JournalscmhVol. 80 , No. 3DOI 10.4171/cmh/30

Path-components of Morse mappings spaces of surfaces

  • Sergey Maksymenko

    National Academy of Science of Ukraine, Kyiv, Ukraine
Path-components of Morse mappings spaces of surfaces cover


Let MM be a connected compact surface, PP be either R1{\Bbb R}^1 or S1S^1, and F(M,P){\cal F}(M,P) be the space of Morse mappings MPM\to P with compact-open topology. The classification of path-components of F(M,P){\cal F}(M,P) was independently obtained by S. V. Matveev and V. V. Sharko for the case P=R1P={\Bbb R}^1, and by the author for orientable surfaces and P=S1P=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=RP=P={\Bbb R}, and for orientable surfaces in the case P=S1P=S^1. We also extend the author's initial proof to non-orientable surfaces.