Characterizing maximal compact subgroups. (English) Zbl 1245.22007
Summary: We prove that for a compact subgroup \(H\) of a locally compact almost connected group \(G\), the following properties are mutually equivalent: (1) \(H\) is a maximal compact subgroup of \(G\), (2) the coset space \(G/H\) is \({\mathbb{Q}}\)-acyclic and \({\mathbb{Z}/2\mathbb{Z}}\)-acyclic in Čech cohomology, (3) \(G/H\) is contractible, (4) \(G/H\) is homeomorphic to a Euclidean space, (5) \(G/H\) is an absolute extensor for paracompact spaces, (6) \(G/H\) is a \(G\)-equivariant absolute extensor for paracompact proper \(G\)-spaces having a paracompact orbit space.
MSC:
| 22D05 | General properties and structure of locally compact groups |
| 22F05 | General theory of group and pseudogroup actions |
| 54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |
Keywords:
locally compact almost connected group; maximal compact subgroup; coset space; acyclic space; contractible space; \(G\)-AEReferences:
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