Van Kampen’s embedding Obstruction is incomplete for $2$-Complexes in $\rz^{4}$
Van Kampen’s embedding Obstruction is incomplete for $2$-Complexes in $\rz^{4}$
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对于 $ z^{4}$ 中的 $2$-复合体，Van Kampen 的嵌入障碍是不完整的
DOI：
10.4310/mrl.1994.v1.n2.a4
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发表时间：
1994
影响因子：
1
通讯作者：
P. Teichner
中科院分区：
文献类型：
--
作者：
M. Freedman;Vyacheslav Krushkal;P. Teichner
1933 年，van Kampen [5] 预见到了正式的上同调理论，对障碍 o(K) ∈ H Z/2(K , Z) 进行了稍微粗略的描述，当且仅当 n 维单纯复形 K 允许分段线性嵌入到 R 中，n ≥ 3 时，该障碍才会消失。这里 K 是复数 K 的删除乘积。所讨论的上同调是 Z/2 等变上同调，其中Z/2 通过交换 K 的因子来作用于空间，并通过与 (−1) 相乘来作用于系数。 Shapiro [3] 和 Wu [6] 在 1950 年代澄清了许多细节。 1991 年，Sarkaria [2] 表明，对于 n = 1，这种阻碍也在该维度上提供了充分必要条件（因此与 Kuratowski 的子图条件相同）。 Sarkaria 最近询问第一作者，o(K) 的消失是否也意味着 n = 2 的可嵌入性。本文的目的是展示一个具有 14 个顶点、43 个 1 单元和 69 个 2 单元的单纯 2 复形 K，其中 o(K) 是微不足道的，但不允许嵌入到 R 中。如果考虑相对设置 (K, L) ⊂ (D, S) 存在类似的障碍和类似于 van Kampen 的高维定理。但设置 n = 2、K = D ⊔D ⊔D 和 L ⊂ S 博罗梅安环给出了一个基本的（众所周知的）示例，其中虽然障碍消失了，但没有相对嵌入。我们的任务是“去相对化”这个简单的例子。在第 2 节中，我们回顾了 van Kampen 的阻碍（推广到相对环境），并给出了一个现代证明，证明它的消失意味着 P.L. 的存在。嵌入到 R 中，n ≥ 3。第 3 节描述了示例，证明了不存在 P.L。嵌入和障碍物消失。在第 4 节中，我们证明 K 没有嵌入到 R 中，即使在拓扑上也是如此。
In 1933, anticipating formal cohomology theory, van Kampen [5] gave a slightly rough description of an obstruction o(K) ∈ H Z/2(K , Z) which vanishes if and only if an n-dimensional simplicial complex K admits a piecewise-linear embedding into R, n ≥ 3. Here K is the deleted product of a complex K. The cohomology in question is the Z/2-equivariant cohomology where Z/2 acts on the space by exchanging the factors of K and acts on the coefficients by multiplication with (−1). Many details were clarified by Shapiro [3] and Wu [6] in the 1950’s. In 1991 Sarkaria [2] showed that for n = 1 this obstruction provides a necessary and sufficient condition in that dimension as well (and is thus identical to Kuratowski’s subgraph condition). Sarkaria recently asked the first named author if it were possible that the vanishing of o(K) might also imply embeddability for n = 2. It is the purpose of this paper to exhibit a simplicial 2-complex K with 14 vertices, 43 1-cells and 69 2-cells for which o(K) is trivial but which does not admit an embedding into R. If one considers relative settings (K, L) ⊂ (D, S) there is an analogous obstruction and a high dimensional theorem analogous to van Kampen’s. But setting n = 2, K = D ⊔D ⊔D and L ⊂ S the Borromean rings gives an elementary (and well known) example where although the obstruction vanishes there is no relative embedding. Our task was to ”unrelativize” this simple example. In section 2 we recall van Kampen’s obstruction (generalized to a relative setting) and give a modern proof that its vanishing implies the existence of a P.L. embedding into R, n ≥ 3. Section 3 describes the example, proves the absence of a P.L. embedding and the vanishing of the obstruction. In section 4 we show that K does not embed, even topologically, into R.