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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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