Special framed Morse functions on surfaces

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ be the space of framed Morse functions, both endowed with the $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookrightarrow\mathbb{F}^1$ is a homotopy equivalence. In the case when at least $\chi(M)+1$ critical points of each function of $F$ are labeled, the homotopy equivalences $\widetilde{\mathbb{K}}\sim\widetilde{\mathcal{M}}$ and $F\sim\mathbb{F}^0\sim\mathscr{D}^0\times\widetilde{\mathbb{K}}$ are proved, where $\mathbb{K}$ is the complex of framed Morse functions, $\widetilde{\mathcal{M}}\approx\mathbb{F}^1/\mathscr{D}^0$ is the universal moduli space of framed Morse functions, $\mathscr{D}^0$ is the group of self-diffeomorphisms of $M$ homotopic to the identity.