Morse homology Michael Landry April 2014 This is a supplementary note for a (hopefully) fun, informal one hour talk on Morse homology, which I wrote after digesting the opening pages of Michael Hutchings great notes [2]. Please contact me with questions, comments, errors, confessions, etc. at michael.landry@yale.edu. 1 Some background Let X be a smooth, compact n-dimensional manifold, and f C (M, R). Let p be a critical point of f, that is df p = 0, and choose local coordinates x 1,..., x n around p. The Hessian matrix of f at p in these coordinates is the symmetric matrix [ 2 ] f H f (p) = x i x j (p). We say that p is a nondegenerate critical point of f if H f (p) is nonsingular; this notion does not depend on our choice of coordinates. If all the critical points of f are nondegenerate, then f is Morse. Since H f (p) is symmetric, its eigenvalues are real. The Morse index of p is the number of negative eigenvalues of H f (p), counted with multiplicity. For more detail, see [4]. Example 1. Let S be a surface embedded in R 3. After perturbing S if necessary, the height function f(x, y, z) = z defines a Morse function on S. The critical points of index 2 are local maxima, the saddle points have index 1, and the local minima have index 0. This is the example to keep in mind while reading this note. 2 The moduli space M(p, q) Equip X with a Riemannian metric g, and recall that the gradient of f, denoted grad f, is the vector field on X defined by df = grad f,. The gradient points in the direction of increase of f, orthogonally to the level curves of f. Let p be a critical point of f. We define the descending manifold D(p) and ascending manifold A(p) of p by D(p) = {x X flowing along grad f takes x to p }, A(p) = {x X flowing along grad f takes x to p }. 1
Figure 1: S 2, with critical points of the height function. In the example when X n is embedded in R n+1, f is the height function and g is the metric induced by the Euclidean one on R n, the index of p can be thought of as the maximum number of linearly independent directions in which grad f points out of p. It follows that in this case D(p) is a disk of dimension ind p and A(p) is a disk of dimension n ind p. This is true in the general case also, and for a proof Hutchings refers us to [1]. Example 2. Let X be the unit sphere in R 3, with f projection to the z coordinate. Then f has 2 critical points: the North Pole N = (0, 0, 1) has index 2, and the South Pole S = (0, 0, 1) has index 0. Furthermore D(N) = X \ {S} and A(S) = X \ {N}. A flow line of (f, g) is an equivalence class of integral curves of grad f, [α : R X] where α β if α(t) = β(t + a) for some a R and all t. We say [α] is a flow line from p to q if lim t α(t) = p and lim t α(t) = q. Definition 3. The pair (f, g) is Morse-Smale if f is Morse and for every pair of critical points p and q, A(q) D(p). It turns out that this condition is generic, and we will henceforth assume that (f, g) is Morse- Smale. Denote by M(p, q) the moduli space of flow lines from p to q. We can identify M(p, q) with (D(p) A(q))/R, where R acts on points in D(p) A(q) by the flow of grad f. By the Morse-Smale condition, D(p) A(q) is a submanifold of X of dimension (n ind q + ind p) n = ind p ind q. The action of R is smooth, free, and proper, so by the quotient manifold theorem (see [3], for example), M(p, q) has the structure of a (ind p ind q 1)-dimensional manifold (as long as that number makes sense). Remark 4. The Morse-Smale condition implies that if ind p ind q, then M(p, q) =. Indeed, if ind p ind q, then dim D(p) + dim A(q) = ind p + n ind q n, so dim(d(p) A(q)) 0 by transversality. If M(p, q) is nonempty then this dimension is 1 since D(p) A(q) would then contain a flow line. Hence M(p, q) =. 2
