manifold M u(x) intersects transversally with f −1(α+ǫ(z)) along a submanifold P. And by Smale condition, P intersects with S+(c) transversally inside f −1(α+ǫ(z)) at finitely many points. Let Φ : Dk → a1 be a chart for P centered at a1 such that ImΦ ∩ S+(z) = {a1} (58) After shrinking the domain, we might assume that D =ImΦ is contained in Ω+(z). By letting D − {a1} descend along the trajectories of X, we have an embedding Ψ : D − {a1} → Ω−(z) (59) Now we claim the following: Proposition 3.5. After shrinking the domain of Ψ, if necessary, the union Q = Im Ψ ∪ S(z) is a manifold of dimension k and its boundary is exactly ∂Q = S−(z). We postpone the proof for this proposition and see how it leads to our

theorem. By Smale condition we have

M s

X(y) ⋔ S−(z)

(60) And note that ImΨ is an open subset of the level set, so M s

X(y) ⋔ ImΨ

(61) So Q intersects with M s

X(y) along a submanifold of dimension 1 with boundary,

and its boundary is ∂Q ∩ M s

X(y) = S−(z) ∩ (M s X(y) = Mz,y.

Let a2 be the point where λ2 meets S−(z). Then a2 ∈ S−(z) ∩ M s

X(y) ⊂

Q ∩ M s

X(y), moreover, it is boundary point. We choose a parametrization of

Q ∩ M s

X(y) around a2:

ψ′ : [0, δ) → Q ∩ M s

X(y)

(62) such that ψ′(0) = a2. Then we might lift this parametrization to M

x,y by

setting ψ : [0, δ) → M

x,y

(63) 19

SLIDE 20

where ψ(0) = (λ1, λ2) and ψ(s) = ‘the flow line through ψ′(s)’. This ψ is a diffeomorphism on (0, δ) and it is continuous at 0. To see this, let ψ(s)± be the points where ψ(s) enters/leaves U(z). Then we have ψ(s)− = ψ′(s) = Ψ(ψ(s)+) (64) Since ψ′(s) converges to a2 as s → 0, we have Ψ−1(ψ′(s)) converges to a1 (by lemma 3.1). So ψ is indeed continuos at 0. It remains to show that this is the

nly ‘branch’ that reaches (λ1, λ2). Suppose ln ∈ Mx,y converges to (λ1, λ2),

let l±

n be the points where ln enters/leaves U(z). By convergence, we have:

lim

n→∞ l+ n = a1

(65) lim

n→∞ l− n = a2

(66) Note that l+

n ∈ M u X(x) ∩ f −1(α + ǫ(z), so l+ n lies in Dk − {a1} for n large. Then

l−

n = Ψ(l+ n ) lies in Q. So ln is in the image of ψ as expected.

Proof for Proposition 3.1. This reduces to inspecting the situation in a Morse

chart. We set the following notions:

U ⊂ Rk × Rn−k (67) f(y, x) = −||y||2 + ||x||2 (68) S+ = {(y, x) : y = 0, ||x||2 = ǫ} (69) S− = {(y, x) : x = 0, ||y||2 = ǫ} (70) Ω± = f −1(±ǫ) ∩ U (71) We restate the required proposition in the local case as follows: Let a ∈ S+ and D ⊂ Ω+ be an embedded dimension k disk that intersects S+ transversally inside Ω+ at a. Let Ψ : Ω+ − S+ → Ω− − S− be the embedding defined by the flow of X. Then (after restricting D to a smaller disk, if necessary) Q = Ψ(D − {a}) ∪ S− (72) is a manifold with boudnary, and its boudnary is exactly S−. The following observation is useful in our proof. X = −∇f = 2(y, −x). Then the flow of X is Φt

X(y, x) = (e2ty, e−2tx)

(73) So the embedding Ψ above can be written down explicitly in coordinate: Ψ(y, x) = (||x|| ||y||y, ||y|| ||x||x) (74) We consider the projection p : (y, x) → y. Note that as a submanifold of Ω+, S+ is defined by y = 0. So the fact that D ⋔ S+ tranlates to the fact that 0 is a 20

