Special equipped Morse functions on surfaces / E. A. Kudryavtseva. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2012. № 4. P. 14-20 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].
Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M, and F1 be the space of framed Morse functions, both endowed with the C∞-topology. The space F0 of special framed Morse functions is defined. We prove that the inclusion mapping F0→F1 is a homotopy equivalence. In the case when at least χ(M)+1 critical points of each function of F are labeled, the homotopy equivalences K∼M and F∼F0∼D0×K are proved, where K is the complex of framed Morse functions, M≈F1/D0 is the universal moduli space of framed Morse functions, D0 is the group of self-diffeomorphisms of M homotopic to the identity.
Key words: Morse function, framed Morse function, complex of framed Morse functions, C∞-topology, universal moduli space.