Path-components of Morse mappings spaces of surfaces
by
Sergey Maksymenko
Kiev, Institute of Mathematics, NAS of Ukraine

Let M be a connected compact surface, P be either R¹ or S¹, and F(M, P) be the space of Morse mappings M ® P with compact-open topology. The classification of path-components of F(M, P) was independently obtained by S. V. Matveev and V. V. Sharko for the case P=R¹, and by the author for orientable surfaces and P=S¹. We present an independent proof of this classification based on the structure of the mapping class group of a surface. The main observation is that "elementary" diffeomorphisms like Dehn twists and Y-diffeomorphisms that generate mapping class group preserves some Morse function. Our approach has a close relationship to the representation of the mapping class group obtained by A.Hatcher and W.Thurston via Morse functions.

Date received: December 5, 2005