$Z_2$-bordism and the Borsuk-Ulam Theorem

TL;DR

This paper classifies free involution manifolds based on whether they satisfy the Borsuk-Ulam property, providing a comprehensive framework for understanding the relation between bordism classes and this topological property.

Contribution

It introduces a classification scheme for $Z_2$-bordism classes of manifolds according to the Borsuk-Ulam property, linking bordism theory with this fundamental topological theorem.

Findings

Classified bordism classes into three categories based on Borsuk-Ulam property satisfaction.

Established criteria to determine if all, some, or none of the representatives satisfy the property.

Provided a framework for analyzing involutions in relation to 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 $X$ with a free smooth involution $\tau$ with respect to the validity of the {\it Borsuk-Ulam property} that for every continuous map $\phi : X \to R^n$ there exists a point $x\in X$ such that $\phi (x)=\phi (\tau (x))$. We will classify a given free $Z_2$-bordism class $\alpha$ according to the three possible cases that (a) all representatives $(X , \tau)$ of $\alpha$ satisfy the Borsuk-Ulam property; \ (b) there are representatives $(X_ 1, \tau_1)$ and $(X_2, \tau_2)$ of $\alpha$ such that $(X_1, \tau_1)$ satisfies the Borsuk-Ulam property but $(X_2, \tau_2)$ does not; \ (c) no representative $(X , \tau)$ of $\alpha$ satisfies the Borsuk-Ulam property.