Let be a connected compact surface, be either or , and be the space of Morse mappings with compact-open topology. The classification of path-components of was independently obtained by S. V. Matveev and V. V. Sharko for the case , and by the author for orientable surfaces and . In this paper we give a new independent and unified proof of this classification for all compact surfaces in the case , and for orientable surfaces in the case . We also extend the author's initial proof to non-orientable surfaces.
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Sergey Maksymenko, Path-components of Morse mappings spaces of surfaces. Comment. Math. Helv. 80 (2005), no. 3, pp. 655–690DOI 10.4171/CMH/30