Topological actions of wreath products

TL;DR

This paper describes the fundamental group of quotient spaces formed by wreath product actions on topological spaces, with applications to the homotopy types of orbits of smooth functions on surfaces.

Contribution

It provides an explicit description of the fundamental group of quotient spaces under wreath product actions, extending known results to non-contractible spaces.

Findings

Explicit formula for (W/(GH)) as a wreath product.

Application to computing homotopy types of orbits of smooth functions.

Extension of classical results to non-contractible spaces.

Abstract

Let $G$ and $H$ be two groups acting on path connected topological spaces $X$ and $Y$ respectively. Assume that $H$ is finite of order $m$ and the quotient maps $p:X\to X/G$ and $q:Y\to Y/H$ are regular coverings. Then it is well-known that the wreath product $G\wr H$ naturally acts on $W = X^m\times Y$, so that the quotient map $r:W \to W/(G\wr H)$ is also a regular covering. We give an explicit description of $\pi_1(W/(G\wr H))$ as a certain wreath product $\pi_1(X/G)\,\wr_{\partial_Y}\pi_1(Y/H)$ corresponding to a non-effective action of $\pi_1(Y/H)$ on the set of maps $H\to\pi_1(X/G)$ via the boundary homomorphism $\partial_{Y}:\pi_1(Y/H) \to H$ of the covering map $q$. Such a statement is known and usually exploited only when $X$ and $Y$ are contractible, in which case $W$ is also contractible, and thus $W/(G\wr H)$ is the classifying space of $G\wr H$. The applications are given to the computation of the homotopy types of orbits of typical smooth functions $f$ on orientable compact surfaces $M$ with respect to the natural right action of the groups $\mathcal{D}(M)$ of diffeomorphisms of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$.