Autumn 2001, MWF 2:30-3:20
Morse theory is also know as Hot Fudge theory: You take your favorite manifold, stand it on end, pour Hot Fudge over it and watch it flow. More precisely, you start from a smooth real-valued function on the manifold with nondegenerate critical points and study the critical-point behaviour, as well as the flow of its gradient vector field. The critical points determine the topology of the manifold to a surprising degree.
The course will be divided into five+ parts (with some flexibility depending on the interests of the audience):
The regular interval theorem, the Morse lemma and Reeb's theorem. In connection with Reeb's theorem, I'll describe Milnor's construction of a 7-manifold that is homeomorphic but not diffeomorphic to the 7-sphere (see Milnor's short paper in the Annals of Math., 1956).
Morse functions. Milnor uses ``focal points'' to prove the existence of Morse functions. One reason he does this is that the focal point approach leads to a proof of the famous Lefschetz theorem on hyperplane sections. Although this theorem is highly recommended to anyone interested in complex manifolds/varieties, we might skip it because at this point we don't have the necessary background in algebraic topology. I'll prove the existence of Morse functions in a different and more standard way, using transversality. Transversality is one of the most important tools in differential topology.
Morse functions and CW-complexes. Here we get to the
main theorem of the subject, which says that passing a critical point of Index k
corresponds homotopically to attaching a cell of dimension k. This provides the crucial link with
algebraic topology and homotopy theory. As an
application, we'll prove the Poincare-Hopf index theorem for vector fields.
Application: The Poincare Conjecture in dimensions
greater than four. This conjecture says that a compact smooth manifold homotopy-equivalent
to Sn must be homeomorphic to
Sn. The conjecture is true
for n=1,2by standard classification theorems, still unsolved for n=3,
proved by Freedman in the early 80's for n=4 and proved by Smale around
1960 for n>4. It would take an entire quarter to go through Smale's
proof in detail. We'll spend two or three weeks at the end giving an outline of
this amazing tour de force.
Further topics: If we stick to I-V above, there won't be any time. But if at all possible I'll at least briefly survey some other directions, including the original applications by Morse to geodesics on Riemannian manifolds. Here one views the path-space of the manifold as a sort of infinite-dimensional manifold in its own right, with geodesics the critical points of an energy function. These ideas were used by Bott to prove his famous periodicity theorem (on the homotopy groups of the classical Lie groups).
Prerequisites: The ``Manifolds'' sequence 544-5-6. Besides the basic notions of smooth manifold theory (tangent bundles, smooth maps, inverse function theorem etc.) the main thing you need from this is the notion of flow associated to a vector field. It helps to know some homology theory, but I won't assume this.
References (no. 1 is the required text; the rest I'll put on on reserve):