Bourgin–Yang version of the Borsuk–Ulam theorem for ℤpk–equivariant maps

Abstract

Let GDZpk be a cyclic group of prime power order and let V and W be orthogonal representations of G with V G D W G D f0g . Let S.V / be the sphere of V and suppose f W S.V /!W is a G –equivariant mapping. We give an estimate for the dimension of the set f f0g in terms of V and W . This extends the Bourgin–Yang version of the Borsuk–Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the G –coincidences set of a continuous map from S.V / into a real vector space W 0 .