Algebraic & Geometric Topology

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

Wacław Marzantowicz, Denise de Mattos, and Edivaldo dos Santos

Full-text: Open access

Abstract

Let G=pk be a cyclic group of prime power order and let V and W be orthogonal representations of G with VG=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.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 4 (2012), 2245-2258.

Dates
Received: 30 April 2012
Revised: 14 August 2012
Accepted: 27 August 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715455

Digital Object Identifier
doi:10.2140/agt.2012.12.2245

Mathematical Reviews number (MathSciNet)
MR3020205

Zentralblatt MATH identifier
1262.55001

Subjects
Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 55M35: Finite groups of transformations (including Smith theory) [See also 57S17] 55N91: Equivariant homology and cohomology [See also 19L47] 57S17: Finite transformation groups

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

Citation

Marzantowicz, Wacław; de Mattos, Denise; dos Santos, Edivaldo. Bourgin–Yang version of the Borsuk–Ulam theorem for $\mathbb{Z}_{p^k}$–equivariant maps. Algebr. Geom. Topol. 12 (2012), no. 4, 2245--2258. doi:10.2140/agt.2012.12.2245. https://projecteuclid.org/euclid.agt/1513715455


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