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Topics in Morse Theory: Lecture Notes i 1 ii TOPICS IN MORSE THEORY: LECTURE NOTES Ralph L. Cohen Kevin Iga Paul Norbury August 9, 2006 1The first author was supported by an NSF grant during the preparation of this work ii Preface These notes are an expanded version of lecture notes for a graduate course given at Stanford University during the Autumn of 1990. The course was geared to students who had completed a one year course in Algebraic Topology and had some familiarity with basic Differential Geometry. There were two basic goals in the course and in these notes. The first was to give an introduction to Morse theory from a topological point of view. Be- sides classical Morse theory on a compact manifold, topics discussed included equivariant Morse functions, and more generally nondegenerate functions having critical submanifolds, as well as Morse functions on infinite dimensional Hilbert manifolds that satisfy the Palais–Smale condition (C). The general theme of these discussions was an attempt to understand, in as precise terms as possi- ble, how the topology of the manifold is determined by the critical points of a Morse function and the gradient flow lines between them. Toward this end we discussed recent work of myself, J. Jones, and G. Segal which defines the concept of the classifying space of a Morse function, which is a simplicial space whose n - simplices are parameterized by n - tuples of gradient flow lines; the main theorem of which being that this space is homeomorphic to the underlying manifold. In order to have the machinery to discuss this work in detail I spent time in the course and in the notes discussing several relevant topics from Al- gebraic Topology, including framed cobordism, simplicial spaces, and spectral sequences. The second goal of the course was to discuss several examples of relatively recent work in Gauge theory where Morse theoretic ideas and techniques have been applied. In these cases critical points tend to form moduli spaces, and we again attempted to emphasize the question of how the topology of the ambient manifold is determined by the topology of the critical sets and the spaces of gradient flows. We discussed several algebraic topological techniques for study- ing these questions including adaptations of the classifying space construction mentioned above. These notes are in a rather rough form. Many details have been omitted, but references to the literature where they may be found are given at the end of every chapter. I would like to take this opportunity to thank the students in the course for their interest and enthusiasm. In particular I would like to thank M. Betz, P. Norbury, and M. Sanders for going through various rough drafts of these iii iv Preface notes and in some cases finding and correcting errors and making suggestions. I am also grateful to S. Kerckhoff, T. Mrowka, and H. Samelson for helpful discussions. R.L. Cohen April, 1991 Introduction Let M be a C∞, compact, Riemannian manifold (without boundary), and f : M −→ R ∞ a C map. A point p ∈ M is a critical point of f if the differential dfp : TpM −→ R is zero. (Here TpM denotes the tangent space of M at p.) Morse theory arises from the recognition that the number of critical points of f is constrained by the topology of M. For instance, since M is compact, then f attains its maximum value, and hence, f has at least one critical point. Morse theory typically deals with more non-trivial examples, where the differentiability of M and f play a larger role. At first, one might guess this is primarily an application of topology to optimization theory. After all, optimization is concerned with finding critical points, and Morse theory would say that even if we understand only the topology of M, we might be able to make important predictions about how many critical points we might have. But the main interest in Morse theory has been the reverse: using a discus- sion of critical points of f to uncover information about the topology of M.A more accurate statement is that we can study general classes of manifolds using the perspective obtained by considering the critical points of various functions f. This is crucial in the program to classify manifolds, though it has also in- creased our understanding of a wide range of subjects from symplectic geometry to non-abelian gauge theory. Typically, we do not simply count the number of critical points, but consider more detailed information, such as the second derivative test, to count which critical points are maxima, which are minima, and so on. The second derivative test at each critical point p ∈ M involves a symmetric bilinear form, the Hessian Hessp(f), on the tangent space Hessp(f) is most properly viewed as a symmetric, bilinear form on the tangent space Hessp(f): TpM × TpM −→ R. We will define Hessp(f) in this context later. However in terms of local coordi- nates {x1, . , xn} of a neighborhood of p ∈ M, Hessp(f) is the familiar matrix of second order partial derivatives ∂2f Hessp(f) = (p) . ∂xi∂xj v vi Introduction A critical point p ∈ M is said to be nondegenerate if the Hessian Hessp(f) is nonsingular. The index of a nondegenerate critical point p, λ(p), is defined to be the dimension of the negative eigenspace of Hessp(f). Thus a local minimum has index 0, and a local maximum has index n. In an undergraduate multivariable calculus class, students typically focus on dimension n = 2, where local minima have index 0, saddles have index 1, and local maxima have index 2. We will use examples from dimension n = 2 to build geometric intuition, but we will be interested in applying Morse theory to arbitrary dimension n. Now, instead of only counting the number of critical points of f, we can be more delicate and ask how many critical points there are of each index. We define the integers c0, . , cn so that ci is the number of critical points of index i. As an example of the sort of relation Morse theory predicts between the ci and the topology of M, we give the following theorem, Morse’s theorem (which will be proved in chapter 5): Theorem 1 (Morse’s Theorem) Let f : M −→ R be a C∞ function so that all of its critical points are nondegenerate. Then the Euler characteristic χ(M) can be computed by the following formula: X i χ(M) = (−1) ci(f) where ci(f) is the number of critical points of f having index i. A function f : M −→ R whose critical points are all nondegenerate is called a Morse function. The kinds of theorems we would like to prove in Morse theory will typically only apply to Morse functions. As we will see in chapter 4, however, “most” smooth functions are Morse. Thus in the hypothesis of the previous theorem, we could have said that f is a C∞ Morse function. Recall that the Euler characteristic of M is X i χ(M) = (−1) bi(M) where bi(M) is the ith Betti number of M (the ith Betti bumber of M is the rank of the ith homology group of M). Thus the above theorem is astonishing in that on the right side of the equation we have an expression that involves f, while the left side of the equation does not depend on f at all, and in fact only depends on topological information about M. There are other theorems, mentioned in chapter 5, that relate numbers of critical points of each index with the Betti numbers of the manifold; and we will see how studying other questions related to f give rise to more delicate topological information about M than just the Betti numbers. There are a number of ways these sorts of theorems can be proved, corre- sponding to the several ways of approaching Morse theory. 0.1 The Classical approach to Morse Theory Historically, Morse’s original idea was to consider f −1((−∞, a]), for various real numbers a, and compare these manifolds as manifolds with boundary. This is Introduction vii already explained clearly in Milnor’s book Morse Theory [?]. This approach proves that compact manifolds have the homotopy type of a CW complex, and the CW complex structure allows us to compute the Euler characteristic. 0.2 Smale’s approach The second approach due to Smale, will be introduced in chapter 6. In this approach, we consider gradient flow lines.A gradient flow line (or integral curve) is a curve γ : R −→ M that satisfies the following differential equation (the flow equation): dγ + ∇ (f) = 0. dt γ Here ∇(f) is the gradient vector field determined by f. Given the Riemannian metric h·, ·i, ∇(f) is determined by the property that h∇p(f), vi = dfp(v) where v is any tangent vector v ∈ TpM. Hence if viewed as trajectories, gradient flow lines are the paths of steepest descent. Note that this approach depends explicitly on a Riemannian metric, unlike the previous approach. Through these flows, we can actually decompose M into cells, and in such a way reconstruct the CW complex itself. In this way, the decomposition arises naturally from the Morse function f. Central to this approach are the stable manifold and unstable manifold of a critical point. Definition 0.1 Let M be a manifold, f : M −→ R a Morse function, and g a metric on M. Let p be a critical point of f. Then the stable manifold of p, W s(p), is the set of points in M that lie on gradient flow lines γ(t) (defined using f and g) so that limt→+∞ γ(t) = p.
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