Morse Homology

Trabajo de Tesis presentado al Departamento de Matem´aticas

Presentado por: Juanita Pinz´on Caicedo Asesor: Bernardo Uribe

Para optar por el t´ıtulo de Matem´atica

Universidad de Los Andes Departamento de Matem´aticas Julio 2007 Contents

Introduction ii

1 Preliminary Notions 1

2 Morse Functions 4 2.1 Gradient Flow Lines ...... 4 2.2 Stable, Unstable submanifolds ...... 6 2.3 Homotopy ...... 7

3 Morse-Smale Functions 15

4 The Morse Homology Theorem 18

5 Examples 26 5.1 Complex Projective Spaces ...... 26 5.2 Real Projective Spaces ...... 29

A Morse-Bott 36

i Introduction

In the present work we study Morse-Smale functions over Riemannian manifolds and the Morse- Smale chain complexes that can be assigned to each one of them. Morse functions are real-valued functions with isolated and nondegenerate critical points, they are the starting point of Morse Theory. The main objective is to prove Morse Homology Theorem, a theorem that relates certain topological properties of a manifold to some information obtained from a real-valued function defined on it. The theorem says that the homology groups found using a triangulation of the manifold defined by the gradient flow lines of Morse-Smale functions are isomorphic to the singular homology groups. Therefore, Morse Homology Theorem allows us to calculate the homology groups of an unknown manifold by studying much simpler objects, namely, a Morse- Smale function and its critical points.

Given their importance, Morse functions, critical points and gradient ﬂow lines are the topic of section two. That section also contains a series of results which follow from a detailed observation of those functions. Some of those results are a CW-decomposition of the manifold, the tangent space split and the local representation of the function.

However rich the theory is until that point, it can be taken to a higher level by making Morse functions satisfy yet another condition: transversality. With this condition in hand one can define a complex formed by the free abelian groups with generators the critical points, graded by their index, and whose boundary operator is defined counting flow lines from critical points of succesive index. Consequently, transversal Morse functions become very useful for calculating homology groups.

The deﬁnition of the mentioned diﬀerential and the abelian groups forming the chain complex are the starting point of the fourth section which is the conclusion of the work. It is in that section that the proof of Morse Homology Theorem is presented.

The work would not be complete if some examples were not included; that is why in section 5 the calculation of the homology groups of both complex and real projective spaces, using Morse Homology, are included. A slightly diﬀerent approach to the calculation of homology is presented in the appendix, Morse-Bott functions with a short mention to its homology complexes.

ii 1 Preliminary Notions

The main goal of this work being to relate topological properties of a manifold with properties of real functions it is essential to know the deﬁnitions of some concepts that although are not going to be mentioned very often, are the minimun requirements for understanding Morse Theory or any theory concerning manifolds.

Obviously, the ﬁrst term to be deﬁned has to be Manifold:

Definition 1.1 A Manifold M of dimension n is a topological Hausdorff paracompact space n such that ∀ m ∈ M ∃ A ⊂ R open ,U ⊂ M open m ∈ U y ϕ : U → A a homeomorphism. In n other words, it is a topological space locally homeomorphic to R . Each pair (U, ϕ) is called a chart or a local coordinate system. A Manifold is called smooth if for (U, ϕ) and (V, ψ) two different charts such that U ∩ V 6= ∅, ϕ ◦ ψ−1 : ψ(U ∩ V ) → ϕ(U ∩ V ) is a C∞ function.

∞ Deﬁnition 1.2 A tangent vector to M at p is a map v : C (M) → R which for all f, g ∈ C∞(M) satisﬁes:

i) v(f + g) = v(f) + v(g), v(af) = av(f) (is linear)

ii) v(fg) = v(f)g(p) + f(p)v(g)

Deﬁnition 1.3 The set of all tangent vectors at a point p ∈ M is called the tangent space at a point p. As it is a vector space of dimension n, its basis is { ∂ ,..., ∂ } where ∂x1 ∂xn p ∂f ∂(f◦ϕ−1) ∞ = (ϕ(p)) for f ∈ C (U), (U, ϕ) a local chart around p and (x1, . . . , xn) the local ∂xi p ∂xi coordinate functions around p. It is denoted as TpM. The disjoint union of the tangent spaces ` at every point of the manifold, p ∈ M TpM is the tangent bundle of M and is denoted by TM

Definition 1.4 A vector field X on M is a map that to each point p ∈ M assigns a tangent ∞ Pn ∂ vector of class C at that point. Using 1.3 it is natural to define Xp = ai(p) where i=1 ∂xi p ∞ each of the ai’s is in C (Up, R), that is, a function that assigns a real number to elements of an open subset of M.

Deﬁnition 1.5

Let f : M → R

1 Then its derivative is a map from a tangent space of M to a tangent space of R which is R itself:

df∗ : T∗M → T∗R = R

v∗ → v(f) with local expression

∂f ∂f df∗ = dx1 + ... + ∂x1 ∂xn where (x1, . . . , xn) are local coordinates around *.

A metric is a linear transformation

g : T∗M × T∗M → R satisfying i) Symmetry: ∀v∗, w∗ ∈ T∗M, g(v∗, w∗) = g(w∗, v∗) ii) Linearity in the ﬁrst variable: ∀a ∈ R, ∀v∗, w∗ ∈ T∗M, g(av∗, w∗) = ag(v∗, w∗). ∀v∗, w∗, z∗ ∈ T∗M, g(v∗ + w∗, z∗) = g(v∗, z∗) + g(w∗, z∗). iii) Nonnegativity: ∀v∗ ∈ T∗M, g(v∗, v∗) ≥ 0 iv) Nondegeneracy: g(v∗, w∗) = 0 ∀w∗ ∈ T∗M if and only if v∗ = 0 which deﬁnes an inner product on T∗M and which can also be thought of as

g : T∗M → Hom(T∗M → R) where Hom(T∗M → R) is by deﬁnition the dual of the tangent space and is called the cotangent space. It is denoted by T ∗M, then

∗ g : T∗M → T M and since

∇f∗ → df∗

−1 df∗ is the dual vector of ∇f∗, the gradient vector ﬁeld of f. In other words, ∇f∗ = g∗ · df∗

2 Deﬁnition 1.6 Let X = Pn a (p) ∂ be a vector ﬁeld over a n-dimensional manifold M. An p i=1 i ∂xi integral curve through a point p ∈ M is a function c :[a, b] ⊂ R → M, c(t) = (c1(t), . . . , cn(t) such that c(0) = p andc ˙(t) = Xc(t) or ai(c(t)) =c ˙i(t).

∗ Considering the relation between T M and T∗M given by the metric of the manifold, we see that the differential equations that define the integral curves of the gradient vector field, if we want to calculate them using df, depend on the metric.

The next example will make explicit the way of calculating the integral curves of the gradient vector ﬁeld on a very simple manifold. Example 1.1 Let f : R2 → R, f(x, y) = −x2 + y2. Then df = −2xdx + 2ydy and ∇f(x, y) = h−2x, 2yi and so, by the previous def- dc dc inition, the integral curves are determined by 1 = −2c and 2 = dt 1 dt 2c2, which in turn give rise to the explicit equation of the curves −2t 2t c1 = k1e and c2 = k2e .

The constants ki depend on the initial condition of the curves, i.e the 2 point p ∈ R such that c(0) = p. In this example df and ∇f have Figure 1: Gradient vector ﬁeld and the same componets because the metric of Rn is the n × n identity integral curves matrix.

3 2 Morse Functions

The previous section contains the basic concepts concerning Manifolds. It is now time to start with the basics of Morse Theory, namely Morse Functions, a special type of functions which can be locally expressed in terms of its critical points and will prove very useful for the calculation of the Homology groups of a manifold. The most important result that will be introduced in this section is that Morse functions, by means of submanifolds determined by them, give rise to a CW complex homotopic to the manifold. Another important fact is that they lead to a particular decomposition of the tangent space on the critical points according to the sign of the eigenvalues of the Hessian matrix.

Let M be a smooth n-dimensional manifold and f : M → R a real valued function.

2.1 Gradient Flow Lines

Deﬁnition 2.1 p ∈ M is a critical point of f if dfp = 0.

By the relation between df∗ and ∇f∗, the critical points are the points in M which at the same time are singularities of the gradient vector ﬁeld and points in which the derivative vanishes.

Deﬁnition 2.2 The Hessian of f at p is a symmetric bilinear form Hp such that for v, w ∈ TpM

Hp(v, w) =v ˜p(w ˜(f)) wherev, ˜ w˜ are the extensions of v, w to vector ﬁelds andv ˜p = v. Pn ∂ Pn ∂ If (x , . . . , x ) is a local coordinate system at p and v = a |p, w = b |p, we 1 n i=0 i ∂xi i=0 i ∂xi Pn ∂ can takew ˜ = b where b denotes a constant function. Then Hp(v, w) = v(w ˜(f))|p = i=0 i ∂xi i Pn ∂f Pn ∂2f n ∂ ∂ o v( bi ) = ajbi (p) so H is locally represented in terms of the basis ,..., i=0 ∂xi i,j=0 ∂xj ∂xi p ∂x1 ∂n 2 by the n × n matrix whose (i, j) entry is ∂ f (p). ∂xi∂xj

Deﬁnition 2.3 Let p be a critical point of f.

i) p is a non-degenerate critical point if the Hessian matrix in p is non-degenerate, that is if det(Hp) 6= 0.

ii) The index of p is the number of negative eigenvalues of Hp. iii) f is a Morse function if all its critical points are isolated and non-degenerate.

