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Measurable Kac cohomology for bicrossed products. (English) Zbl 1062.22009

Let \(G\) be a locally compact group. In Definition 2.1 a matched pair of locally compact groups is a pair of closed subgroups \(G_1\), \(G_2\) such that \(G_1 \cap G_2 = \{e\}\) and \(G\setminus G_1G_2\) has Haar measure zero. Define the measurable cochain complex \(\{L(G^n,A)\}\) consisting of measurable functions on \(G^n\) with values in a Polish \(G\)-module \(A\) with the ordinary co-boundary map \(d=\sum_{i=0}^{n+1} (-1)^i d_i\) as the alternative sum of co-face operators \(d_i : L(G^n,A) \to L(G^{n+1},A)\) such that \[ d_iF(g_0,\dots,g_{n+1})= \begin{cases} g_0F(\partial_0(g_0,\dots,g_{n+1})) &\text{ if } i=0\\ F(\partial_i(g_0,\dots,g_{n+1})) &\text{ if } i=1,\dots, n+1 \end{cases}. \] The cohomology of the cochain complex is called the measurable cohomology of the locally compact group \(G\) with coefficients in the Polish \(G\)-module \(A\) and was studied by C. C. Moore [Trans. Am. Math. Soc. 221, 1–33 (1976; Zbl 0366.22005) and Trans. Am. Math. Soc. 221, 35–58 (1976; Zbl 0366.22006)] and D. Wigner [Trans. Am. Math. Soc. 178, 83–93 (1973; Zbl 0264.22001)]. In the paper under review the authors study a measurable version of the Kac cohomology for bicrossed products of matched pairs and prove results on the exact sequence and provide methods of computations, especially the explicit calculation in particular cases of some examples.

MSC:

22D05 General properties and structure of locally compact groups
55N99 Homology and cohomology theories in algebraic topology
20J06 Cohomology of groups
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