  Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse
functions on $M$ having fixed number of critical points of each index, moreover
at least $\chi(M)+1$ critical points are labeled by different labels
(enumerated). A notion of a skew cylindric-polyhedral complex, which
generalizes the notion of a polyhedral complex, is introduced. The skew
cylindric-polyhedral complex $\mathbb{\widetilde K}$ (the "complex of framed
Morse functions"), associated with the space $F$, is defined. In the case when
$M=S^2$, the polyhedron $\mathbb{\widetilde K}$ is finite; its Euler
characteristic is evaluated and the Morse inequalities for its Betti numbers
are obtained. A relation between the homotopy types of the polyhedron
$\mathbb{\widetilde K}$ and the space $F$ of Morse functions, endowed with the
$C^\infty$-topology, is indicated.