Taking the index to be the number of negative eigenvalues of Hf is equivalent to taking it as the dimension of the vector space spanned by the negative eigenvalues of Hf . Also, the eigenvalues of the hessian matrix are real numbers because the matrix is symmetric and the spectral theorem states that the eigenvalues of a symmetric matrix are all real.

4 Lemma 2.4 (Morse Lemma) Let f : M → R be a Morse function and p ∈ M a critical point of index k. There exists a local coordinate system (x1, . . . , xn, ϕ) with ϕ(p) = 0 such that if ϕ(m) = (x1, . . . , xn), then k n X 2 X 2 f(m) = f(p) − xi + xi i=1 i=k+1 .

For the complete proof of Morse Lemma see [9].

As a way of relating homology groups of a manifold with critical points of a Morse function, Morse compared the number of critical points of index k (mk) with the dimension of the k-th homology group (bk) which resulted in the Morse inequalities. These inequalities establish that mk ≥ bk for all k or that the number of critical points of index k of f is greater than the dimension of the k-th homology group of the manifold. n n P k P k Considering the polynomials M(t) = mkt and P (t) = bkt , Morse inequalities say that k=1 k=1 the polynomial M(t) − P (t) has positive coeﬃcients. In fact, there is a much stronger result giving a more precise expresion for that polynomial, it says that M(t)−P (t) = (1+t)Q(t) where Q(t) is a ploynomial with positive coeﬃcients. (see [5] for further explanation)

Since we only want a ﬁnite number of isolated critical points, only compact manifolds will be considered in the present text. Also, all the functions mentioned here will be Morse functions unless otherwise stated.

Lemma 2.5 Let γ be a gradient ﬂow line, then the function f ◦ γ : R → R is decreasing, its image is a bounded set and d lim (f ◦ γ)(t) = 0 t→±∞ dt

Proof. Since M is compact and f is continuous, Im(f ◦ γ) is a bounded set. Now, the derivative d of the function dt (f ◦ γ)(t) is such that d d (f ◦ γ)(t) = df ◦ γ(t) = df (−∇f) = − h∇f , ∇f i = − k∇f k2 ≤ 0 dt γ(t) dt γ(t) γ(t) γ(t) γ(t) γ(t)

Then f ◦ γ is decreasing and so, for all t0 ∈ we have lim (f ◦ γ)(t) > (f ◦ γ)(t0) and R t→−∞ lim (f ◦ γ)(t) < (f ◦ γ)(t0). This together with the fact that its image is a bounded set implies t→∞ d 2 d 2 that lim (f ◦ γ)(t) = lim − k∇fγ(t)k = 0 and lim (f ◦ γ)(t) = lim − k∇fγ(t)k = 0. t→∞ dt t→∞ t→−∞ dt t→−∞ H

Lemma 2.6 Gradient ﬂow lines start and end at critical points.

5 Proof. Let γ be a gradient ﬂow line and {tn} a sequence of real numbers such that lim tn = ∞ n n→∞ and suppose lim γ(tn) = p. We want to prove dfp = 0 n→∞ 2 2.5 gives us lim − k∇fγ(t )k = 0 and since f is a continuous function, lim (f ◦ γ)(tn) = f(p) n→∞ n n→∞ 2 2 2 and lim − k∇fγ(t )k = − k∇fpk , that is k∇fpk = 0 and because the metric is nondegenerate n→∞ n this implies that ∇fp = 0. Given that df = g · ∇f, we have dfp = 0, that is, p is a critical point and γ ends at a critical point. A similar argument shows that lim γ(t) is a critical point, that is that γ starts at a t→−∞ critical point. H

2.2 Stable, Unstable submanifolds

Deﬁnition 2.7 Let p be a critical point of f and c an integral curve of −∇f.

i) The stable manifold of p is the set

S(p) = W s(p) = {x ∈ M | lim c(t) = p} t→∞

ii) The unstable manifold of p is the set

U(p) = W u(p) = {x ∈ M | lim c(t) = p} t→−∞

In other words, the stable manifold is the set of all points in a flow line that ends in p whereas the unstable manifold is the set of all points in a flow line that starts at p. It should be remarked that this definition works for any vector field but, since the one that is important for Morse theory is the gradient, the definition is restricted to it.

Lemma 2.8 If p is a critical point of index k the dimension of U(p) is k and the dimension of S(p) is n − k.

Proof. The proof of this result is not easy at all. It depends on the choice of a suitable metric that allows us to locally express the gradient vector ﬁeld as

k n X ∂ X ∂ −∇fp = 2xi − 2xi ∂xi ∂xi i=0 i=k+1

It can be proved that such a metric always exists [4], and so there are k directions in which the gradient ﬂow lines enter p and n − k directions in which they leave p and those are exactly the dimensions of U(p) and S(p) respectively. H

6 Another important fact that follows from the previous deﬁnitions is the tangent space split. Let p be a critical point of f. If we deﬁne

s Tp M = {X ∈ TpM | Hp(X,X) ≥ 0} = span{ positive eigenvectors of Hp}

u Tp M = {X ∈ TpM | Hp(X,X) ≤ 0} = span{ negative eigenvectors of Hp} then we have s u TpM = Tp M ⊕ Tp M since u s Tp M ∩ Tp M = {0} and s u Tp M ∪ Tp M = TpM

u s Remark 2.9. Tp M = TpU(p) and Tp = TpS(p) For the proof of this see [1].

This result is essential for the deﬁnition of the homology complex and for the proof of the Main Theorem. Note that 2.8 can also be proved using 2.9 since the dimension of the tangent space of a manifold at a given point is the same dimension of the manifold.

a Theorem 2.10 M = U(p) p∈C(f)

Proof. It is clear that the union of the unstable submanifolds with the critical points is contained in M. Now, that union covers the manifold because Picard-Lindelöftheorem guarantees the existence of a solution for an ordinary differential equation with a certain initial condition, that is, for every point in the manifold which is not a critical point, we can find a flow line that passes through it. The union is disjoint because each point either belongs to a unique flow line or is a critical point. Also, critical points are part of that union because flow lines join critical points and therefore every critical point belongs to both its stable and unstable manifolds. H

2.3 Homotopy

Deﬁnition 2.11 The set M a = {m ∈ M | f(m) ≤ a} where a is a real number is called a half-space of M

−1 Theorem 2.12 Let a, b ∈ R such that a < b and consider f ([a, b]) ⊂ M

7 i) If f −1([a, b]) has no critical points of f, then M a is diﬀeomorphic to M b and M a is a deformation retract of M b.

ii) If there is only one critical point of f in f −1([a, b]), then M b is homotpy equivalent to a k M ∪g D where k is the index of the critical point.

Proof. i) For the first part of the theorem we will prove first that M a is diffeomorphic to M b and after that, that M a is a deformation retract of M b. ∇f −1 For that, take X = ||∇f||2 , since f ([a, b]) contains no critical points, X is well defined inside it. −1 −1 Let ϕt : f ([a, b]) → f ([a, b]) be defined by ϕt(m) = cm(t) where cm is an integral curve of the vector field X such that cm(0) = m. ϕt is thus an element of the one parameter group of diffeomorphisms. M a diffeomorphic to M b: Consider the function α : R → R defined by α(t) = f(ϕt(m)). Then α(0) = f(m) and d d α(t) = f(ϕ (m)) = df ◦ dϕ (m) dt dt t ϕt(m) t

= df(X)|(ϕt(m)) = X(f)|(ϕt(m)) = hX, ∇fi |(ϕt(m)) = 1

Z t dα(s) Z t From this follows that ds = ds = t and from the fundamental theorem of 0 ds 0 Z t d calculus we have α(s)ds = α(t) − α(0). It is clear then that α(t) − α(0) = t and 0 ds α(t) = α(0) + t = f(m) + t. −1 Let m ∈ f (a), then α(t) = a + t and α(b − a) = a + b − a = b = f(ϕb−a(m)) and f(ϕ0(m)) = a. a Consider ϕb−a : M → M and let x ∈ M ,

(a) if f(x) = a then f(ϕb−a)(x) = b

(b) if f(x) < a then f(ϕb−a(x)) = f(x) + b − a and so f(ϕb−a)(x) < b

b a From a) and b) follows that ϕb−a(x) ∈ M for all x ∈ M and using the fact that ϕb−a is a diffeomorphism, the first part is proved, that is, M a is diffeomorphic to M b. M a is a deformation retract of M b: In order to prove this part of the theorem, we have to prove that there exists continuous b b functions rt : M → M with 0 ≤ t ≤ 1 such that:

b b (a) r0(M ) = M

(b) rtkM b = id b a (c) r1(M ) = M

(d) The map M × [0, 1] → M which sends (m, t) to rt(m) is continuous.

