Caractérisation des groupes de cohomologie d’un groupe topologique en termes de suites exactes. (English) Zbl 0537.22002

Gérard, Yves

Publ. Dep. Math., Nouv. Sér., Univ. Claude Bernard, Lyon 6/A, 22 p. (1982).

Let G be a topological group and let A be a smooth G-module, i.e., a G- module in which every element has an isotropy group open in G. The author considers the complex whose n-cochains are the mappings \(c:G^ n\to A\) such that, for \(0\leq i\leq n\) and for each \((g_ 1,g_ 2,...,g_ i)\in G^ i,\) there exists a neighborhood U of \(1_ G\) in G with \[ c(g_ n,...,g_{i+2},g_{i+1}u,\quad g_ i,...,g_ 1)=c(g_ n,...,g_{i+2},g_{i+1},g_ i,...,g_ 1) \] for all \((g_ n,...,g_{i+1})\in G^{n-i}\) and all \(u\in U\) [J. Algebra, 49, 422-440 (1977; Zbl 0368.22002)].
The author shows that, for each \(n\geq 2\), there exists an isomorphism of the n-th cohomology group \(H^ n(G,A)\) of this complex onto a cohomology group defined in terms of a set of exact sequences which begin by A and end by G.

Reviewer: U.Cattaneo

MSC:

22A05	Structure of general topological groups
20J05	Homological methods in group theory
16W20	Automorphisms and endomorphisms

Keywords:

topological group, cohomology group, Galois group, cohomological functor; smooth G-module

Citations:

Zbl 0368.22002