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Poincaré duality in P.A. Smith theory


Authors: Christopher Allday, Bernhard Hanke and Volker Puppe
Journal: Proc. Amer. Math. Soc. 131 (2003), 3275-3283
MSC (2000): Primary 57S10, 57P10, 55N10; Secondary 55N91
DOI: https://doi.org/10.1090/S0002-9939-03-06856-4
Published electronically: February 6, 2003
MathSciNet review: 1992869
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Abstract: Let $G=S^1$, $G=\mathbb{Z}/p$ or more generally $G$ be a finite $p$-group, where $p$ is an odd prime. If $G$ acts on a space whose cohomology ring fulfills Poincaré duality (with appropriate coefficients $k$), we prove a mod $4$ congruence between the total Betti number of $X^G$ and a number which depends only on the $k[G]$-module structure of $H^*(X;k)$. This improves the well known mod $2$ congruences that hold for actions on general spaces.


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Additional Information

Christopher Allday
Affiliation: Department of Mathematics, University of Hawaii, 2565 Mc Carthy Mall, Honolulu, Hawaii 96822
Email: chris@math.hawaii.edu

Bernhard Hanke
Affiliation: Department of Mathematics, Universität München, Theresienstr. 39, 80333 München, Germany
Email: hanke@rz.mathematik.uni-muenchen.de

Volker Puppe
Affiliation: Department of Mathematics, Universität Konstanz, 78457 Konstanz, Germany
Email: Volker.Puppe@uni-konstanz.de

DOI: https://doi.org/10.1090/S0002-9939-03-06856-4
Keywords: Group action, Betti number, Poincar\'e duality space
Received by editor(s): September 20, 2001
Received by editor(s) in revised form: May 3, 2002
Published electronically: February 6, 2003
Additional Notes: The second author holds a DFG research grant. He thanks the University of Notre Dame for its hospitality during the work on this paper
Communicated by: Paul Goerss
Article copyright: © Copyright 2003 American Mathematical Society

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