SLIDE 21

regular value of p|D. By implicit function theorem, after shrinking D, we may assume D = {(y, h(y)) : y ∈ D′} (75) where D′ ⊂ Rk ×{0} is some k-disk centered at 0 and h : D′ → Rn−k is smooth with h(0) = a. Because D ⊂ Ω+, we have: ||h(y)||2 = ||y||2 + ǫ (76) Define g =

h ||h||, we write

D = {(y,

||y||2 + ǫg(y) : y ∈ D′}

(77) Then Ψ(D − {a}) = {(

||y||2 + ǫ

||y||2 y, ||y||g(y) : y ∈ D′ − {0} (78) We expect to piece together S− and Ψ(D − {a}) along the ‘inner side’ - as ||y|| tends to 0. To do so , we introduce the polar coordinate (ρ, u) ∈ (0, δ) × Sk−1

n D′ − {0}. Under this coordinate, Ψ(D − {a}) is like

H : (0, δ) × Sk−1 → Ω− (ρ, u) → (

ρ2 + ǫu, ρg(ρ, u))

(79) Then we extend domain of H to [0, δ) × Sk−1 by setting H(0, u) := (√ǫu, 0) ∈ S− (80) Thus Q = H([0, δ) × Sk−1) is a manifold with boundary. And its boundary is exactly S−. Remark 3.5.

We haven’t justify that M

x,y is a compactification of Mx,y.

It still remains to show that Mx,y is dense in M

x,y.

This is done by following the same argument we used in the above proof.

In fact the same argument shows that for ind(x) − ind(y) = k, the com-

pactified space M

x,y admits a structure of manifold with corners. And its

l dimensional parts are exactly those Mx,z1 × ... × Mzk−l−1,y. Corollary 3.3. ∂2 = 0.

Proof. For each critical point x, we compute

∂2(x) =

ind(z)=ind(x)−1
ind(y)=ind(z)−1

n(x, z)n(z, y)y =

ind(y)=ind(x)−2
ind(z)=ind(x)−1

n(x, z)n(z, y)y (81) 21

SLIDE 22

For x, y such that ind(y) =ind(x) − 2, we have

ind(z)=ind(x)−1

n(x, z)n(z, y) = #

ind(z)=ind(x)−1

Mx,z × Mz,y (82) But

ind(z)=ind(x)−1 Mx,z × Mz,y is the boundary of M x,y, a compact 1-

manifold. Note that a compact 1-manifold with nonempty boundary is disjoint

union of some closed intervals and circles. So the number of its boundary points will be even, in other words, 0 in Z/2. Definition 3.6. The complex (Cf

∗ , ∂X) defined above is called Thom-Smale

complex. Its homology will be called Morse homology and written Hf

∗ (both

considered associated to the Morse function f and pseudo gradient field X). We compute a few examples below. Example 3.4. For the embedded sphere (S2,f,X), the associated complex Cf

is zero except for k = 0, 2. For these two dimensions, we have Cf

k ∼

= Z/2. There are no two consecutive dimensions with nonzero chain groups so the differential ∂X = 0. And Hf

k =

if k = 0, 2

therwise

Example 3.5. For the torus (T 2, f = sin 2πx+sin 2πy, X), we have one critical point of index 0, two critical points of index 1 and one critical point of index 2. Inspect again the flow lines we draw above. Note that for each pair of points, the flow lines connecting them come up in pair, so we have ∂X = 0. Then Hf

k =

   Z/2 if k = 0, 2 Z/2 ⊕ Z/2 if k = 1

therwise

3.4 Manifold with boundary (or corners)

A manifold with corners is locally modeled by Rn−k × [0, ∞)k. The special case where k = 1 is often called a manifold with boundary. [0, 1]k is a good first example to think about when working with manifold with corners. For these manifolds, we have seen that doing Thom-Smale complex can be

complicated. For instance, a flow line might end up at some boundary point,

while on closed manifolds a flow line always connects two critical points. How- ever, if we can avoid such bad circumstances, the above constructoin still makes

sense. To be more precise, we have the following:

Theorem 3.4. Let M be a compact manifold with corners, f a Morse function

n M and X a pseudo gradient field on M adapted to f. We assume

22

SLIDE 23

The critical points of f all stay away from the boundary of M.
If M is a manifold with boundary, then X is transversal to the bound-

ary ∂M. More generally, if M is a manifold with corners, X would be transeversal to ‘each faces’. Then the construction in this section carries over to this situation.

Proof. The assumption guarantees that those flow lines connecting two critical

points stay away from the boundary, so the proof above is still valid in this case.

