On the homotopy type of spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ with a fixed number of critical points of each index such that at least $\chi(M)+1$ critical points are labelled by different labels (numbered). The notion of a skew cylindric-polyhedral complex is introduced, which generalizes the notion of a polyhedral complex. The skew cylindric-polyhedral complex $\widetilde{\mathbb K}$ (“the complex of framed Morse functions”) associated with the space $F$ is defined. In the case $M=S^2$ the polytope $\widetilde{\mathbb K}$ is finite; its Euler characteristic $\chi(\widetilde{\mathbb K})$ is calculated and the Morse inequalities for its Betti numbers $\beta_j(\widetilde{\mathbb K})$ are obtained. The relation between the homotopy types of the polytope $\widetilde{\mathbb K}$ and the space $F$ of Morse functions equipped with the $C^\infty$-topology is indicated.
Bibliography: 51 titles.