Abstract
Smith theory says that the fixed point set of a semi-free action of a group
$G$
on a contractible space is
${\mathbb {Z}}_p$
-acyclic for any prime factor
$p$
of the order of
$G$
. Jones proved the converse of Smith theory for the case
$G$
is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain
$K$
-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of
$K$
-theoretical obstructions.
Publisher
Cambridge University Press (CUP)
Cited by
1 articles.
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1. Fixed point sets and the fundamental group II: Euler characteristics;Proceedings of the Royal Society of Edinburgh: Section A Mathematics;2023-10-10