Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups

Abstract

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G=W^G=\{0\}$. Let $S(V)$ be the sphere of $V$ and $f:S(V)\to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $\dim V$ and $\dim W$, if $G$ is the torus ${\mathbb{T}}^k$, or the $p$-torus ${\mathbb{Z}}_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V)\to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f=\infty$.

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