Homotopy dimension of orbits of Morse functions on surfaces

Sergiy Maksymenko

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arXiv · 0710.4437

Homotopy dimension of orbits of Morse functions on surfaces

Sergiy Maksymenko

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Abstract

Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the connected components of $O$ have the homotopy type of a finite-dimensional CW-complex. Actually, these connected components are homotopy equivalent to a certain covering space of the $n$-th configuration space of the interior of $M$. As a consequence we obtain that the fundamental group of $O$ is a subgroup of the $n$-th braid group of $M$.

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Sergiy Maksymenko. 2007-10-24. Homotopy dimension of orbits of Morse functions on surfaces. https://arxiv.org/abs/0710.4437

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