Topological actions of wreath products

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 \(p:X\rightarrow X/G\) $p:X\to X/G$ and \(q:Y\rightarrow Y/H\) $q:Y\to Y/H$ are regular coverings. Then it is well known that the wreath product \(G\hspace{1.111pt}{\wr }\hspace{1.111pt}H\) $G\wr H$ naturally acts on \(W = X^m\hspace{1.111pt}{\times }\hspace{1.111pt}Y\) $W=X^m\times Y$ , so that the quotient map \(r:W \rightarrow W/(G\hspace{1.111pt}{\wr }\hspace{1.111pt}H)\) $r:W\to W/(G\wr H)$ is also a regular covering. We give an explicit description of \(\pi _1(W/(G\hspace{1.111pt}{\wr }\hspace{1.111pt}H))\) ${\pi }_1(W/(G\wr H))$ as a certain wreath product \(\pi _1(X/G)\hspace{1.111pt}{\wr }_{\partial _Y}\pi _1(Y/H)\) ${\pi }_1(X/G){\wr }_{{\partial }_Y}{\pi }_1(Y/H)$ corresponding to a non-effective action of \(\pi _1(Y/H)\) ${\pi }_1(Y/H)$ on the set of maps \(H\rightarrow \pi _1(X/G)\) $H\to {\pi }_1(X/G)$ via the boundary homomorphism \(\partial _{Y}:\pi _1(Y/H) \rightarrow H\) ${\partial }_Y:{\pi }_1(Y/H)\to H$ 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 \(W/(G\hspace{1.111pt}{\wr }\hspace{1.111pt}H)\) $W/(G\wr H)$ is the classifying space of \(G\hspace{1.111pt}{\wr }\hspace{1.111pt}H\) $G\wr H$ . 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 of diffeomorphisms of M on \(\mathscr {C}^{\infty }(M,\mathbb {R})\) ${\mathcal{C}}^{\infty }(M,\mathbb{R})$ .