MATH 7530 - Algebraic Topology II: Morse Theory and More

Jim West, fall 2016.

Consider the height function $f$ on a torus $T$ sitting on edge. The structural decomposition of $T$ that you see given by the level sets is in fact an instance of a general phenomenon controlled by the gradient of $f$. Morse theory is about this kind of description. If we have time, we'll investigate either the $h$-cobordism theorem or the Bott periodicity theorem as an application.

[The $h$-cobordism theorem says (in this version) that if $W$ is a simply connected smooth $n+1$ dimensional manifold, $n>4$, $W$ has 2 boundary components, $M$ and $N$, and if the inclusion of each into $W$ is a homotopy equivalence, then $W$ is diffeomorphic to $M \times I$ (and thus $M$ is diffeomorphic to $N$). It is the foundation of the theory of surgery in $n$ dimensions. The Bott periodicity theorem describes periodicity of the homotopy groups of the stable Unitary and Orthogonal groups and is fundamental to the study of vector bundles (topological K-theory.)]