8 b b Let rt : M → M be deﬁned by ( x if x ∈ M a r (x) = t −1 ϕt(a−f(x))(x) if x ∈ f ([a, b])

then ( f(x) if x ∈ M a f(r (x)) = t −1 f(ϕt(a−f(x))(x)) = (1 − t)f(x) + ta if x ∈ f ([a, b])

b Also, as (1 − t)f(x) + ta < (1 − t)b + tb = b for all t ∈ [0, 1] we have rt(x) ∈ M . Let us check the properties mentioned above:

(a) ( x if x ∈ M a r (x) = 0 −1 ϕ0(x) = x if x ∈ f ([a, b])

that is, r0 = ı the identity map.

(b) rt|M a = id follows immediatly from the deﬁnition of rt. b a (c) r1(M ) = M ( x if x ∈ M a r (x) = 1 −1 ϕa−f(x)(x) if x ∈ f ([a, b]) and so ( f(x) if x ∈ M a f(r1(x)) = a if x ∈ f −1([a, b])

b a that is, r1(M ) = M (d) continuity b Fix x ∈ M , then rt(x) = cx(t) wich is continuous by deﬁnition. −1 Fix t ∈ [0, 1], then it is enough to prove that for x ∈ f (a), rt is continuous. On the one hand, rt(x) = x, on the other, rt(x) = ϕt(a−f(x))(x) = ϕ0(x) = x then rt is continuous ii) Let p be a critical point of f of index k such that f(p) = c and p is the only critical point in f −1[c − , c + ], by Morse Lemma there exists a local chart (U, ϕ) such that for every point m ∈ U, f can be expressed as

2 2 2 2 f(m) = c − x1 − ... − xk + xk+1 + ... + xn

where (x1, . . . , xn) = ϕ(m). What we will prove here is that M c+ is homotopy equivalent to M c− with a k-cell attached by proving that if we let the points in f −1[c − , c + ] slide over the ﬂow lines to which

9 they belong, the points in flow lines which end at p will reach p and stay there, the points in flow lines that start in p will remain unchanged and will form a cell of dimension k and the points in flows that start in critical points with image greater than c, will reach points in f −1(c − ). 2 2 2 2 2 Let (x1, . . . , xk) = u,(xk+1, . . . , xn) = v, x1 + ... + xk = u and xk+1 + ... + xn = v , then, f(m) = c − u2 + v2 and

M c− = {m ∈ U | f(m) ≤ c − } = m ∈ U | −u2 + v2 ≤ − M c+ = {m ∈ U | f(m) ≤ c + } = m ∈ U | −u2 + v2 ≤ identifying u with the first k elements of the chart and v with the last n − k makes it 2 possible to think of ϕ as a function from U to R instead of a diffeomorphism of U with n n R . The fact that U is diffeomorphic to an open set of R means that if we zoom in the n manifold, we would be looking at a subset of R , and by the previous identification, at a 2 2 subset of R . That way we can think of f as a function from R to R and so we can draw 2 the function’s level curves in R which will give us the following image.

−1 n 2 2 f (c + ) = x ∈ R | u − v = −1 n 2 2 f (c − ) = x ∈ R | v − u = −1 n 2 2 f (c) = x ∈ R | u = v

In the picture, the gray lines are some of the gradient ﬂow lines of f over which the points in M c+ will slide and the thick black line represents the set

ek = (u, 0) ∈ U | u2 ≤ since −∇f = (2u, −2v) it is easy to see that the points in ek are precisely the points in ﬂows that start at p.

10 Following the gray lines in the picture we get from f −1(c + ) to f −1(c − ), but if we start at the point (0, ), we would stop at p = (0, 0) without passing to the ﬂow starting at p which shows us that the thick black line is left ﬁxed and so M c+ is homotopy equivalent to M c− with ek attached to it. H This theorem allows us to recreate the shape of a manifold by successively attaching k-cells to the half-spaces of M.

Deﬁnition 2.13 A topological space X is a CW-complex if there are subspaces X(n) with [ X(0) ⊆ X(1) ⊆ · · · ⊆ X = X(n) n∈N such that: i) X(0) is a discrete set of points, a 0-cell.

(n+1) (n) ii) For all n ∈ N, X is obtained from X by a attaching (n + 1)-cells. (n) (n) iii) A subspace U of X is open if and only if for all n ∈ N, U ∩ X is open in X .

t Theorem 2.14 If M is compact for all t ∈ R, then M has the homotopy type of a CW-complex that has one cell of dimension k for each critical point of index k.

Since only compact manifolds are under consideration, all the half-spaces M t are compact. Two diﬀerent lemmas are needed for the proof of the theorem

k Lemma 2.15 Let h0, h1 : S → M be homotopic attaching maps. Then the identity function of M can be extended to a homotopy equivalence

k k H : M ∪ h1 D → M ∪ h0 D

Lemma 2.16 Let g : Sk → M be an attaching map. If

h : M → X is a homotopy equivalence, then it can be extended to a homotopy equivalence

k k H : M ∪ gD → X ∪ hgD

k k k Restated: M ∪g D ∧ M ∼ X ⇒ M ∪g D ∼ X ∪ ghD

The proof for both Lemmas can be found in [9] and [1].

11 Proof. (of Theorem 2.14) Let c1 < c2 < . . . cm−1 < cm be the critical values of the Morse function f. Since the manifold is compact, there is a ﬁnite number of critical points and so a ﬁnite number of critical values. The theorem will be proved using induction on the critical values. Base Step a Clearly c1 is the minimun value of the function so, for a < c1 M = ∅

c1 X0 = M = {p ∈ C(f) | f(p) = c1}

Inductive Step −1 Let ai ∈ R be such that ci−1 < ai < ci and let f (ci)∩C(f) = {p1, . . . , pni } be the set of critical points with image ci. a Suppose Xi ' M i where Xi is some CW-complex. By theorem 2.12 there exists such that

ci+ ci− λ1 λ2 λn M ' M ∪gi D ∪gi D ∪ · · · ∪gi D i 1 2 ni

−1 i i where λk = λ i is the index of the k-th critical point of f (ci) and g , . . . , g are the attaching pk 1 ni maps of theorem 2.12 Moreover, by that same theorem, M ai and M ci are diﬀeomorphic, that is, there is a homotopy equivalence ci+1− ai hi : M → M

Also, for every k = 1, . . . , ni, the map

i λk−1 Hi ◦ hi ◦ gk : S → Xi is homotopic to a map λk−1 (λk−1) ψk : S → Xi

(λk−1) where Xi denotes the (λk − 1)-skeleton of Xi. S λ1 S S λn c + Thus Xi+1 := Xi D ··· D i is a CW-complex homotopy equivalent to M i ψ1 ψ2 ψni H

The next example will explain the decomposition of the 2-dimensional Torus.

Example 2.1 Take M = T 2 and ϕ : [0, 2π) × [0, 2π) ⊂ R2 → R3 deﬁned by ϕ(u, v) = ((R + r cos u) sin v, r sin u, (R + r cos u) cos v) as a local chart. It is not hard to see that ϕ is the parametrization of the torus standing on the xy-plane. Let f : M → R such that f(x, y, z) = z or (f ◦ ϕ)(u, v) = (R + r cos u) cos v. Then ∇(f ◦ ϕ) = (−r sin u cos v, −(R + r cos u) sin v) and the integral curves take the form shown in the image.

12 Since the torus has dimension 2, the torus has the same metric as R2 -the 2 × 2 identity matrix- and the ﬂow lines or integral curves can be drawn in 2 dimensions. The critical points of f are those which satisfy: r sin u cos v = 0 and (R+r cos u) sin v = 0 that is, sin u = 0 or cos v = 0, and sin u = 0. However, cos v = 0 and sin v = 0 never happens so we need u = 0, π, 2π and v = 0, π, 2π. The following table gives the critical points, its index and the Hessian matrix evaluated at them. (u, v) (0, 0) (π, 0) (π, π) (0, π)

ϕ(u, v) (0, 0,R + r) = p4 (0, 0,R − r) = p3 (0, 0, −(R − r)) = p2 (0, 0, −(R + r)) = p1 −r 0 r 0 r 0 −r 0 Hessian matrix 0 −R − r 0 −R + r 0 R − r 0 −R + r Index 2 1 1 0

Let us call c1 = f(p1), c2 = f(p2), c3 = f(p3) and c4 = f(p4) and let ai be such that ci < ai < ci+1. c1 a0 Clearly, M = {p1} and M = ∅.

[ [ M c1− D0 = ∅ {·} ∼ M c1+ =∼ M a1

g1 g1 The figure at the right can be deformed into the figure at the left by pushing its sides until they encounter. Also, the figure at the left deformes into a point if the lines are pushed. The space M a1 is homotopy S 0 equivalent to {p1} D and the latter is homotopy equivalent to {p1}

[ M c2− D1 ∼ M c2+ =∼ M a2

Again, if we were to push the top extremes of the ﬁgure at the right (M a2 ), at some point we would reach the critical point p2. If we kept pushing, the point p2 and some points below would remain unchanged but every other point would reach M c2−.

