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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Orientation preserving actions of finite abelian groups on spheres


Author: Ronald M. Dotzel
Journal: Proc. Amer. Math. Soc. 100 (1987), 159-163
MSC: Primary 57S17; Secondary 57S25
DOI: https://doi.org/10.1090/S0002-9939-1987-0883421-6
MathSciNet review: 883421
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Abstract: If $ G$ is a finite Abelian group acting as a $ {{\mathbf{Z}}_{(\mathcal{P})}}$-homology $ n$-sphere $ X$ (where $ \mathcal{P}$ is the set of primes dividing $ \vert G\vert)$, then there is an integer valued function $ n(,G)$ defined on the prime power subgroups $ H$ of $ G$ such that $ {X^H}$ has the $ {{\mathbf{Z}}_{(p)}}$-homology of a sphere $ {S^{n(H,G)}}$. We prove here that there exists a real representation $ R$ of $ G$ such that for any prime power subgroup $ H$ of $ G,\dim (S({R^H})) = n(H,G)$ where $ S({R^H})$ is the unit sphere of $ {R^H}$, provided that $ n - n(H,G)$ is even whenever $ H$ is a $ 2$-subgroup of $ G$.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0883421-6
Article copyright: © Copyright 1987 American Mathematical Society

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