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Morse Homology 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).
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