Existence and deformability of topological Morse functions

TL;DR

This paper investigates the existence and deformability of topological Morse functions, providing a simple construction of continuous families, which enhances understanding of their properties on topological manifolds.

Contribution

It introduces a straightforward method to construct continuous families of topological Morse functions, addressing key issues of existence and deformability.

Findings

Constructed continuous families of topological Morse functions

Demonstrated existence of topological Morse functions in various contexts

Provided insights into deformability properties of these functions

Abstract

In the 1950s Morse defined the analogue of Morse functions for topological manifolds. In many instances, when mathematicians are using techniques on topological manifolds that appear to be Morse-theoretic in nature, there is a topological Morse function implicit in the argument. Topological Morse functions are known to inherit most of the familiar properties of the usual (smooth) Morse functions, with two crucial exceptions: existence and deformability. This paper gives a simple construction of continuous families of topological Morse functions.