Deformations of functions on surfaces

TL;DR

This paper reviews recent advances in understanding how smooth functions on compact surfaces deform, focusing on homotopy types of stabilizers and orbits, and provides a new proof for properties of Morse maps.

Contribution

It offers a comprehensive review of deformational properties of functions on surfaces and introduces a novel proof regarding the homotopy equivalence of Morse map orbits.

Findings

Homotopy types of stabilizers and orbits described

Connected components of Morse map orbits are homotopy equivalent to products of circles

New direct proof for properties of generic Morse maps

Abstract

The paper contains a review on recent progress in the deformational properties of smooth maps from compact surfaces $M$ to a one-dimensional manifold $P$. It covers description of homotopy types of stabilizers and orbits of a large class of smooth functions on surfaces obtained by the author, E. Kudryavtseva, B. Feshchenko, I. Kuznietsova, Yu. Soroka, A. Kravchenko. We also present here a new direct proof of the fact that for generic Morse maps the connected components their orbits are homotopy equivalent to finite products of circles.