Reconstruction of Manifolds from Their Morse Functions

Abstract

This paper describes how to recover the topology of a closed manifold $M$ from a good Morse function $f$ on $M$. The essential method was suggested by Cohen, Jones and Segal. They constructed a topological category $C_{f}$ and claimed that the classifying space $BC_{f}$ is homeomorphic to $M$. We prove it from a different viewpoint with them using a cell decomposition of $M$ associated to $f$. The cell complex $M_{f}$ equipped with the decomposition induces a topological category $C(M_{f})$ whose classifying space $BC(M_{f})$ is homeomorphic to $M$. We show that $C(M_{f})$ is isomorphic to $C_{f}$ as a topological category.