[ M c3− D1 ∼ M c3+ =∼ M a3

13 At this point we see that the same argument used for the previous steps apply to this and the next step. It is the same argument used in the proof of part 2 of 2.12.

[ M c4− D2 ∼ M c4+ = M

g4 Since we have already recreated the whole torus, we need no more steps.

14 3 Morse-Smale Functions

In this section a new condition will be set upon Morse functions so as to develop a wider range of results without which we would not be able to prove Morse Homology Theorem.

Deﬁnition 3.1 A Morse function is called a Morse-Smale function if for every point m ∈ U(p) ∩ S(q) TmM = TmU(p) ⊕ TmS(q) for p, q ∈ C(f), the set of critical points of f. In other words, if U(p) is transverse to S(q).

The height function considered in example 2.1 is an example of a Morse function which is not Morse-Smale. From the graph of the integral curves or gradient ﬂow lines it can be seen that U(q) = S(p) for q, p the critical points with index 1 and therefore, for any m in a ﬂow line between p and q, TmU(p) and TmS(q) do not span TmM.

It is a known result from diﬀerential topology that if N,Z are two immersed transversal submanifolds of a manifold M, then the dimension of N ∩ Z is equal to the dimension of N plus the dimension of Z minus the dimension of M. From this follows that if we take W (p, q) = U(p) ∩ S(q)

dim W (p, q) = dim(U(p) ∩ S(q)) = λp − λq.

It should be noted that Morse-Smale flow lines do not start and end at critical points with the same index for if they did, W (r, p), where λr = λp, would be of dimension 0, that is, it could only contain a finite number of points but no lines. This in turn would mean that two different flow lines intersect at a single point. Since flow lines either not intersect at all or are the same, there are no flow lines joining critical points with the same index. This observation tells us that dim W (p, q) > 0, in other words, that flow lines go from critical points to critical points decreasing the index.

Transversality condition allowed Smale to introduce an order relation on C(f) determined by the existence of a flow line between two critical points: for p and q critical points of f, p q if there is a flow line that joins them. Such an order is partial: its reflexiveness lies in the fact that the flow which stays static in p joins the point with itself (γ(t) = p∀t ∈ R), its antisymmetry follows from the fact that if there is a flow joining p and q starting at p, there is not a flow joining them and starting at q unless p and q are the same point. Now, suppose r is a critical point such that W (p, r) and W (r, q) are both non-empty. Since f is Morse-Smale, dimW (p, r) = λp − λr > 0

15 and dimW (r, q) = λr − λq > 0, that is, λp > λr > λq so dimW (p, q) = λp − λq > 0 and W (p, q) is therefore nonempty and so the order is transitive.

For the forthcoming results the moduli space of ﬂow lines is needed. Such a space is deﬁned by

M(p, q) = W (p, q)/R where R acts over M via the one parameter group of diffeomorphism (the gradient flow lines), then the elements of the moduli space are all the flows from p to q which obviously do not go through any other critical point. Now, the ends of the moduli spaces would be

M(p, q) = {γ ∈ C∞( ,M) | γ is a piecewise integral curve of − ∇f , γ(f(p)) = p and γ(f(q)) = q} R k∇fk2 that is, the set of the continuous curves in M smooth on M − C(f). They are piecewise because those which go through critical points between p and q are also included in the set. Such a description of the ends of the Moduli Space is possible thanks to the transitivity property of the order on C(f). There is a composition law in M(p, q), which is understood better when regarded as concatenation, it is deﬁned as

M(p, r) × M(r, q) → M(p, q)

(γ1,γ2) → γ1 ◦0 γ2 where r is a critical point such that p r q. Here, the 0 as a subscript represents a measure inside M(p, q) deﬁned as follows: for γ1 ∈ M(p, r), γ2 ∈ M(r, q) and t ∈ [0, ], γ1 ◦t γ2 is an element of M(p, q) which joins p with q and stays t units away from r. This composition law will be used for the proof of the following lemma.

Lemma 3.2 Let p, q ∈ C(f) such that p q, then for every r ∈ C(f) p r q

{p, q, r} ∪ W (p, r) ∪ W (r, q) ⊆ W (p, q)

Proof. Consider the map

ψ :[f(p), f(q)] × M(p, q) → M (t, γ) → γ(t)

It follows from the way ψ is deﬁned that W (p, q) ⊂ Imψ, for M(p, q) is a subset of M(p, q). Also, p, q ∈ Imψ since for all γ ∈ M(p, q), γ(f(p)) = p and γ(f(q)) = q. Moreover, Imψ is the closure of W (p, q), i.e, it is the smallest closed set containing W (p, q). Being the continuous image of a compact set, Imψ is a compact set of M and therefore closed.

16 All that is left to prove is that it is the smallest one. Let us suppose W (p, q) ⊂ Imψ strictly. Take m ∈ Imψ − W (p, q), then there would exist U ⊂ M an open neighborhood of m such that its intersection with W (p, q) was empty. We know that there is a γ ∈ M(p, q) − M(p, q) (a continuous curve close to at least one critical point between p and q) such that m = γ(t) and so γ([f(p), f(q)]) ∪ U would be an open neighborhood of γ that does not intersect M(p, q) which is a contradiction. Therefore we have Im ψ = W (p, q). Now, let r be a critical point of f between p and q and such that W (p, r) and W (r, q) are non- empty. Then both M(p, r) and M(r, q) are also non-empty and so we can take x1 ∈ W (p, r), γ1 ∈ M(p, r), t1 ∈ (f(p), f(r)) and x2 ∈ W (r, q), γ2 ∈ M(r, q), and t2 ∈ (f(r), f(q)) such that γ1(t1) = x1 and γ2(t2) = x2, by the composition or concatenation law deﬁned we know that γ1 ◦0 γ2 is an element of M(p, q) such that ψ(t1, γ1 ◦0 γ2) = x1 and ψ(t2, γ1 ◦0 γ2) = x2, that is, x1, x2 belong to Im ψ = W (p, q) and so W (p, r) ∪ W (r, q) ∪ {p, q, r} ⊆ W (p, q). H

Corollary 3.3 If p and q are such that λp − λq = 1, W (p, q) = {p, q} ∪ W (p, q)

Proof. The previous lemma tells us that {p, q} ∪ W (p, q) ⊆ W (p, q). Since there are no critical points between p and q, the only curves satisfying the conditions for belonging to M(p, q) are the ﬂow lines joining p with q then we have M(p, q) = M(p, q) and ψ gives us the equality. H

Lemma 3.4 The number of ﬂow lines joining succesive critical points is ﬁnite.

Proof. If p and q are succesive critical points, then dimM(p, q)=0 and M(p, q) is a set with finitely many elements. If p and q are succesive critical points, then Imψ = W (p, q) = W (p, q) ∪ {p, q}. Now, let Up and Uq be neighborhoods of p and q respectively. The flow lines from p to q form an open cover for W (p, q) so the family of sets consisting of the union of Up, Uq and a flow line is an open cover of W (p, q). By definition W (p, q) is a compact set, therefore, every open cover has a finite subcover and taking into account that every subcover of the cover taken will consist of the same flow lines together with smaller neighborhoods of the critical points, the number of flow lines has to be finite. H

17 4 The Morse Homology Theorem

In the present section we finally reach the peak of the present work, namely, the Morse Homology Theorem which will allow us to find the homology groups of a compact manifold by studying a Morse-Smale function defined on it. It was proved by Smale in [10] that the set of Morse-Smale gradient vector fields is dense in the set of smooth gradient vector fields so that Morse-Smale functions can always be found. It is our job now to exhibit the relation between that analytic result and the topological properties of the manifold, for that, throughout this section we will assume that the function f : M → R is a Morse-Smale function and that M is a compact oriented manifold. Let us start by defining the Morse-Smale chain complex, the structure whose homology will calculate the homology of M. The two entities needed for a chain complex are abelian groups and a boundary operator that squares to zero. The groups will then be

Ck(f) = Z{[U(p)] | p a critical point of index k} the free abelian group generated by the oriented unstable manifolds of critical points of index k.

Before defining the differential let us first choose positive orientations for TpU(p) for all p ∈ C(f). As TpU(p) ⊕ TpS(p) = TpM, these orientations impose positive orientations on TpS(p) for each critical point p. Thus, all the stable and unstable manifolds are oriented. Now, the definition of the boundary operator

∂ : Ck(f) → Ck−1(f) is dependent of the exact sequence

0 → TmGm → TmU(p) ⊕ TmS(q) → TqM → 0 0 → (g) → (g, u1, . . . , uk−1), (s1, . . . , sn−k+1) → (u1, . . . , uk−1, s1, . . . , sn−k+1) → 0 here Gm is the gradient ﬂow line γ that starts at p ∈ Critk, ends at q ∈ Critk−1 and passes through m regarded as a 1-dimensional submanifold. Also, g is a basis for TmGm,(g, u1, . . . , uk−1) a completed positive basis for TmU(p) and (s1, . . . , sn−k+1) a positive basis for TmS(q). Now, ∂ is deﬁned as follows: for a critical point p of index k the boundary of the element generated by its unstable manifold is X ∂[U(p)] = nγ (p, q)[U(q)].

q∈Critk−1 γ∈M(p,q) where nγ (p, q) is 1 if the orientation of TmM given by the basis thus obtained is positive and nγ (p, q) is -1 if the resulting orientation is negative. Note that this deﬁnition would be not be

18 possible if we had not proved that there is a ﬁnite number of ﬂow lines joining critical points of consecutive index.

