Actions of totally disconnected groups and equivariant singular homology

作者：Illman Soren^*

来源：Topology and Its Applications, 2010, 157(17): 2659-2678.

DOI：10.1016/j.topol.2010.07.016

• Group action
• Totally disconnected locally compact group
• Equivariant singular homology
• p-adic group
• Hilbert-Smith conjecture

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.

• 出版日期2010-11-1