The Borsuk-Ulam theorem for 3-manifolds
Algebraic Topology
2021-02-02 v3
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.
Keywords
Cite
@article{arxiv.2001.03680,
title = {The Borsuk-Ulam theorem for 3-manifolds},
author = {Chahrazade Matmat and Christian Blanchet},
journal= {arXiv preprint arXiv:2001.03680},
year = {2021}
}