Reconstruction of Manifolds from Their Morse Functions

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 $B{C}_f$ 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.