#### Volume 12, issue 4 (2012)

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Bourgin–Yang version of the Borsuk–Ulam theorem for $\mathbb{Z}_{p^k}$–equivariant maps

### Wacław Marzantowicz, Denise de Mattos and Edivaldo L dos Santos

Algebraic & Geometric Topology 12 (2012) 2245–2258
##### Abstract

Let $G={ℤ}_{{p}^{k}}$ be a cyclic group of prime power order and let $V$ and $W$ be orthogonal representations of $G$ with ${V}^{G}={W}^{G}=\left\{0\right\}$. Let $S\left(V\right)$ be the sphere of $V$ and suppose $f:S\left(V\right)\to W$ is a $G$–equivariant mapping. We give an estimate for the dimension of the set ${f}^{-1}\left\{0\right\}$ 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 $S\left(V\right)$ into a real vector space ${W}^{\prime }$.

##### Keywords
equivariant maps, covering dimension, orthogonal representation, equivariant $K$–theory
##### Mathematical Subject Classification 2010
Primary: 55M20
Secondary: 55M35, 55N91, 57S17
##### 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/