The Borsuk-Ulam theorem for 3-manifolds

Chahrazade Matmat; Christian Blanchet

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arXiv · 2001.03680

The Borsuk-Ulam theorem for 3-manifolds

Chahrazade Matmat·Christian Blanchet

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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.

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BibTeX RIS

Chahrazade Matmat, Christian Blanchet. 2020-01-10. The Borsuk-Ulam theorem for 3-manifolds. https://arxiv.org/abs/2001.03680

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