Topological actions of wreath products
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arXiv
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866913321185181696 |
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| author | Maksymenko, Sergiy |
| author_facet | Maksymenko, Sergiy |
| contents | 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 $\mathcal{C}^{\infty}(M,\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1409_4319 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Topological actions of wreath products Maksymenko, Sergiy Geometric Topology Algebraic Topology 20E22, 55P15, 37E30, 55Q05 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 $\mathcal{C}^{\infty}(M,\mathbb{R})$. |
| title | Topological actions of wreath products |
| topic | Geometric Topology Algebraic Topology 20E22, 55P15, 37E30, 55Q05 |
| url | https://arxiv.org/abs/1409.4319 |