# Marzantowicz, Waclaw, Kolokvij junij 2016

In 1933 S. Ulam posed and K. Borsuk showed that if n > m then it is impossible to map $f: S^n \to S^m$ 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 $\dim f^{-1}(0) \geq n-m-1$.

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 VG = WG = {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 Zf = 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 $\dim Z_f= \infty$.