Deformations of circle-valued Morse functions on surfaces

TL;DR

This paper classifies the connected components of the space of certain Morse functions from a surface to the circle, extending previous results to functions with specific boundary behavior.

Contribution

It extends the classification of Morse functions on surfaces to include those with boundary restrictions involving constant or covering maps.

Findings

Connected components of the function space are classified.

Extension of previous classifications to boundary conditions.

Provides a comprehensive understanding of Morse functions on surfaces with boundary.

Abstract

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.