The Borsuk-Ulam theorem for 3-manifolds

TL;DR

This paper investigates the Borsuk-Ulam theorem for 3-manifolds with involutions, determining the Z_2-index based on cohomological and linking matrix methods, with applications to both orientable and non-orientable cases.

Contribution

It provides a complete characterization of the Z_2-index for 3-manifolds with involutions using cohomological operations and linking matrices, extending the theorem's understanding.

Findings

The Z_2-index can be 1, 2, or 3 depending on the manifold.

Cohomological operations determine the index in the non-orientable case.

Linking matrices from surgery presentations express the index in the orientable case.

Abstract

We study the Borsuk-Ulam theorem for triple (M;\tau; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution \tau. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index and in our case it takes value 1, 2 or 3. We fully discuss this index according to cohomological operations applied on the characteristic class x \in H^1(N; Z_2), where N = M/\tau is the orbit space. In oriented case, we obtain an expression of the index from the linking matrix of a surgery presentation of the orbit space. We illustrate our results with examples, including a non orientable one.