4 A priori: the invariance of Thom-Smale com- plex

In this subsection, we will show that up to chain homotopy, the Thom-Smale complex doesn’t depend on the choice of Morse function f or pseudo gradient field X. The theorem we intend to prove is the following: Theorem 4.1. Given a compact manifold without boundary M, let (f, X) and (f ′, X′) be two pairs of Morse functions and pseudo gradient flow adapted to it. Then we have an chain homotopy: Φ(f,X),(f ′,X′) : (Cf

∗ , ∂X) → (Cf ′ ∗ , ∂X′)

(83) Historically, this follows from the classical result that Morse homology is isomorphic to the singular homology of the manifold M. Without this result at hand, here we mimic the strategy of Floer [3]. His method of constructing ‘con- tinuation map’ (the map Φ below) plays an important role in the development

f the infinite dimensional analogy of Morse homology. The scheme of proof is

as follows:

For two such pairs (f, X) and (f ′, X′), we construct a chain map Φ(f,X),(f ′,X′).
We show that

Φ(f,X),(f,X) = Id

We show that for three such pairs (fi, Xi)

Φ(f2,X2),(f3,X3) ◦ Φ(f1,X1),(f2,X2) = Φ(f1,X1),(f3,X3) where the equality holds up to chain homotopy.

Proof. We carry out the construction above as follows:

23

SLIDE 24

Step 1 Given (f, X) and (f ′, X′). We choose a smooth function h(t) on [−1/3, 4/3] such that h([−1/3, 1/3]) = 1 and h([2/3, 4/3]) = 0. Define a function F : M × [−1/3, 4/3] Ft(x) = F(x, t) = h(t)f(x) + (1 − h(t))f ′(x) (84) Then we may choose another function g(t) on [−1/3, 4/3] whose critical points are 0 (maximum) and 1 (minimum). We require that it is increasing on [−1/3, 0] and [1, 4/3]; and suffciently decreasing on [0, 1] such that ∀x ∈ M, ∀t ∈ (0, 1), d dtF(x, s) + g′(s) < 0 (85) This can be done since M is compact. F = F + g is a Morse function whose critical points are Crit(F) = Crit(f) × {0} ∪ Crit(f ′) ∪ {1} (86) Moreover, for a ∈ Crit(f) and b ∈ Crit(f ′) IndF (a, 0) = Indf(a) + 1 IndF (b, 1) = Indf ′(b) Using a partition of unity, we might choose a pseudo gradient field X on M × [−1/3, 4/3] adapted to F such that X = X − ∇g on [−1/3, 1/3] X = X′ − ∇g on [2/3, 4/3] After perturbation, we might assume X satisfies Smale condition, and moreover, transversal to the section M × t for t ∈ {−1/3, 1/3, 2/3, 4/3}. We are gonna study the Thom-Smale complex associated to (f, X) on M × [−1/3, 4/3]. Since X is transversal to 1/3 and, the restriction of (f, X) can still be used to define a Thom-Smale complex on M × [−1/3, 1/3]. To understand this complex, we need the following lemma. Lemma 4.1. Fix a compact manifold M and f a Morse function on M. Let X and X′ be pseudo gradient fields adapted to f that satisfy Smale property. If they are C1 close enough, then we have: ∂X = ∂X′

Proof. Consider the family of vector fields Xs = sX + (1 − s)X′ interpolating

between these two vector fields. Then for each s, Xs would also be a pseudo gradient field adapted to f. Since X satisfies Smale condition, if X′ is sufficiently C1−close to X, then Xs will also satisfy Smale condition for each s. Choose a critical point x of index k and y of index k − 1. Fix some ǫ > 0, we may assume 24

SLIDE 25

that Sn−k ∼ = f −1(f(y) + ǫ) ∩ M s

Xs(y) is some fixed embedded n − k sphere for

all s. We also have a map F : Dk × I → M such that F(·, s) is the embedding of M u

Xs(x) into M for each s. We claim that

F −1Sn−k is a compact 1 dimensional submanifold. Note that for each s, Xs is a pseudo gradient field that satisfies Smale condition, so we have that F ⋔ Sn−k. Thus by transversality theorem, F −1Sn−k is a manifold of dimension 1, with its boundary = F −1(·, 0)Sn−k ∪ F −1(·, 1)Sn−k. Also note that when a point approaches the boundary of Dk, it will converge to some critical point. This is absurd for a point on Sn−k. So F −1Sn−k must be compact. By the classification

f compact 1-fold, we see that the points in F −1(·, 0)Sn−k and F −1(·, 1)Sn−k

must come up in pairs, so counting they give the same number mod 2. This proves the lemma. Now on M × [−1/3, 1/3] (CF |M×[−1/3,1/3]

∗

, ∂X) = (C

F |[−1/3,1/3] ∗

, ∂X−∇g) = (Cf

∗ , ∂X)

(87) and likewise on M × [2/3, 4/3] (CF |M×[2/3,4/3]

∗