For the theorem’s proof, completely taken from [3], some notions of classifying spaces for finite dimensional manifolds are needed. A perspective which, according to the authors, when used for infinite dimensional manifolds is adequate for the infinite dimensional version of Morse Theory known as Floer homology. The results and definitions included in [3] are of utmost importance for the next results, however, only the final result will be used here, namely, that we can work with the decomposition of M k a 1 Rf = U(p)

p∈Critj j≤k 0 1 k−1 k n which defines a filtration Rf ⊂ Rf ⊂ · · · ⊂ Rf ⊂ Rf ⊂ · · · Rf = M. By the previous k equality, Rf is the space of points m ∈ M belonging to a flow line starting at a critical point k k−1 of index at most k. Then Rf − Rf is the set of points in M belonging to flows starting at critical points of index exactly k, that is,

k k−1 a Rf − Rf = U(p)

p∈Critk

k k−1 Now, in Rf /Rf the critical points of index stricly less than k and all the points in flow lines starting at any of them are identified so that the only points which are not ’killed’ in that quotient are those which belong to flows starting at critical points of index exactly k. That gives us k k−1 k k−1 a ∼ a k ∼ _ k Rf /Rf = Rf − Rf ∪ {·} = U(p) ∪ {·} = D ∪ {·} = S

p∈Critk Critk Critk

Theorem 4.1 ( k k−1 ⊕Critk Z if s = 0 Hk+s(Rf , Rf ) = 0 if s > 0

Proof. By the previous result

  ( k k−1 _ k ⊕Critk Z if s = 0 Hk+s(Rf /Rf ) = Hk+s  S  = 0 if s 6= 0 Critk

1As it was Ralph Cohen who suggested the deﬁnition given, John Jones chose the letter R for denoting the ﬁltration.

19 To prove the theorem it is enough to prove the equality

k k−1 k k−1 Hk(Rf , Rf ) = Hk(Rf /Rf )

k For that, using a result from algebraic topology, is enough to prove that Rf is a topological k−1 k k space and Rf is a closed subspace of Rf . For that, we are going to prove that Rf can be expressed as a ﬁnite union of closed sets which in turn will prove that it is a closed subspace of k M, in fact, we are going to prove that Rf can be expressed as the union of the closures of the spaces W (p, q) where p ∈ Critk and q ∈ Crit0, that is

k a Rf = W (p, q)

p∈Critk q∈Crit0

k a k i) Rf ⊆ W (p, q) Let m ∈ Rf then either m ∈ Critj or m ∈ U(pj) where

p∈Critk q∈Crit0 pj ∈ Critj. Since W (p, q) contains every critical point of index less than k and 0 ≤ j ≤ k, if m ∈ Critj, m ∈ W (p, q). Now, if m ∈ U(pj) then m ∈ W (pj, pi) for some pi ∈ Criti with i < j and p pj pi q. By 3.2 W (pj, pi) ⊆ W (pj, q) ⊆ W (p, q) so m ∈ W (p, q).

k a ii) Rf ⊇ W (p, q) Let p ∈ Critk and q ∈ Crit0 such that W (p, q) 6= ∅. Take

p∈Critk q∈Crit0 m ∈ W (p, q), then there exist r ∈ C(f), p r q such that m ∈ W (p, r), m ∈ W (r, q) or k m = r. If we have the ﬁrst case, then we also have m ∈ U(p) ⊆ Rf , similarly, if we have k k the second we also have m ∈ U(r) ⊆ Rf and if we have the third then clearly m ∈ Rf . k That way we have that for every k such that 0 ≤ k ≤ n, Rf is a closed subspace of M, in k−1 k particular Rf is a closed subspace of M and therefore of Rf .

H The proof for the so-mentioned theorem, theorem 4.3, will follow from the following theorem and some results from the theory on Serre’s Spectral Sequences. —

Theorem 4.2 In the index spectral sequence

1 k k−1 Ek,s = Hk+s(Rf , Rf ) the exact sequence d1 1 d1 1 d1 1 d1 ... −→ Ek+1,0 −→ Ek,0 −→ Ek−1,0 −→ ... is the Morse complex

∂ ∂ ∂ ∂ ... −→ Ck−1(f) −→ Ck(f) −→ Ck+1(f) −→ ...

20 Proof. The ﬁrst thing we have to do is study the attaching map

k−1 k−1 k−1 k−2 _ k−1 S −→ Rf −→ Rf /Rf ' S

Critk−1

k Let p ∈ Critk, then U(p) ' D , a k-dimensional disk with p located at its center, and ∂U(p) ' Sk−1, in other words, Sk−1 represents the boundary of the unstable manifold of p. Also, k−1 k−2 _ k−1 a as Rf /Rf ' S , the previous map is a map between ∂U(p) and (U(q) ∪ {·})

Critk−1 q∈Critk−1 with the same homotopy type as the attaching map relative to the CW-complex studied in 2.12. Let us now deﬁne the degree of the map to later see that calculating it is the same as calculating ∂ using n(p, q) and in this way that d1 and ∂ are the same application.

For each q ∈ Critk−1 consider the map

ρ : Sk−1 → Sk−1

Since Sk−1 (left) is the boundary of U(p) and there is a finite number of flows joining p and q (3.4), Sk−1 contains a finite number of points in W (p, q). In addition, the number of positively oriented points in ρ−1(y) for any regular value y ∈ Sk−1 (right) is the same, hence we can take any regular value in Sk−1 and think of it as if it were q. The elements on the preimage of the chosen point are therefore the elements in W (p, q) which, as we saw in 2.12, determine the attaching map. That said it is time to define the sign of ρ at x ∈ ρ−1(y) as ( 1 if dρx preserves orientation sign(dρx) = −1 otherwise and the degree of ρ as X deg(ρ) = sign(dρx) x∈ρ−1(y)

The degree can be lifted to a homomorphism

k−1 k−1 ρ˜ : Hk(S , Z) = Z → Hk(S , Z) = Z m → deg(ρ) · m which gives us the value of d in one of the generators of L (the generator [U(p)]) 1 Critk Z

If the process is repeated for every critical point of index k − 1 we get → L where Z Critk−1 Z X m → deg(ρq) · m.

Critk−1

To ﬁnish the proof, let us see that indeed sign(ρx) = nγx (p, q).

21 k−1 k−1 TxGx ⊕ TxS can be thought of as TmU(p) because dimTxS = k − 1 and dimTxGx = k−1 k−1 k−1 1. Also, Tρ(x)S can be taken as TmU(q), now, TxS is attached to Tρ(x)S via ρ. k−1 If we set (g, u1, . . . , uk−1) as a positive basis for TxGx ⊕ TxS and therefore for TmU(p), (g, dρx(u1, . . . , uk−1)) is another basis for that space which is obviously positive if dρx preserves orientation and negative otherwise. We can complete the basis obtained with a positive basis (s1, . . . , sn−k+1) of TmS(q) to get a generating set of TmM:(g, dρx(u1, . . . , uk−1), s1, . . . , sn−k+1). That set is not a basis for TmM because (g, dρx(u1, . . . , uk−1)) and (s1, . . . , sn−k+1) share the direction given by the gradient ﬂow line. However, (dρx(u1, . . . , uk−1), s1, . . . , sn−k+1) is a basis of TmM. We chose (s1, . . . , sn−k+1) to be positive, that is, to make (u1, . . . , uk−1, s1, . . . , sn−k+1) a positive basis of TmM, thus, (dρx(u1, . . . , uk−1), s1, . . . , sn−k+1) is a positive basis only if sign(dρx) = 1. We have in this manner proved what we wanted, let us see a picture that I hope will clarify what has been said. Let us suppose that x is the only point with image q and take as positive orientation the clockwise one. Also, let us take (g, up) a positive basis of TxU(p), uq = dρx(up) and sq a positive basis of TxS(q), then If dρx preserves orientation we get

where the picture in the right represents the two spheres already attached. In this case, after the attaching takes place up = uq. By the proof just exposed (up, sq) and (uq, sq) are basis of TxM. We chose the ﬁrst one to be positive and since in this case up = uq,(uq, sq) is also positive, that is n(p, q) = 1 = signdρx. Now, if dρx does not preserve orientation we then get

and up becomes −uq when the spheres get glued together. Then, as (up, sq) was chosen to be a positive basis of TxM,(uq, sq) = (−up, sq) is obviously a negative basis and n(p, q) = −1 =

22 signdρx.

