Functions on surfaces and incompressible subsurfaces
Abstract
Let $M$ be a smooth connected compact surface, $P$ be either a real line $\mathbb R$ or a circle $S^1$. Then we have a natural right action of the group $D(M)$ of diffeomorphisms of $M$ on $C^\infty(M,P)$. For $f\in C^\infty(M,P)$ denote respectively by $S(f)$ and $O(f)$ its stabilizer and orbit with respect to this action. Recently, for a large class of smooth maps $f:M\to P$ the author calculated the homotopy types of the connected components of $S(f)$ and $O(f)$. It turned out that except for few cases the identity component of $S(f)$ is contractible, $\pi_i O(f)=\pi_i M$ for $i\geq3$, and $\pi_2 O(f)=0$, while $\pi_1 O(f)$ it only proved to be a finite extension of $\pi_1D_{Id}M\oplus\mathbb Z^{l}$ for some $l\geq0$. In this note it is shown that if $\chi(M)<0$, then $\pi_1O(f)=G_1\times\cdots\times G_n$, where each $G_i$ is a fundamental group of the restriction of $f$ to a subsurface $B_i\subset M$ being either a $2$-disk or a cylinder or a Mobius band. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are studied.
Full Text
