Bourgin-Yang theorems for the $p$-toral groups

In 1933 S. Ulam posed and K. Borsuk showed that if $n>m$ \newline then {\bf{it is impossible}} to map $f: S^n \to S^m$

\centerline{preserving symmetry: $f(-x)= -f(x)$ .}

Next in 1954-55, C. T. Yang, and D. Bourgin, showed that if $f: \mathbb{S}^n\to \mathbb{R}^{m+1}$ preserves this symmetry then

\centerline{$ \dim f^{-1}(0) \geq n-m-1$.}

We will present versions of the latter for some other groups of symmetries and also discuss the case $n=\infty$

Let $V$ and $W$ be orthogonal representations of a compact Lie group $G$ 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 estimate the dimension of set $Z_f=f^{-1}\{0\}$ in terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, the $p$-torus $\mathbb Z_p^k$, or the cyclic group $\mathbb Z_{p^k}$, $p$-prime.

Finally, we show that for any $p$-toral group: $\;e\hookrightarrow \mathbb{T}^k \hookrightarrow G \to \mathcal{P}\to e$, $P$ a finite $p$-group and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$

Bourgin-Yang_Primorska.pdf