## Algebraic & Geometric Topology

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

#### Abstract

Let $G=ℤpk$ 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 $f−1{0}$ 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 $W′$.

#### Article information

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

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

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

#### 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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