Bourgin-Yang versions of the Borsuk-Ulam theorem for p-toral groups

2013/02/06 by Marzantowicz, Wacław, de Mattos, Denise, Santos, Edivaldo L. dos
#55M20 (Primary) #55M35 #55N91 #57S17 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1302.1494

Abstract

Let V and W be orthogonal representations of G with V^G= W^G=\0\. Let S(V ) be the sphere of V and f : S(V ) → W be a G-equivariant mapping. We 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 extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any p-toral group G and a G-map f:S(V) → W, with dim V=∞ and dim W

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