Framed Morse functions on surfaces

Abstract

Let be a smooth, compact, not necessarily orientable surface with (maybe empty) boundary, and let be the space of Morse functions on 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 . In the case of an orientable surface this is a closed 1-form on with punctures at the critical points of local minimum and maximum of such that in a neighbourhood of each critical point the pair has a canonical form in a suitable local coordinate chart and the 2-form does not vanish on punctured at the critical points and defines there a positive orientation. Each Morse function on is shown to have a framing, and the space endowed with the -topology is homotopy equivalent to the space of framed Morse functions. The results obtained make it possible to reduce the problem of describing the homotopy type of to the simpler problem of finding the homotopy type of . As a solution of the latter, an analogue of the parametric -principle is stated for the space .Bibliography: 41 titles.