Bourgin-Yang versions of the Borsuk-Ulam theorem for \(p\)-toral groups (Q2409483)

Let \(G\) be a compact Lie group and let \(V\) and \(W\) be two orthogonal representations of \(G\), such that \(V^{G}=W^{G}=\left\{0\right\}\) for the sets of fixed points of \(G\). Let \(f:S(V)\to W\) be a \(G\)-equivariant mapping. In this paper the authors 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 result extends the classical Bourgin-Yang problem for this class of groups. Finally, they show that for an action of a \(p\)-toral group \(G\), dim\(Z_f=\infty\) if dim\(V=\infty\) and dim\(W<\infty\).