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Morse Functions and Morse Homology

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Morse Functions and Morse Homology Morse Functions and Morse Homology Abhineet Agarwal/Sabrina Victor April 2019 1 Morse Functions Let X be a finite dimensional compact smooth manifold, and let f : X ! R be a smooth function. Definition 1.1. A critical point of f is a point p 2 X such that dfjp= 0 : TpX ! R: Also define Crit(f) to be the set of critical points of f. Figure 1: Critical Points of Torus and Sphere ∗ If p is a critical point, we also define the Hessian H(f,p): TpX ! Tp X as follows: Let r be any connection on TX and if v 2 TpX, define: H(f; p)(v) = rv(df) Recall that the connection defines the notion of parallel transport on a vector bundle. Also note that in our definition of the Hessian does not depend on our choice of connection. This is because df vanishes at p and the difference 1 between any two connections is a tensor. Next let x1; :::xn are local coordinates @ ∗ for X near p, then with respect to the bases f g and fdxig for TpX; T X @xi p 2 respectively, the Hessian is given by the matrix @ (f) . This is exactly what we @xi@xj expect from the formula for the hessian above. Remark. Note that the hessian matrix is obviously a symmetric matrix. Then it is a known fact that the hessian is a self-adjoint map from TpX to itself. Definition 1.2. A critical point is nondegenerate if the Hessian does not have zero eigenvalue. Definition 1.3. We also define Morse Index ind(p) to be the number of negative eigenvalues of the Hessian Lemma 1.1. If p is a non-degnerate critical point with index i(the hessian has i negative eigenvalues at p), then there exists local coordinates x1; ::::xn for X near p such that 2 2 2 2 f(p) = −x1 − :::: − xi + xi+1 + ::::xn Definition 1.4. A function f is morse if all of its critical points are non- degenerate. In fact, a generic smooth function on X is morse. I will give more examples of Morse Functions later, but for now lets just take a function to be the height of a torus or a sphere. Then just by looking at the images, we see that its critical points are non-degenerate, and so we see that our function is indeed morse. Prop: Critical points of Morse functions are isolated. Pf: Let p be a non-degenerate critical point with f(p) = c. Then in some neighbourhood of p with coordinates u1; :::un, we can write via Morses's lemma that 2 2 2 2 f = c − u1 − :::uλ + uλ+1::: + un Here, λ is the index of p as define above. Then we can compute the gradient easily and see that @f = ±2ui @ui @f Therefore = 0 for all i only when ui = 0, which occurs only at p. Hence, @ui the only critical point of f within U is p. 2 Gradient Flow Let g be a metric on X, and let V denote the negative gradient of f with respect to g. Here a good example to keep in mind is simply the euclidean metric. 2 The flow of the vector field V defines a one-parameter group of diffeomorphisms R dΨs Ψs : X ! X for s 2 with Ψ0 = id and dt = V . Recall that diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Just to recall, I will define a one-parameter group of diffeomorphisms. Definition 2.1. A one-parameter group of diffeomorphisms G on a manifold V of class Ck is a Ck map Ψ:(t; x) 2 R × V ! Ψt(x) 2 v such that k • For all real t, the map Ψt : x 2 V ! Ψt(x) 2 V is a C diffeomorphism of V • For all real t,s and for all points in V, we have that Ψt + s(x) = Ψt(Ψs(x)) Definition 2.2. If p is a critical point, we define the descending manifold. D(p) = fx 2 Xjlims→−∞Ψs(x) = pg Similarly, we define the ascending manifold: A(p) == fx 2 Xjlims!