Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1981-0620005-3
MathSciNet review: 620005
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57S17, 55M35

Retrieve articles in all journals with MSC: 57S17, 55M35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0620005-3
Article copyright: © Copyright 1981 American Mathematical Society