Topology of the spaces of functions with prescribed singularities on surfaces

TL;DR

This paper investigates the topological structure of the space of smooth functions on a surface that share the same prescribed singularity types as a given function, analyzing their homotopy types and symmetry group actions.

Contribution

It characterizes the homotopy type of the space of functions with prescribed $A_$-type singularities on surfaces and describes its decomposition under coordinate transformations.

Findings

Determines the homotopy type of the function space.

Describes the orbit decomposition under coordinate change group.

Provides a detailed topological classification of these function spaces.

Abstract

Let $M$ be a smooth connected orientable closed surface and $f_0\in C^\infty(M)$ a function having only critical points of the $A_\mu$-types, $\mu\in\mathbb N$. Let ${\mathcal F}={\mathcal F}(f_0)$ be the set of functions $f\in C^\infty(M)$ having the same types of local singularities as those of $f_0$. We describe the homotopy type of the space $\mathcal F$, endowed with the $C^\infty$-topology, and its decomposition into orbits of the action of the group of "left-right changings of coordinates".