Higher dimensional links in a simplicial complex embedded in a sphere
Higher dimensional links in a simplicial complex embedded in a sphere
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嵌入球体的单纯复形中的高维链接
DOI：
10.2140/pjm.2000.194.465
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发表时间：
2000
影响因子：
0.6
通讯作者：
Kouki Taniyama
中科院分区：
文献类型：
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作者：
Kouki Taniyama
在本文中，我们工作在分段线性范畴。Conway和Gordon在文献[1]中证明了6个顶点上的完全图在3-空间中的任何嵌入都包含一对非平凡连通的圈。我们建议读者参考[6]、[2]、[4]、[3]等相关著作。本文将Conway和Gordon的结果推广到高维情形。设σ ij是j维单形σj = ε v1，v2，. . .，vj+1，其中v1，v2，. . .，vj和vj+1是σj的0-单形。设Sk为k维单位球面。设X和Y是嵌入S2 n +1的不相交的n维球面。然后定义连接数k（X，Y）∈ Z直到符号，例如见[7]。则k（X，Y）的模2约简k2（X，Y）∈ Z/2 Z是有定义的.我们注意到k2（X，Y）= k2（Y，X）（mod 2）。设Ln是σ n2 n +3的所有不交子复形的无序对的集合，其中每个子复形同胚于一个n维球面。我们注意到，Ln的每个元素（J，K）可以写为：
Throughout this paper we work in the piecewise linear category. Conway and Gordon showed in [1] that any embedding of the complete graph over six vertices into the 3-space contains a pair of nontrivially linked circles. We refer the reader to [6], [2], [4], [3] etc. for related works. In this paper we generalize the result of Conway and Gordon to higher dimensions. Let σi j be the i-skeleton of a j-dimensional simplex σj = 〈v1, v2, . . . , vj+1〉 where v1, v2, . . . , vj and vj+1 are the 0-simplices of σj . Let Sk be the kdimensional unit sphere. Let X and Y be disjoint n-dimensional spheres embedded in S2n+1. Then the linking number k(X, Y ) ∈ Z is defined up to sign, see for example [7]. Then the modulo 2 reduction k2(X, Y ) ∈ Z/2Z of k(X, Y ) is well-defined. We note that k2(X, Y ) ≡ k2(Y, X) (mod 2). Let Ln be the set of all unordered pairs of disjoint subcomplices of σn 2n+3 each of which is homeomorphic to an n-dimensional sphere. We note that each element (J, K) of Ln can be written as