Recognition of linear actions on spheres

Author:
Sören Illman

Journal:
Trans. Amer. Math. Soc. **274** (1982), 445-478

MSC:
Primary 57S17; Secondary 57Q30, 57S25

MathSciNet review:
675064

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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