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 $f: mathbb{S}^nto mathbb{R}^{m+1}$ 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: $;ehookrightarrow mathbb{T}^k hookrightarrow G to mathcal{P}to e$, $P$ 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