The Borsuk-Ulam theorem for 3-manifolds

Abstract

We study the Borsuk-Ulam theorem for triple (M, τ, ℝ ⁿ ), 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 ∈ H ¹ (N, ℤ₂), where N = M/τ is the orbit space. In the 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.