Note that the previous argument gives us an easier way for calculating nγ (p, q) since ( 1 up and uq give the same orientation for U(q) nγ (p, q) = −1 up and uq give opposite orientations for U(q)

The following diagram shows the relations we have proved.

a a a ∂U(p) w (U(q) ∪ {·}) w (U(q) ∪ {·}) ∂U(p) U(q) ∂U(p) w p∈Critk q∈Critk−1 q∈Critk−1

u u u u u u ∂ X M X w ∂ ∂ [U(p)] [U(q)] [U(p)] w n(p, q)[U(q)] [U(p)] w n(p, q)[U(q)] q∈Critk−1 p∈Critk q∈Critk−1

_ k _ k−1 _ k−1 S w S ρ k w S k−1 w k−1 S Critk Critk−1 S S Critk−1

u u u u u ρ˜ u M w w Z M M ZZ Z Z d1 w Z Critk−1 Critk Critk−1 H

Theorem 4.3 The homology of the Morse-Smale chain complex of f (C∗(f), ∂∗) is isomorphic to the singular homology H∗(M; Z)

Proof. To prove the theorem, we are going to prove that ( H (M) if s = 0 E2 = k k,s 0 if s > 0 using some theory on Serre Spectral Sequences. A Serre spectral sequence is a way of computing homology groups by expressing the homology groups of a manifold M in terms of relative homology groups of a filtration of M. It can be thought of as a book consisting of a sequence of pages containing a two-dimensional array of relative homology groups and differentials. The group located at the position (r, s) of the i-th i i page is denoted by Er,s and if either r or s is negative, Er,s is the trivial group. Other facts from these sequences are that the differentials of the i-th page go i positions to the left and i − 1

23 positions up, and that the groups of the i-th page are the homology groups calculated from the groups of the i − 1-th page. Indeed, the fact that the groups in negative positions are 0 together with the fact that differentials (maps) after every step (page) move to the left and up, imply that for each position (r, s) there is a group which after a finite number of steps remains unchanged. i i+1 The reason for that is that for some i, the differentials become the zero map so that Er,s = Er,s . ∞ This group is the (r, s)-stabilized group and is denoted by Er,s. The k-th homology group of the manifold with coefficients from a field is found by summing the (r, k − r)-stabilized groups, more ∞ precisely Hk(X,F ) = ⊕rEr,k−r for F a field. Let us now return to our sequence. In 4.2 we defined an exact sequence based on the index k k−1 filtration of M, that is, given by the relative homology groups Hk(Rf , Rf ). Then, we already have the entries of the first step or page and so, its diagram looks as follows:

0u 0u 0 u ··· u 0u 0u 0 0u 0u 0 u ··· u 0u 0u 0 . . 0u 0u 0 u ··· u 0u 0u 0 d d d d d u 1 u 1 u u 1 ⊕ u 1 u 1 0 ⊕Crit1 Z ⊕Crit2 Z ··· Critn−1 Z ⊕Critn Z 0 Given that the homology groups that can be expressed as the sum of limiting spaces of a spectral sequence are those which have coefficients in a field and Z is not a field, it may appear that for the spectral sequence being considered the sum of the k-th limiting spaces is not the k-th homology group of the manifold. Nonetheless, we will later see that our sequence is so special that it is possible to use the mentioned construction to find the homology groups.

Now, to find the entries of the second page we have to find the homology groups of the entries 2 1 1 of the first page which gives us Er,0 = Kerdr/Imdr+1 and 0 for every other group. This time the diagram looks like this:

0ADAAA 0ADAAA 0 0 ... 0ADAAA 0ADAAA 0 0 AAAA AAAA AAAA AAAA 0 0 0 0 ... 0 0 0 0 . . AD AD AD AD 0AAA 0AAA 0 0 ... 0AAAA 0AAA 0 0 AAAA AAAA AAA AAA 2 2 2 ... 2 2 2 A 0 E1,0 E2,0 E3,0 En−2,0 En−1,0 En,0 0

3 2 2 Calculating the homology groups of the second page we get Er,0 = Kerdr/Imdr+1 = Ker0/Im0 = 2 Er,0 and the following diagram:

24 NQ NQ NQ 0NNN 0 0 0 ... 0NNN 0NNN 0 0 0 NNN NNN NNN 0 0NNN 0 0 ... 0 0NNN 0NNN 0 0 NNN NNN NNN 0 0 0 0 ... 0 0 0 0 0 . . NQ ... NQ NQ 0NNN 0 0 0 0NNN 0NNN 0 0 0 NNN NNN NNN 0 0NN 0N 0 ... 0 0NN 0N NNN 0 0 NN NN NNN 2 2 2 ... 2 2 2 2 0 E1,0 E2,0 E3,0 En−3,0 En−2,0 En−1,0 En,0 0 We have thus seen that our sequence stabilizes in the second page, in other words that from the second step onwards the differentials are the zero map. This together with the fact that for 2 ∞ 2 p 6= k the group Ep,k−p is the trivial group gives us that ⊕pEp,k−p = Ek,0. We have seen that the differentials of the first page are the attaching maps from the CW-complex, that implies that the second level of the sequence is the CW-homology, a kind of homology which is isomorphic to singular homology. Thus, the sum of the limiting spaces gives us the homology groups, that ∞ 2 is, Hk(M, Z) = ⊕pEp,k−p = Ek,0. H

25 5 Examples

5.1 Complex Projective Spaces

n n+1 n+1 ∗ CP = V ⊆ C | V is a vector space and dim(V ) =1 = (C \{~0})/C

The n-th dimensional complex projective space as a manifold

We will begin by checking that the complex is in fact a manifold which by deﬁnition amounts to 2n n n showing that it is locally diﬀeomorphic to R = C , that is, there exists an atlas over CP . n Local Charts of CP · Open sets:

Uj ={[(z1, z2, . . . , zn)] | zj 6= 0} is the complement of the closed set {[(z1, z2, . . . , zn)] | zj = 0} n+1 (one of the axis) and therefore an open subset of C · Diﬀeomorphisms:

n −1 n ϕj : Uj → C ϕj : C → Uj z0 zj−1 zj+1 zn [~z] → ,..., , ,..., (z1, . . . , zn) → [z1, . . . , zj, 1, zj+1 . . . , zn] zj zj zj zj

∞ Since ϕj ∈ C and Uj is an open subset, (Uj, ϕj) is a chart. Charts Compatibility

n Consider (Ui, ϕi) and (Uj, ϕj) two charts of CP and suppose i < j. For them to be compatible, −1 ∞ n the composition ϕi ◦ ϕj : ϕj(Ui ∩ Uj) → ϕi(Ui ∩ Uj) has to be a C function over C . n Let ~z = (z0, . . . , zi, . . . , zj, . . . , zn−1) be an element of ϕj(Ui ∩ Uj) ⊂ C . Then

−1 ϕi ◦ ϕj (~z) = ϕi ([z0, . . . , zi, . . . , zj−1, 1, zj . . . , zn−1]) z z z z 1 z z = 0 ,..., i−1 , i+1 ,..., j−1 , , j ,..., n−1 zi zi zi zi zi zi zi

−1 and ϕi ◦ ϕj is an inﬁnitely diﬀerentiable function.

We are now in place to start Morse Theory, the first step is then to find a Morse-Function which in this case is µ, the moment map which will be next defined.

n Morse function over CP

26 n µ : CP −→ R j−1 n+1 P 2 P 2 λi|zi| + λj + λi+1|zi| i=1 i=j+1 [(z1, z2, . . . , zn+1)] −→ j−1 n+1 P 2 P 2 |zi| + 1 + |zi| i=1 i=j+1 j−1 n+1 P 2 2 P 2 2 λi(xi + yi ) + λj + λi+1(xi + yi ) i=1 i=j+1 −→ j−1 n+1 P 2 2 P 2 2 (xi + yi ) + 1 + (xi + yi ) i=1 i=j+1

Taking

j−1 n+1 P 2 P 2 f(z1, z2, . . . , zn+1) = λi|zi| + λj + λi+1|zi| i=1 i=j+1 j−1 n+1 P 2 2 P 2 2 g(z1, z2, . . . , zn+1) = (xi + yi ) + 1 + (xi + yi ) i=1 i=j+1 will make easier the calculation of the derivatives of µ since

f(z1, z2, . . . , zn+1) µ(z1, z2, . . . , zn+1) = g(z1, z2, . . . , zn+1)

First Derivative ∂f g − f ∂g ∂(µ ◦ ϕj) ∂xk ∂xk 2xk = 2 = 2 (λkg − f) for k < j ∂xk g g ∂f g − f ∂g ∂(µ ◦ ϕj) ∂xk ∂xk 2xk = 2 = 2 (λk+1g − f) for k ≥ j ∂xk g g

Critical points of µ

2x1 2xj−1 2xj 2xn+1 dµ = g2 (λ1g − f) dx1+...+ g2 (λj−1g − f) dxj−1+ g2 (λj+1g − f) dxj +...+ g2 (λn+2g − f) dxn+1 ∂(µ ◦ ϕj) and = 0 ⇐⇒ xk = 0, ∂xk n n therefore the critical points of µ taken as elements of C are (0, 0,..., 0) and as elements of CP [(0,..., 0, 1, 0,..., 0)] where 1 is at the j-th position.

