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Actions of totally disconnected groups and equivariant singular homology

Authors
Journal
Topology and its Applications
0166-8641
Publisher
Elsevier
Publication Date
Volume
157
Issue
17
Identifiers
DOI: 10.1016/j.topol.2010.07.016
Keywords
  • Group Action
  • Totally Disconnected Locally Compact Group
  • Equivariant Singular Homology
  • P-Adic Group
  • Hilbert–Smith Conjecture
Disciplines
  • Mathematics

Abstract

Abstract We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert–Smith conjecture. The Hilbert–Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.

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