Marzantowicz, Waclaw, Kolokvij junij 2016

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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.

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