Path-components of Morse mappings spaces of surfaces

Abstract

Let $M$ be a connected compact surface, $P$ be either ${\mathbb{R}}^1$ or $S^1$, and $\mathcal{F}(M,P)$ be the space of Morse mappings $M\to P$ with compact-open topology. The classification of path-components of $\mathcal{F}(M,P)$ was independently obtained by S. V. Matveev and V. V. Sharko for the case $P={\mathbb{R}}^1$, and by the author for orientable surfaces and $P=S^1$. In this paper we give a new independent and unified proof of this classification for all compact surfaces in the case $P=P=\mathbb{R}$, and for orientable surfaces in the case $P=S^1$. We also extend the author's initial proof to non-orientable surfaces.