Free groups and handlebodies
Free groups and handlebodies
复制标题
自由组和手柄
DOI：
--
复制
发表时间：
1965
期刊：
影响因子：
--
通讯作者：
M. Curtis
中科院分区：
文献类型：
--
作者：
J. J. Andrews;M. Curtis
本文提出了一个关于自由群的猜想，并给出了该猜想成立时的一些拓扑结果。该猜想似乎是Nielsen定理的自然推广，其主要拓扑结果是组合5流形的可收缩二维子复形的正则邻域是5胞的定理。这与三维和四维庞加莱猜想有一些有趣的结果。对于这个猜想的弱化，以及拓扑问题与一个更一般问题的关系，裁判员评论如下：“本文的结果来自于一个稍弱的猜想：假设P是（x1, ** *, xn: r1, **, rn），并且是平凡群的表示；在本文中定义(i), (ii), (iii), （iv）类型的操作，作为在相关集合上应用本文中定义的操作的结果。类型(v)的操作将包括添加一个额外的生成器，例如y，以及额外的相对器y；类型（vi）的操作将是类型(v)的逆操作。较弱的猜想是，P可以通过(i)-(vi)类型的有限运算序列简化为空表示。“这个问题与以下问题有关：如果二维多面体P和Q具有相同的简单同伦类型，它们是否都嵌入到三维多面体Z中，使得Z几何收缩为P， Z几何收缩为Q?”如果你用“n”和“n+ 1”代替“2”和“3”，根据J. H. C. Whitehead在简单空间、核和m群中的说法，这个问题对于n大于2有肯定的答案。当P是一个可收缩的二维多面体，Q是一个点时，它等价于上述猜想。
In this note we state a conjecture about free groups and give some topological consequences which would follow if the conjecture is true. The conjecture seems to be a natural extension of a theorem of Nielsen, and the main topological consequence is the theorem that regular neighborhoods of contractible 2-dimensional subcomplexes of combinatorial 5-manifolds are 5-cells. This has some interesting consequences relative to the 3and 4-dimensional Poincare conjectures. The referee has remarked on a weakening of the conjecture and on the relation of the topological problem to a more general problem as follows: "The results of the paper follow from a somewhat weaker conjecture: Suppose P is (x1, * * *, xn: r1, **, rn), and is a presentation of the trivial group; define an operation of type (i), (ii), (iii), (iv) on this presentation to be the result of applying such an operation as defined in the paper on the set of relators. An operation of type (v) will consist of adding an additional generator, say y, and the additional relator y; an operation of type (vi) will be the inverse of an operation of type (v). The weaker conjecture would be that P can be reduced to the empty presentation by a finite sequence of operations of types (i)-(vi). "The problem is related to the question: If the 2-dimensional polyhedra P and Q have the same simple homotopy type, can they both be embedded in a 3-dimensional polyhedron Z such that Z geometrically contracts to P and Z geometrically contracts to Q? If you replace "2" and "3" by "n" and "n+ 1," this question has the affirmative answer for n greater than 2, according to J. H. C. Whitehead in Simplicial spaces, nuclei, and m-groups. It is equivalent to the above conjecture when P is a contractible 2-dimensional polyhedron and Q is a point."