27 Second Derivatives of µ

2 ∂ (µ ◦ ϕj) 2 2 2 = 3 (g − 4xk)(λkg − f) if k < j ∂xk g 2 ∂ (µ ◦ ϕj) 2xk ∂g ∂f = 3 (2f − λkg) − g ∂xk∂xl g ∂xk ∂xl 2 ∂ (µ ◦ ϕj) 2 2 2 = 3 (g − 4xk)(λk+1g − f) if k ≥ j ∂xk g 2 ∂ (µ ◦ ϕj) 2xk ∂g ∂f = 3 (2f − λk+1g) − g ∂xk∂xl g ∂xk ∂xl

Hessian Matrix at critical points Since the derivatives are here expressed in terms of f and g, it is mandatory to ﬁrst obtain the values of the functions at critical points before evaluating the derivative functions.

f(0,..., 0) = λj g(0,..., 0) = 1

Therefore the value of the second derivative at a critical point is

2 ∂ (µ ◦ ϕj) 2 = 2(λk − λj) if k < j ∂xk 2 ∂ (µ ◦ ϕj) 2 = 2(λk+1 − λj) if k ≥ j ∂xk ∂2(µ ◦ ϕ ) j = 0 if k 6= l ∂xk∂xl

This results give us the Hessian at the critical points which in this case is the diagonal matrix of size 2n × 2n.

Hµ = diag(λ1 − λj, . . . , λj−1 − λj, λj−1 − λj, λj+1 − λj , λj+1 − λj, . . . , λn+2 − λj) | {z } position (2(j+1),2(j+1))

n Q 2 Then det Hµ = (λk − λj) and as we are focused on Morse functions, it shall be required k=0,k6=j that det Hµ 6= 0 that is, λk 6= λj for all k 6= j. Moreover, considering the characteristic polynomial n Q 2 of Hµ, det(Hµ − αI) = (λk − λj − α) , it follows that the index of the j-th critical point, k=0,k6=j i.e the one with 1 at the j-th position, is equal to the cardinality of the set {λk | λk < λj}. An easy way of ﬁnding the index of each critical point is taking λi < λi−1. That way, the index of the j-th critical point would be 2j.

28 Homology Calculating the homology groups in this case is easier than in many others. All the critical points of this function have even index, then ∂, the diﬀerential of the Morse-Smale complex is zero. The homology complex is then

0 → Z → 0 → Z → ... → Z → 0 → Z → 0 and the homology groups are ( n 0 for k odd Hk(CP , Z) = Z for k even, 0 ≤ k ≤ 2n

5.2 Real Projective Spaces

n n+1 n+1 ∗ n RP = V ⊆ R | V is a vector space and dim(V ) =1 = R /R = S /Z2

Let (λ1, . . . , λn) be an increasing sequence of positive real numbers and consider the function

n+1 X 2 f(x1, . . . , xn+1) = λjxj j=1 using the standard parametrization of the n-sphere we then have:

R Sn i−1 n f p 2 2 w X 2 X 2 (u1, . . . , ui−1, 1 − u1 − ... − un, ui, . . . , un) (λj − λi)uj + (λj+1 − λi)uj + λi j=1 j=i (x , . . . , x , x , x , . . . , x ) 1 i−1 i u i+1 n+1 556 55 ϕi 55 55 55 Rn (u1, . . . , un)

29 Note that the function was deﬁned over the sphere invariant under the action of Z2, that suggests that using the same function we can calculate homology groups of the n-sphere, in particular of S2. In S2 the function has 6 critical points: (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1) each of index 0, 1, 2 respectively. We thus get the following Morse complex:

0 → Z ⊕ Z → Z ⊕ Z → Z ⊕ Z → 0 and all we need to calculate the homology groups is to ﬁnd ∂ and for that we have to study the ﬂow lines joining the critical points of succesive index. Let us begin with the critical points of index 2 and 1. We thus need to observe the orientations of the stable and unstable manifolds of a point in W (p, q) where p can be either (0, 0, 1) or (0, 0, −1) and q either (0, 1, 0) or (0, −1, 0). Let us take p1 = (0, 0, 1), p2 = (0, 0, −1), q1 = (0, 1, 0) and q2 = (0, −1, 0) and consider the following diagram.

p1 →q1 p1 →q2 p1 →q1, q2 p2 →q1 p2 →q2 p2 →q1, q2 Z →ZZ →ZZ →Z ⊕ ZZ →ZZ →ZZ →Z ⊕ Z 1 →1 1 →1 1 →(1, 1) 1 →1 1 →1 1 →(1, 1)

1 1 Then the function ⊕ → ⊕ is given by the matrix with Z Z Z Z 1 1 1 1 1 1 Ker = h(1, −1)i Im = h(1, 1)i 1 1 1 1

It is now time to examine ﬂow lines joining critical points of index 1 and 0. For that, let us call r1 = (1, 0, 0) and r2 = (−1, 0, 0).

q1 →r1 q1 →r2 q1 →r1, r2 q2 →r1 q2 →r2 q2 →r1, r2 Z →ZZ →ZZ →Z ⊕ ZZ →ZZ →ZZ →Z ⊕ Z 1 → − 1 1 →1 1 →(−1, 1) 1 →1 1 → − 1 1 →(1, −1)

−1 1 And the diﬀerential ⊕ → ⊕ is given by and so Z Z Z Z 1 −1 −1 1 −1 1 Ker = h(1, 1)i Im = h(1, −1)i 1 −1 1 −1

That way we have the following homology complex and homology groups.

0 w Z ⊕ Z w Z ⊕ Z w Z ⊕ Z w 0 h(1, −1)i /0 h(1, 1)i / h(1, 1)i h(1, −1), (1, 0)i / h(1, −1)i Z 0 Z 2 2 2 H2(S , Z) H1(S , Z) H0(S , Z)

30 2 RP Charts and Function

p 2 2 2 2 ϕ1(u, v) =( 1 − u − v , u, v)(f ◦ ϕ1)(u, v) =u + 2v + 1 p 2 2 2 2 ϕ2(u, v) =(u, 1 − u − v , v)(f ◦ ϕ2)(u, v) = − u + v + 2 p 2 2 2 2 ϕ3(u, v) =(u, v, 1 − u − v )(f ◦ ϕ2)(u, v) = − 2u − v + 3

Derivative

d(f ◦ ϕ1) = 2udu + 4vdv d(f ◦ ϕ2) = −2udu + 2vdv d(f ◦ ϕ3) = −4udu − 2vdv Riemannian Structure, Metric The vectors that span the Tangent Space to S2 at a point (u, v) are −u −v Tu = (1, 0, √ ) and Tv = (0, 1, √ ) 1 − u2 − v2 1 − u2 − v2

3 the inner product of them in R is therefore

1 − v2 1 − u2 uv hT ,T i = hT ,T i = hT ,T i = u u 1 − u2 − v2 v v 1 − u2 − v2 u v 1 − u2 − v2 and the induced metric in S2 is given by the matrix

1 1 − v2 uv g = 1 − u2 − v2 uv 1 − u2 which has as its inverse the matrix 1 − u2 −uv u2 − 1 uv g−1 = = − −uv 1 − v2 uv v2 − 1

Gradient Vector Field u2 − 1 uv 2u −∇ (f ◦ ϕ ) = −g−1 · d (f ◦ ϕ ) = 1 1 uv v2 − 1 4v u2 − 1 uv −2u −∇ (f ◦ ϕ ) = −g−1 · d (f ◦ ϕ ) = 2 2 uv v2 − 1 2v u2 − 1 uv −4u −∇ (f ◦ ϕ ) = −g−1 · d (f ◦ ϕ ) = 3 3 uv v2 − 1 −2v

Gradient Flow Lines

31 The ﬂow lines in this case can be found by solving the following systems of diﬀerential equations

ϕ1 ϕ2 ϕ3 dc 1 2c (t) c2(t) − 1 + 4c (t)c2(t) −2c (t) c2(t) − 1 + 2c (t)c2(t) −4c (t) c2(t) − 1 − 2c (t)c2(t) dt 1 1 1 2 1 1 1 2 1 1 1 2 dc 2 2c2(t)c (t) + 4c (t) c2(t) − 1 −2c2(t)c (t) + 2c (t) c2(t) − 1 −4c2(t)c (t) − 2c (t) c2(t) − 1 dt 1 2 2 2 1 2 2 2 1 2 2 2 And look as shown in the images

Figure 2: Flow lines found using ϕ1

Figure 3: Flow lines found using ϕ2

32 Figure 4: Flow lines found using ϕ3

Homology Complex Remark 5.1. Even though homology groups of non-orientable manifolds with coefficients in Z can not be found, there is a way of calculating them using coefficients in Z2 instead. The method consists in claculating the number of flow lines joining succesive critical points mod 2. Just as in the case of orientable manifolds, this number will give us the map between the groups of the homology complex. In our example we have 2 ≡ 0 mod2 flows between critical points of index 2 and 1 and 2 ≡ 0 mod2 between critical points of index 1 and 0, therefore the homology complex with coefficients in Z2 is 0 0 0 0 → Z2 → Z2 → Z2 → 0 which gives us the following homology groups: 2 2 2 H2(RP , Z2) = Z2 H1(RP , Z2) = Z2 H0(RP , Z2) = Z2 2n This method can be used for all the non-orientable projective spaces RP .

