The Borsuk-Ulam theorem for closed $3$-manifolds having geometry $S^2\times \R$

TL;DR

This paper classifies free involutions on closed 3-manifolds with $S^2 imes $ geometry and determines their Borsuk-Ulam indices, extending classical topological results to this specific geometric setting.

Contribution

It provides a complete classification of free involutions and computes the Borsuk-Ulam index for these manifolds, a novel extension in geometric topology.

Findings

All free involutions on such manifolds are classified.

The Borsuk-Ulam index for each involution is explicitly determined.

The results extend classical Borsuk-Ulam theorems to $S^2 imes $ geometries.

Abstract

Let $M$ be a closed 3-manifold which admits the geometry $S^2\times \R$. In this work we determine all the free involutions $\tau$ on $M$, and the Borsuk-Ulam index of $(M,\tau)$.