Reconstruction of Manifolds from Their Morse Functions

TL;DR

This paper demonstrates how to reconstruct the topology of a closed manifold from a Morse function by establishing a cell decomposition and showing the equivalence of associated topological categories, confirming the manifold's topology.

Contribution

The paper provides a new proof that the classifying space of a topological category derived from a Morse function is homeomorphic to the manifold, using cell decompositions.

Findings

Classifying space of the constructed category is homeomorphic to the manifold

Cell decomposition induces a topological category equivalent to Cohen-Jones-Segal's

New proof confirms the topological reconstruction from 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.