Path-components of Morse mappings spaces of surfaces

Abstract

Let $M$ be a compact surface and $P$ be a one dimensional manifold without boundary, that is the line $\mathbb{R}^1$ or a circle $S^1$. The classification of path-components of the space of Morse maps from $M$ into $P$ was recently obtained by S. V. Matveev and V. V. Sharko for the case $P=\mathbb{R}$. For $P=S^1$ the classification was obtained by the author. All this results can be reformulated as one theorem: "Two Morse maps $f,g:M \to P$ belong to the same path component of a space of Morse mappings from $M$ into $P$ if and only if they are homotopic and have the same number of crutucal points in each index and the same sets of positive and negative boundary circles". Here we give another independent proof of this theorem based on Lickorish's theorem on generators of homeotopy group of surface.