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 element of F. In the case of an orientable surface M this is a closed 1-form α 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,α) has a canonical form in a suitable local coordinate chart and the 2-form df and α 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∞-topology is homotopy equivalent to the space 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 F. As a solution of the latter, an analogue of the parametric h-principle is stated for the space F. Bibliography: 41 titles.