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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

Let G = pk be a cyclic group of prime power order and let V and W be orthogonal representations of G with V G = WG = {0}. Let S(V ) be the sphere of V and suppose f : S(V ) W is a G–equivariant mapping. We give an estimate for the dimension of the set f1{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.

equivariant maps, covering dimension, orthogonal representation, equivariant $K$–theory
Mathematical Subject Classification 2010
Primary: 55M20
Secondary: 55M35, 55N91, 57S17
Received: 30 April 2012
Revised: 14 August 2012
Accepted: 27 August 2012
Published: 5 January 2013
Wacław Marzantowicz
Faculty of Mathematics and Computer Science
Adam Mickiewicz University of Poznań
ul. Umultowska 87
61-614 Poznań
Denise de Mattos
Instituto de Ciências Matemáticas e de Computação
Departamento de Matemática
Universidade de São Paulo
Caixa Postal 668
13560-970 São Carlos
Edivaldo L dos Santos
Departamento de Matemática
Universidade Federal de São Carlos
Caixa Postal 676
13565-905 São Carlos