${\mathbb{Z}_2}$-bordism and the Borsuk–Ulam Theorem

Abstract

The purpose of this work is to classify, for given integers ${m,\, n\geq 1}$, the bordism class of a closed smooth ${m}$-manifold ${X^m}$ with a free smooth involution ${\tau}$ with respect to the validity of the Borsuk–Ulam property that for every continuous map ${\phi : X^m \to \mathbb{R}^n}$ there exists a point ${x\in X^m}$ such that ${\phi (x)=\phi (\tau (x))}$. We will classify a given free ${\mathbb{Z}_2}$-bordism class ${\alpha}$ according to the three possible cases that (a) all representatives ${(X^m, \tau)}$ of ${\alpha}$ satisfy the Borsuk–Ulam property; (b) there are representatives ${({X_{1}^{m}}, \tau_1)}$ and ${({X_{2}^{m}}, \tau_2)}$ of ${\alpha}$ such that ${({X_{1}^{m}}, \tau_1)}$ satisfies the Borsuk–Ulam property but ${({X_{2}^{m}}, \tau_2)}$ does not; (c) no representative ${(X^m, \tau)}$ of ${\alpha}$ satisfies the Borsuk–Ulam property.