Framed Morse functions on surfaces

Let $M$ be a smooth, compact, not necessarily orientable surface with (maybe empty) boundary, and let $F$ be the space of Morse functions on $M$ that are constant on each component of the boundary and have no critical points at the boundary. The notion of framing is defined for a Morse function $f\in F$. In the case of an orientable surface $M$ this is a closed 1-form $\alpha$ on $M$ with punctures at the critical points of local minimum and maximum of $f$ such that in a neighbourhood of each critical point the pair $(f,\alpha)$ has a canonical form in a suitable local coordinate chart and the 2-form $df\wedge\alpha$ does not vanish on $M$ punctured at the critical points and defines there a positive orientation. Each Morse function on $M$ is shown to have a framing, and the space $F$ endowed with the $C^\infty$-topology is homotopy equivalent to the space $\mathbb F$ of framed Morse functions. The results obtained make it possible to reduce the problem of describing the homotopy type of $F$ to the simpler problem of finding the homotopy type of $\mathbb F$. As a solution of the latter, an analogue of the parametric $h$-principle is stated for the space $\mathbb F$.
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