n RP

First derivative:

i−1 n X X d(f ◦ ϕi)x = 2(λj − λi)xjdxj + 2(λj+1 − λi)xjdxj j=1 j=i critical points: (0,..., ±1,..., 0) in Sn which becomes de same class in the quotient. (Because f is invariant under the action of Z2 over the sphere)

33 Riemannian Structure: n+1 Let us remember that, in order to ﬁnd the local matrix expression of the metric induced by R in Sn we have to compute the inner product of the vectors which form a basis for the tangent space of the sphere at a given point, these vectors are

xj Tx = ej − ei j p 2 2 1 − x1 − ... − xn and the inner product between them takes the values

2 2 2 2 2 xj 1 − x1 − ... − xj−1 − xj+1 − ... − xn Txj ,Txj = 1 + 2 2 = 2 2 1 − x1 − ... − xn 1 − x1 − ... − xn x x T ,T = j k xj xk 2 2 1 − x1 − ... − xn and the matrix which deﬁnes the metric in the sphere would then be given by

 2 2 2 2 1−x1−...−xj−1−xj+1−...−xn  1−x2−...−x2 if j = k gjk = 1 n xj xk  2 2 otherwise 1−x1−...−xn we would now be in place to ﬁnd ∇f if we were given a point in Sn since

∇f = g−1 · df

All we would have to do would be to calculate g−1 which in this case is given by

( 2 ˆ2 2 −1 1 − x1 − ... − xj − ... − xn if j = k gjk = xjxk if j 6= k

Second derivative:

( ∂f 2 2(λ − i) if j = l = j ∂xj∂xl 0 if j 6= l

Hessian matrix for the i-th critical point:

Hfi = 2diag(1−i, 2−i, . . . , (i−1)−i, (i+1)−i, . . . , (n+1)−i) the critical point (0,..., ±1,..., 0) has index i − 1. (clearly non-degenerate) Up until this point, the procedure is the same for every n, nonetheless, since the orientability of the manifold plays a central role in ﬁnding the attaching maps and calculating the homology, the

34 steps in finding such groups is different for n = 2k (non-orientable) and n = 2k+1 (orientable). In n 5.1 we saw a way of calculating homology groups for RP for an even n but a way for calculating homology groups of orientable projective spaces has not been exposed. It is now time to do that. The flow lines between succesive critical points form maximun circles in Sn the difference is that the attaching map from level 2k + 1 to level 2k is 0 whereas the attaching map between levels 2k and 2k−1 is 2. This is just a generalization of what happens in S2. If we go back to the calculation n of the differentials, we see that the attaching map from p1 to both q1 and q2 was 1. In RP 2 q1 = q2 but the orientation of RP in those points is opposite, so the attaching map is found by n n substracting the attaching maps in S . We thus have 0 as our attaching map in RP . In the same example, we found that the attaching map from q1 to r1 was -1 but when we considered the flow n between q1 and r2 the attaching map was 1. So we have that the attaching map in RP is -2 or 2. Note that the reason behind all this is that ∂2 = 0, that is, if we have two flow lines from a critical point p to a critical point q and another two flow lines from q to r with λp −λq = 1 = λq −λr, we need to have n1(p, q)n1(q, r) + n1(p, q)n2(q, r) + n2(p, q)n1(q, r) + n2(p, q)n2(q, r) = 0 where the subscripts represent the flow line considered. The only way to get this is to have either n1(p, q) and n2(p, q) having the same sign but n1(q, r) and n2(p, q) having opposite sign or the other way around. n As was pointed, this can be generalized for even and odd critical points in RP , therefore we have: 0 2 2 0 0 0 → Z → Z → Z → · · · → Z → Z → Z → 0 n And so, the homology groups of RP for odd n are given by  Z for k = n, 0 n  Hk(RP , Z) = Z2 for odd k, 0 < k < n 0 for even k, 0 < k < n

35 A Morse-Bott

n After considering the Morse-Smale function over CP , I wanted to generalize it to the Lens Spaces, however, this was not possible because it turned out to be not a Morse-Smale function. Its critical points are not isolated. Nevertheless, there is a way to obtain the homology of Lens Spaces by reforming some of the results of Morse Theory. In this section we will show how this can be done.

Deﬁnition A.1 i) V a connected submanifold of M such that all is elements are critical points of f : M → R is a non-degenerate critical submanifold of f if for every v ∈ V , TvV is equal to the null-space of the Hessian of f on TvM ii) f is a Morse-Bott function if it is such that every critical point belongs to a non-degenerate critical manifold.

By deﬁnition, a critical submanifold V is connected, thus, all of its points have the same index and this number is the index of V . (The index is calculated using the form induced by the Hessian over the normal bundle of the critical manifold. In terms of matrices, the matrix resulting from taking away the zero entries of the diagonal Hessian matrix.)

Lens Spaces

2n−1 n 2 2 S = {(z1, . . . , zn) ∈ C | |z1| + ... + |zn| = 1} The action

2n−1 2n−1 Zp × S → S (λ, z1, . . . , zn) → (λz1, . . . , λzn)

2n−1 2n−1 determines a relation ∼ in S such that (z1, . . . , zn) ∼ (λz1, . . . , λzn) therefore, S /Zp is well deﬁned. Take

2n−1 f : S → R   n n n X 2  X 2 X 2 (z1, . . . , zn) → i|zi| + k 1 − |zi|  = (i − k) |zi| + k i=1 i=1 i=1 i6=k i6=k i6=k

36 Let zi = xi + iyi, then df is given by the following equations

∂f = 2 (i − k) xi for 1 ≤ i ≤ n, i 6= k ∂xi ∂f = 2 (i − k) yi for 1 ≤ i ≤ n, i 6= k ∂yi ∂f = 0 for i = k ∂xk ∂f = 0 ∂yk

2 and the critical points of f are (0, . . . , zk,..., 0) where |zk| = 1 is a circle located at one of the n axis of C .

Also, the Hessian matrix will be determined by the second partial derivatives which in this case are

∂2f = 0 if i = k or i 6= j ∂xi∂xj ∂2f 2 = 2 (i − k) if i = j 6= k ∂xi then we have

Hf = 2 diag(1 − k, 1 − k, 2 − k, 2 − k, . . . , −1, −1, 0, 0, 1, 1, . . . , n − k, n − k)

Since the critical points are not isolated and the Hessian matrix has determinant zero, f is not a Morse function in the sense previously deﬁned, however, it is a Morse-Bott function: (x1, y1, . . . , xk, yk, . . . , xn, yn) is an element of TmM if and only if xi = 0 = yi for all i 6= k, that 2n 2 2 n is, it has the form (0, 0, . . . , xk, yk,..., 0, 0) ∈ R where xk + yk = 1 or (0, . . . , zk,..., 0) ∈ C 2 where |zk| = 1.

Although it is not possible to calculate the homology of Lp in the same fashion of Morse functions, it is possible to obtain a CW-complex with the same homotopy type of the manifols from Morse- Bott functions. This complex will be such that each critical manifold V of index k contributes with a ﬁbre-bundle over V with ﬁber Dk.

3 3 Let us consider the case of S /Zp. A not so difficult way of understanding S is to think of it 3 1 as the point compactification of R . The critical points of the studied function are S resting on the xy-plane (x2 + y2) and the z axis which becomes a circle in S3 after the compactification takes place. They have index 2 and 0 respectively. Then, the Morse-Bott complex consists of a fibre-bundle over S1 with fiber D2 and a fibre-bundle over the z-axis with fiber D0 = {·}.

37 References

[1] BANYAGA, A. y HURTUBISE, D. Lectures on Morse Homology. Dordrecht; Boston : Kluwer Academic Publishers, 2004. 324 p.

[2] BOTT, Raoul. Morse theory indomitable. In: Inst. Hautes tudes Sci. Publ. Math. No. 68, 1988; pp. 99-114.

[3] COHEN, Ralph L.; JONES, John y SEGAL, Graeme B. Floer’s inﬁnite-dimensional Morse theory and homotopy theory. Geometric aspects of inﬁnite integrable systems (Japanese) (Kyoto, 1993). In: Surikaisekikenkyusho Kokyuroku. No. 883, 1994; pp. 68-96.

[4] COHEN, Ralph. Private Communication. July, 2007.

[5] GUEST, Martin A. Morse theory in the 1990s: Invitations to geometry and topology. In: Oxf. Grad. Texts Math., 7, Oxford Univ. Press, Oxford, 2002; pp. 146-207.

[6] HATCHER, Allen. Algebraic Topology. Cambridge : Cambridge University Press, 2002. 550 p.

[7] HATCHER, Allen. The Serre Spectral Sequence. In: Spectral Sequences in Algebraic Topol- ogy. [online] Available at http://www.math.cornell.edu/ hatcher/#SSAT

[8] JOST, J¨urgen. Riemannian Geometry and Geometric Analysis. 4th edition. Berlin : Springer-Verlag, 2005. 566 p.

[9] MILNOR, John W. Morse Theory. Princeton: Princeton University Press, 1963. 153 p.

[10] SMALE, Stephen. On Gradient Dynamical Systems. In: The Annals of Mathematics. 2nd Series, Vol. 74, No. 1. (Jul., 1961), pp. 199-206.