Path-components of Morse mappings spaces of surfaces

Let $M$ be a compact surface and $P$ be a one dimensional manifold withoutboundary, that is the line $\mathbb{R}^1$ or a circle $S^1$. The classificationof path-components of the space of Morse maps from $M$ into $P$ was recentlyobtained 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 bereformulated as one theorem: "Two Morse maps $f,g:M \to P$ belong to the samepath component of a space of Morse mappings from $M$ into $P$ if and only ifthey are homotopic and have the same number of crutucal points in each indexand the same sets of positive and negative boundary circles". Here we giveanother independent proof of this theorem based on Lickorish's theorem ongenerators of homeotopy group of surface.Comment: LaTex2e, 33 pages, 26 figures (eps) Now the proof is given for all compact orientable and non-orientable surface