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

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

Zbl 0368.22002