Actions of finite groups and smooth functions on surfaces

Abstract

Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and ${\mathcal{E}}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and ${\mathcal{S}}_V(f)=\{h\in \mathcal{S}(f)|h(V)=V\}.$ The group ${\mathcal{S}}_V(f)$ acts on the set ${\pi }_0\partial {\mathcal{E}}_V$ of connected components of the boundary of ${\mathcal{E}}_V.$ Therefore we have a homomorphism $\varphi :\mathcal{S}(f)\to \mathrm{Aut}({\pi }_0\partial {\mathcal{E}}_V)$. Let also $G=\varphi (\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}({\pi }_0\partial {\mathcal{E}}_V).$ Suppose that the inclusion $\partial {\mathcal{E}}_V\subset M\setminus V$ induces a bijection ${\pi }_0\partial {\mathcal{E}}_V\to {\pi }_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to {\mathcal{S}}_V(f)$ of the homomorphism $\varphi ,$ so, the action of $H$ on $\partial {\mathcal{E}}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial ${\mathbb{R}}^2\to \mathbb{R}$ without multiple linear factors.

Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=883