Recognition of linear actions on spheres
Author:
Sören Illman
Journal:
Trans. Amer. Math. Soc. 274 (1982), 445-478
MSC:
Primary 57S17; Secondary 57Q30, 57S25
DOI:
https://doi.org/10.1090/S0002-9947-1982-0675064-4
MathSciNet review:
675064
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a finite group acting smoothly on a homotopy sphere
. We wish to establish necessary and sufficient conditions for the given
-action on
to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a
-homeomorphism
, where
denotes the unit sphere in an orthogonal representation space
for
. In order for a
-action on
to be topologically equivalent to a linear action it is clearly necessary that:
(i) For each subgroup of
the fixed-point set
is homeomorphic to a sphere, or empty.
(ii) For any subgroups and
, of
the pair
is homeomorphic to a standard pair
, where each
, is a standard
-subsphere of
.
In this paper we consider the case where the fixed-point set is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need:
(ii) For any subgroups
and
, of
such that
, the pair
is homeomorphic to a standard pair
, where each
, is a standard
-subsphere of
.
Our main results are then that, in the case when is abelian, conditions (i) and (ii)
are necessary and sufficient for a given
-action on
to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1982-0675064-4
Keywords:
Smooth actions on spheres,
linear actions on spheres,
equivariant engulfing
Article copyright:
© Copyright 1982
American Mathematical Society