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

Abstract

Let $G={\mathbb{Z}}_{p^k}$ 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 $W^{\prime }$.