p where p \in C_k-1 and

counts the number of gradient flow lines from p to q with a sign, depending on whether the orientation of U_q \int S_p agrees with the direction of flow. ----------------------------------------------------------------------- Complete the details of the "toothy" S^2 example with 6 critical points, and work out the example of torus S^1 x S^1 with 4 critical points (here you will need to wiggle the metric to get a generic gradient flow). ----------------------------------------------------------------------- Now try the general case (you may want to forget signs, and do just a Z/2 version for a start): Show that \d_k-1 \d_k = 0, and thus there is a well defined homology; show that this homology is independent of the metric and the choice of f (in fact, it computes the singular homology of M, by the Mayer-Vietoris principle). ======================================================================= 5/6 The Morse inequalities are a consequence of the fact that the Morse-... complex computes the singular homology of M: i) c_k >= b_k for k = 0, 1, ..., n ii) c_k - c_k-1 + ... +/- c_0 >= b_k - b_k-1 + ... +/- b_0 where c_k = dim C_k are the Morse numbers, and b_k = dim H_k = dim Z_k - dim B_k are the Betti numbers. Both i) and ii) are purely algebraic facts about a complex C_* and its homology H_*; in fact 1) is rather obvious since Z_k is a subspace of C_k. It's nice to think of these in terms of attaching handles to make M: i) there are at least as many k-handles as b_k ii) 1) the number of 1-handles is at least as many as b_1 plus the number of 1-handles needed to attach the 0-handles: c_1 >= b_1 + [c_0 - b_0] ii) 2) the number of 2-handles is at least as many as b_2 plus the number of 2-handles needed to attach the 1-handles: c_2 >= b_2 + [(c_1 - b_1) - (c_0 - b_0)] and so forth, noting that the preceding inequality implies [...] >= 0. ----------------------------------------------------------------------- The Morse inequalities are used both ways: to bound the topology of M, and prove the existence of critical points. For instance, a Morse function on RP^n must have n+1 critical points. (We noted in an earlier problem that a quadratic form on R^n+1 defines a function on RP^n, and it is Morse when it has distinct eigenvalues - the critical points correspond to the eigenlines.) ----------------------------------------------------------------------- How many critical points must a Morse function have on a surface of genus g? On the n-torus S^1 x ... S^1? On CP^n? ======================================================================= 5/8 Fill in the details of the Gauss-Bonnet-Kroenecker-Lipschitz-Killing- Allendoerfer-Weil-Chern-Lashof theorem for an embedded hypersurface e:M^n -> R^n+1: deg(\nu:M -> S^n) = Euler(M)/2. You need to see that the Gauss (or rather, the Lipschitz-Killing) curvature det(d\nu) at p has sign which agrees with (-1)^k where k is the index of the critical point p for the height function f_v(p) = v.e(p) in the direction v, and use the change of variables formula to express deg(\nu) as (half) the average over v in S^n of the (signed) number of critical points for f_v (because, the critical points of f_v are the same as those for f_-v = -f_v, this double counts). ----------------------------------------------------------------------- Note that the integral of |det(d\nu)|, the absolute total curvature, is bounded below by the (no longer alternating) sum of the Betti numbers. ======================================================================= Reading/Exam Period Short Course on Riemannian Geometry (extrinsic and intrinsic): Covariant derivatives, connections and parallel transport** Second fundamental form and extrinsic curvature Intrinsic curvature, holonomy and the Gauss/Coddazzi equation Geodesics and curvature Harmonic forms and curvature Curvature (Ricci, etc.) flows to canonical metrics (**not to be confused with alternative transport ;) ======================================================================= Have a great summer, year, and career! *"Copyleft" means that you have permission to use this material for non-commercial purposes as long as you acknowledge its source. Any use, in whole or in part, must include the "Copyleft 2007 by Rob Kusner" message or its equivalent. Commercial use without prior approval of the copyleft holder is strictly forbidden, and is punishable by methods requiring extraordinary rendition!!