2015/04/15 by Crabb, Michael C., Goncalves, Daciberg L., Libardi, Alice K. M. +1
#55M20 #57R20 #57R85 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1504.03929
The purpose of this work is to classify, for given integers m, n≥ 1, the bordism class of a closed smooth m-manifold X with a free smooth involution τ with respect to the validity of the \it Borsuk-Ulam property that for every continuous map ϕ: X → Rn there exists a point x∈ X such that ϕ(x)=ϕ(τ(x)). We will classify a given free Z2-bordism class α according to the three possible cases that (a) all representatives (X , τ) of α satisfy the Borsuk-Ulam property; (b) there are representatives (X_ 1, τ1) and (X2, τ2) of α such that (X1, τ1) satisfies the Borsuk-Ulam property but (X2, τ2) does not; (c) no representative (X , τ) of α satisfies the Borsuk-Ulam property.