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The topological structure of Polish groups and groupoids of measure space transformations. (English) Zbl 0851.22001

Soit \((X,\mu)\) un espace mesuré sans atome et \(\sigma\)-fini. Soit \(\text{Part} (X,\mu)\) le groupoïde des \(\mu\)-isomorphismes d’un sous-ensemble de \(X\) sur un autre sous-ensemble de \(X\), muni de la topologie faible. L’auteur montre que \(\text{Part}(X, \mu)\) est contractile par une homotopie laissant invariant le sous-groupoïde des isomorphismes partiels conservant \(\mu\), ainsi que le groupe \(\text{Aut} (X,\mu)\) des isomorphismes de \((X, \mu)\). Pour un sous-groupe dénombrable \(\Gamma\) de \(\text{Aut}(X, \mu)\), soit \([\Gamma] = \{\theta\in \text{Aut}(X,\mu)/\theta(x) \in \Gamma x\) pour presque tout \(x\}\). L’auteur étudie la contractilité du groupe \([\Gamma]\), muni de la topologie uniforme, ainsi que le type d’homotopie de son normalisateur dans \(\operatorname{Aut}(X,\mu)\).
Reviewer: R.Cauty (Paris)

MSC:

22A05 Structure of general topological groups
55P10 Homotopy equivalences in algebraic topology
22A22 Topological groupoids (including differentiable and Lie groupoids)
28D15 General groups of measure-preserving transformations
46L55 Noncommutative dynamical systems
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