If ind p = ind q + 1, then M(p, q) is 0-dimensional and, as the following theorem implies, compact. The nice thing about compact 0-dimensional manifolds is that they are finite! This trivial observation is the germ of the upcoming construction of the Morse chain complex. Theorem 5. The moduli space M(p, q) can be compactified by adding in broken flow lines from p to q, i.e. paths from p to q which are the concatenation of flow lines through intermediate critical points. The compactification M(p, q) is a smooth manifold with corners, and the k-times broken flowlines form the codimension k parts of the boundary. In particular, if ind p = ind q + 1 there are no possible intermediate critical points so M(p, q) = M(p, q) is compact and therefore finite. Figure 2: A sequence of flow lines converging to a broken flow line (dark). 3 The Morse chain complex We now orient M(p, q). First, choose an orientation of D(p) for all critical points p. Let x be a point on a flow line [α] M(p, q). Then at x im α there is an isomorphism T D(p) = T M(p, q) T α T D(q) and we orient T M(p, q) such that this isomorphism is orientation-preserving. This is confusing but it is easier to see what is going on in an example; I recommend trying to see the isomorphism for yourself in Example 1. Define C i to be the free abelian group generated by the critical points of index i, and i (p) = #M(p, q)q ind q=i 1 3
where the sum is taken over all critical points of index i 1 and #M(p, q) is the signed number of flowlines from p to q, the sign being determined by the orientation on M(p, q) as described above. Proposition 6. = 0. Proof. Let p, q be critical points of index i and i 2. The coefficient of q in 2 p is ind r=i 1 #M(p, r) #M(r, q) = # ind r=i 1 M(p, r) M(r, q) = # M(p, q) = 0 where the last equality follows from the fact that M(p, q) is an oriented 1-manifold with boundary, so its signed number of boundary points is 0. So we have a chain complex, and we can compute its homology groups. But this chain complex depends on a lot of things: namely the Morse function f, the metric g, and the orientations we chose for all but descending manifolds to orient the M(p, q) s. Unless we re working in the category of manifolds with a Morse function, Riemannian metric, and oriented descending submanifolds, = 0 does not seem like anything to jump over the moon about. However, as you probably suspected already, the following is true: Fact 7. The homology groups of this Morse complex are isomorphic to the singular homology groups of X. Perhaps you have now noticed a pattern of me citing big results in order to bail myself out of trouble. Oh well! This is a fact for another day, since one hour is a short amount of time. In the fall I might try to learn about Floer homology with others, and we would probably go over the proof of this fact then. Let me know if you are interested. 4 Examples: torus and Klein bottle Let s compute the Morse homology groups of some example surfaces. Example 8. Take the standard 2-torus T 2 and stand it on its end. This situation is not Morse- Smale since there are 2 flow lines between the saddle points, so we perturb it slightly as in Figure 3a. Consider 2 (p). The 2 flow lines from p to q have opposite sign so #M(p, q) = 0. Similarly #M(p, r) = 0 so 2 = 0. In fact, 1 = 0 also, so when we take the homology of the Morse chain complex we get H 2 (T 2 ) = Z, H 1 (T 2 ) = Z Z, and H 0 (T 2 ) = Z. Example 9. We can embed the Klein bottle K in R 4. Let f(w, x, y, z) = z; after perturbation, f and the metric R 4 induces on K are Morse-Smale. The situation is as shown in Figure 3b by cheating and projecting to R 3. We have C 2 = Zp, C 1 = Zq Zr, C 0 = Zs. Up to sign, we see that 2 (p) = 2r because the 2 elements in M(p, q) have opposite signs, and the 2 elements in M(p, r) have the same sign. Finally, 1 (q) = 1 (r) = 0. Hence H 2 (K) = 0, H 1 (K) = Zq Zr/(2r) = Z Z/(2), H 0 (K) = Z. The difference between our computations for the torus and Klein bottle lies in 2, and one can see that this stems from the fact that A(r) {p} represents a singular homology class of finite order. 4
(a) (b) Figure 3: The 2-torus and Klein bottle References [1] R. Abraham, J. Robbins. Transversal mappings and flows. W.A. Benjamin, 1967. [2] M. Hutchings. Lecture notes on Morse homology. Available on the author s website: math.berkeley.edu/ hutching/ [3] J. Lee. Introduction to smooth manifolds. Second edition. Springer, 2013. [4] J. Milnor. Morse theory. Princeton University Press, 1969. 5