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Topological actions of wreath products

Sergiy Maksymenko

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 $π_1(W/(G\wr H))$ as a certain wreath product $π_1(X/G)\,\wr_{\partial_Y}π_1(Y/H)$ corresponding to a non-effective action of $π_1(Y/H)$ on the set of maps $H\toπ_1(X/G)$ via the boundary homomorphism $\partial_{Y}:π_1(Y/H) \to H$ 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 $\mathcal{C}^{\infty}(M,\mathbb{R})$.

Topological actions of wreath products

Abstract

Let and be two groups acting on path connected topological spaces and respectively. Assume that is finite of order and the quotient maps and are regular coverings. Then it is well-known that the wreath product naturally acts on , so that the quotient map is also a regular covering. We give an explicit description of as a certain wreath product corresponding to a non-effective action of on the set of maps via the boundary homomorphism of the covering map . Such a statement is known and usually exploited only when and are contractible, in which case is also contractible, and thus is the classifying space of . The applications are given to the computation of the homotopy types of orbits of typical smooth functions on orientable compact surfaces with respect to the natural right action of the groups of diffeomorphisms of on .

Paper Structure

This paper contains 24 sections, 15 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Homotopies in spaces of maps
  5. Exponential law
  6. Paths and homotopies
  7. Homotopy groups and sets
  8. Fundamental group $\pi_1(X,x)$
  9. The set $\pi_1(X,A,x)$
  10. The set $\pi_0(X,x)$
  11. Group structure on $\pi_1(X,A,x)$
  12. Weakly discontinuous actions
  13. Group structure on $\pi_0\mathcal{P}(X,x,Gx)$
  14. Dynamical systems
  15. Wreath products
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose $X$ and $Y$ are path connected topological spaces, the actions of $G$ and $H$ are properly discontinuous, that is the corresponding quotient maps $p\colon X\to X/G$ and $q\colon Y\to Y/H$ are regular coverings, and $H$ is finite and consists of $m$ elements. Let also $\partial_{{}_{Y}}\!\col see Example exmp:wreath_prod_of_homomorphism for the definition of the right-hand side wreath produ

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1.1: e.g. Engelking:GT:1989
  • Corollary 2.1.2
  • proof
  • Corollary 2.1.3
  • proof
  • Corollary 2.1.4
  • proof
  • Example 2.2.1
  • Example 2.2.2
  • ...and 39 more