Z2-bordism and the Borsuk–Ulam Theorem

Abstract

The purpose of this work is to classify, for given integers m,n≥1, the bordism class of a closed smooth m-manifold Xm with a free smooth involution τ with respect to the validity of the Borsuk–Ulam property that for every continuous map φ: Xm→ Rn there exists a point x∈ Xm 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 (Xm, τ) of α satisfy the Borsuk–Ulam property; (b) there are representatives (X1m,τ1) and (X2m,τ2) of α such that (X1m,τ1) satisfies the Borsuk–Ulam property but (X2m,τ2) does not; (c) no representative (Xm, τ) of α satisfies the Borsuk–Ulam property.