+1Ψs(x) = pg Fact: If p is a non-degenerate point critical point, then D(p) is an embedded open disc in X with dimension D(p) = ind(p) In fact, the tangent space TpD(p) ⊆ TpX. The intuition for this is that we should think of the descending manifold as lines flowing at the manifold that end up at p. Since they end up at p and are "descending", we have the idea that as you flow along this path you need the help of negative eigenvectors, which is precisely the negative eigenspace of the Hessian(f,p). Simiarly, we have a complimentary result for ascending manifolds. A(p) is an embedded open disc witht the complimentary dimension: dimA(p) = dimX − ind(p) = dimX − dimD(p) Definition 2.3. A function f and a metric g is Morse-smale if f is Morse, and for every pair of critical points p and q, the descending manifold is transverse to the ascending manifold. Definition 2.4. If p and q are critical points, a flow line from p to q is a path γ : R ! X with the following conditions: 3 • γ0(s) = V (γ(s)) • lims→−∞γ(s) = p • lims→−∞γ(s) = p Figure 2: Flow Lines on a manifold Note that R acts on the set of flow lines from p to q by precomposition with translations of R. Intutively what this means is that we should think of R as a group action on the set of paths. So if we apply a real number to a path, then we are shifting it along the path.We then let M(p; q) denote the moduli space of flow lines from p to q, modulo translation. We can identity: A D M (q) \ (p) (p; q) = R What is a moduli space? Think of the following problem we want to classify all lines in the x-y plane that pass through the origin. Clearly this is parametrised by the angle that the lines makes with the origin. In other words, the set of lines is one to one correspondence with the interval [0; π). Therefore, in our case the moduli space is assigning each flow line to a point that we can use to characterise the flow lines. In fact, we have the following result: M(p; q) is a smooth manifold with M(p; q) = ind(p) − ind(q) − 1 3 Compactification of Manifolds When ind(p)-ind(q) = 1, then the moduli space M(p; q) has dimension zero, and so ideally we would like to count the number of points in it. In order to do we need to know that M(p; q) is compact. But first we need a few definitions: Definition 3.1. A smooth manifold with corners is a topological space that is n n locally isomorphic to an Hi = fx 2 R jx0; ::::xn ≥ 0g; 0 ≤ i ≤ n 4 Definition 3.2. Compactification is the process of turning a topological space into a compact space. Definition 3.3. A stratification of a topological space X is a decomposition i=n X = [i=0 Si where each of the Si are smooth manifolds of dimension i and so that: i=n−1 Sk n Sk ⊂ [i=0 Si . Here the closure Sk is called the stratum of dimension k. With all these definition, we are finally able to state our first theorem(obviously without proof) Theorem 3.1. If X is closed and the pair (f,g) is Morse-smale, then for any two critical points (p,q), the Moduli space M(p; q) has a natural compactification to a smooth manifold with corners whose dimension k stratum is: M M M M (p; q)k = [r1:::rk2Crit(f) (p; r1) × (r1; r2) × ::::: (rk; q) (1) If k = 1, then the boundary of our new and better compactified manifold is given by: ind(p)+ind(r)+1 @M(p; q) = [r2Crit(f)(−1) M(p; r) × M(r; q) Note that in the theorem it is also required that all the points are distinct. The take away of all of this is simply that we can compactify moduli spaces of flow lines into compact manifolds with corners. An example of this is if ind(p) = i and ind(q) = i - 2. Then M(p; q) is a compact one manifold with boundary given given by the formula above. 4 The Chain Complex Definition 4.1. A module is a generalisation of a vector space. In a vector space we require the set of scalar that multiply the vectors to be from a field. A module only requires the scalars to be drawn from a ring. Definition 4.2. A free module is a module that has a basis { that is, a gener- ating set consisting of linearly independent elements. For example, every vector space is a free module Definition 4.3. Let Criti(f) denote the set of index i critical points of f. Then the chain module Ci is the free Z module generated by this finite set: Morse Z Ci (f; g) = Criti(f) Morse Definition 4.4. The differential @ (p): Ci ! Ci−1 counts gradient flow lines. That is, if p 2 Criti(f) then X @Morse(p) = M(p; q):q q2Criti−1(f) 5 Here the M(p; q):q 2 Z denotes the number of points in M(p; q) counted with the signs given by the oreintation on M(p; q) Morse Definition 4.5.
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