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.

摘要

我们研究三重（M，τ，ⁿ）的Borsuk-Ulam定理，其中M是一个紧凑的，连通的，具有无定点对合τ的3流形。的最大价值ñ为其博苏克-乌拉姆定理成立被称为ℤ ₂ -index并在我们的情况下，它取值1，2或3，根据同调的操作，我们充分讨论这个指标应用在特色班X ∈ ^ h ¹（ñ，ℤ ₂），其中ñ =中号/ τ 是轨道空间。在定向情况下，我们从眼眶空间的手术表现的链接矩阵中获得索引的表达式。我们通过示例来说明我们的结果，包括一个不可定向的示例。