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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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 $ G$ be a finite group acting smoothly on a homotopy sphere $ \Sigma^m$. We wish to establish necessary and sufficient conditions for the given $ G$-action on $ \Sigma$ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a $ G$-homeomorphism $ \gamma:\Sigma\to S^m(\rho)$, where $ {S^m}(\rho ) \subset {\mathbf{R}^{m + 1}}(\rho )$ denotes the unit sphere in an orthogonal representation space $ \mathbf{R}^{m + 1}(\rho )$ for $ G$. In order for a $ G$-action on $ \Sigma$ to be topologically equivalent to a linear action it is clearly necessary that:

(i) For each subgroup $ H$ of $ G$ the fixed-point set $ \Sigma^H$ is homeomorphic to a sphere, or empty.

(ii) For any subgroups $ H$ and $ H \subsetneq {H_i},\,1 \leq i \leq k$, of $ G$ the pair $ (\Sigma^{H},\,\cup_{i=1}^{k}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $ (S^{n},\,\cup_{i=1}^{k}S_{i}^{n_{i}})$, where each $ S_i^{{n_i}},\,1 \le i \le k$, is a standard $ n_i$-subsphere of $ S^n$.

In this paper we consider the case where the fixed-point set $ \Sigma^G$ 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)$ ^{\ast}$ For any subgroups $ H$ and $ H \subsetneq {H_i},\,1 \leq i \leq p$, of $ G$ such that $ {\operatorname{dim}}\,{\Sigma ^{{H_i}}} = {\operatorname{dim}}\,{\Sigma ^H} - 2$, the pair $ \Sigma^{H},\,\cup_{i=1}^{p}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $ ({S^n},\, \cup _{i = 1}^pS_i^{n - 2})$, where each $ S_i^{n - 2},\,1 \le i \le p$, is a standard $ (n-2)$-subsphere of $ S^n$.

Our main results are then that, in the case when $ G$ is abelian, conditions (i) and (ii)$ ^{\ast}$ are necessary and sufficient for a given $ G$-action on $ \Sigma $ 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: http://dx.doi.org/10.1090/S0002-9947-1982-0675064-4
PII: S 0002-9947(1982)0675064-4
Keywords: Smooth actions on spheres, linear actions on spheres, equivariant engulfing
Article copyright: © Copyright 1982 American Mathematical Society