Bourgin–Yang version of the Borsuk–Ulam theorem for ℤpk–equivariant maps

Let $G = ℤ_{p^{k}}$ be a cyclic group of prime power order and let $V$ and $W$ be orthogonal representations of $G$ with $V^{G} = W^{G} = {0}$ . Let $S (V)$ be the sphere of $V$ and suppose $f : S (V) \to W$ is a $G$ –equivariant mapping. We give an estimate for the dimension of the set $f^{- 1} {0}$ in terms of $V$ and $W$ . This extends the Bourgin–Yang version of the Borsuk–Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the $G$ –coincidences set of a continuous map from $S (V)$ into a real vector space $W^{'}$ .

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Keywords

equivariant maps, covering dimension, orthogonal representation, equivariant $K$–theory

Mathematical Subject Classification 2010

Primary: 55M20

Secondary: 55M35, 55N91, 57S17

References

Publication

Received: 30 April 2012

Revised: 14 August 2012

Accepted: 27 August 2012

Published: 5 January 2013

Authors

Wacław Marzantowicz
Faculty of Mathematics and Computer Science Adam Mickiewicz University of Poznań ul. Umultowska 87 61-614 Poznań Poland

Denise de Mattos
Instituto de Ciências Matemáticas e de Computação Departamento de Matemática Universidade de São Paulo Caixa Postal 668 13560-970 São Carlos Brazil
http://www.icmc.usp.br/~topologia/
Edivaldo L dos Santos
Departamento de Matemática Universidade Federal de São Carlos Caixa Postal 676 13565-905 São Carlos Brazil
http://www2.dm.ufscar.br/~edivaldo/

Volume 12, issue 4 (2012)

Wacław Marzantowicz, Denise de Mattos and Edivaldo L dos Santos

Abstract