The Borsuk-Ulam theorem for 3-manifolds

2020/01/10 by Chahrazade Matmat, Matmat, Chahrazade, Christian Blanchet +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2001.03680

Abstract

We study the Borsuk-Ulam theorem for triple (M;τ; \Rⁿ), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution τ. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z₂-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 ∈ H¹(N; Z₂), where N = M/τis 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.

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