On Two‐Dimensional Aspherical Complexes
On Two‐Dimensional Aspherical Complexes
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关于二维非球面配合物
DOI：
10.1112/plms/s3-4.1.375
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发表时间：
1954
影响因子：
1.8
通讯作者：
W. Cockcroft
中科院分区：
文献类型：
--
作者：
W. Cockcroft
设L是连通的二维CW复形，f设K表示复形LU {e？}，其中，{e 1}，i= 1，2，.，是一组2-胞室，以通常的方式邻接于L的1-截面。JI在这篇笔记中将关注的问题是：如果L是非球面的，那么K是否可以是非球面的？由于证明0-胞和1-胞与二维非非球面复形的适当部分的连接不能使其成为非球面是一件简单的事情，这个问题的解决方案将回答这个问题：非球面二维复形的子复形本身是非球面的吗？问题的部分解由下面两个定理给出，这两个定理将在下面证明。定理1.若L是有限连通非球面二维复形，且TTX {L}是阿贝尔群，有限群，自由群，则复形LU {e？}也是非球面的，其中{e？}，t= 1，2，...，是一组邻接于L的l-截面的2-胞室。
LET L be a connected two-dimensional CW complex, f Let K denote the complex L U {e?}, where {e\}, i= 1, 2,..., is a set of 2-cells adjoined to the 1-section of L in the usual way. JI shall be concerned in this note with the problem: if L is non-aspherical,^ can K be aspherical? Since it is a simple matter to prove that the adjunction of 0-cells and 1-cells to the appropriate sections of a two-dimensional non-aspherical complex cannot make it aspherical, a solution of this problem would answer the question: is a subcomplex of an aspherical two-dimensional complex itself aspherical? A partial solution of the problem is given by the following two theorems which will be proved below.THEOREM 1. If L is a finite connected non-aspherical two-dimensional complex, and if TTX {L) is either (i) an Abelian group, or (ii) a finite group, or (iii) a free group, then the complex L U {e?} is also non-aspherical, where {e?}, t= 1, 2,..., is a set of 2-cells adjoined to the l-section of L.