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Abelian $ p$-group actions on homology spheres

Author: Ronald M. Dotzel
Journal: Proc. Amer. Math. Soc. 83 (1981), 163-166
MSC: Primary 57S17; Secondary 55M35
MathSciNet review: 620005
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Abstract: The Borel formula is extended to an identity covering actions of arbitrary Abelian $ p$-groups. Specifically, suppose $ G$ is an Abelian $ p$-group which acts on a finite $ {\text{CW}}$-complex $ X$ which is a $ {Z_p}$-homology $ n$-sphere. Each $ {X^H}$ must be a $ {Z_p}$-homology $ n(H)$-sphere and then

$\displaystyle n - n(G) = \sum {(n(K)} - n(K/p))$

where the sum is over $ {A_0} = \{ \left. K \right\vert G/K\;{\text{is}}\;{\text{cyclic}}\} $ and the group $ K/p$ is defined by

$\displaystyle K/p = \{ g \in \left. G \right\vert pg \in K\} .$

This result is an immediate corollary of Theorem 2, whose converse Theorem 1, is also proven. Thus actions of Abelian $ p$-groups on homology spheres resemble linear representations.

References [Enhancements On Off] (What's this?)

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  • [4] R. Dotzel, A converse to the Borel formula, Trans. Amer. Math. Soc. 250 (1979), 275-287. MR 530056 (80g:55004)
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