Stabilizers and orbits of circle-valued smooth functions

Abstract

Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M)\times Diff(P)$ on the space of smooth mappings $C^{\infty }(M,P)$. For $f\in C^{\infty }(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author proved that under mild conditions on $f\in C^{\infty }(M,R^1)$ the corresponding stabilizers and orbits are homotopy equivalent: $S_{MR}\sim S_M$ and $O_{MR}\sim O_M$. These results are extended here to the actions on $C^{\infty }(M,S^1)$. It is proved that under the similar conditions (that are rather typical) we have that $S_{MS}\sim S_M$ and $O_{MS}\sim O_M\times S^1$.

11 pages, 1 figure. This paper is an extension of author's preprint http://xxx.lanl.gov/math.FA/0411612 to circle-valued functions