ON TOPOLOGICALLY UNKNOTTED SPHERES
ON TOPOLOGICALLY UNKNOTTED SPHERES
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论拓扑无结球体
DOI：
10.2307/1970127
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发表时间：
1963
影响因子：
4.9
通讯作者：
J. Stallings
中科院分区：
文献类型：
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作者：
J. Stallings
1.B.Mazur[10]和M.Brown[2]建立了关于n-球面中(n-1)-球面解结的一般定理。在最近的形式[4]中，它指出，如果X是n球面S_n的一个子集，其中X同胚于S_n-1，并且如果X满足局部光滑性的拓扑条件，则存在S_n的自同胚，它将X带到S_n的赤道(n-1)-球面上。人们想知道，对于k不是k=n-1的值，关于S_n的子集Y，是否有类似的说法成立，其中Y与S_k同胚。这是一个经典的事实，当k=n-2时，会出现有结的光滑球体。但除此之外，除了存在某种局部病理时，没有已知的S_k在S_n中真正有结球体的例子。在这里，我们将证明嵌入在SO中的局部光滑k-球确实是拓扑不打结的，如果k=5，则S4中的局部光滑1-球的情况未解决。将给出一个条件，该条件确保光滑地包含在锡中的(n-2)球面X是解结的。它是n>；5和SnX具有S1同伦型。S4中2-球体的情况尚未解决。对于S3中的1-球面，用三维拓扑学中的一个特殊结果[1]和Dehn引理[11]解决了这个问题。当S_n中包含的k-球面X是光滑的(可能在一点除外)，且k为5时，则证明了X是解结的。这种方法不能处理可能在两个点处不光滑的结的情况。这种奇怪的情况似乎与同构猜想中的困难有关，即锡本身的一次同胚应与单位映射同位。这些结果的证明相当复杂。研究光滑纽结补的同伦性质是十分必要的。对吞没定理[12]进行了巧妙的应用。通过应用关于开锥并的一个结果来完成证明[13]。与E.C.Zeeman[15]的分段线性解结定理的比较表明了这一点。塞曼定理和我们的定理有着相似的外观和重叠的区域。此外，这两种方法都被一些相似的分段线性方法证明。当应用于分段线性嵌入时，Zeeman的结果比我们的结果强得多
1. B. Mazur [10] and M. Brown [2] have contributed to a general theorem on the unknotting of (n - 1)-spheres in the n-sphere. In its most recent form [4], it states that if X is a subset of the n-sphere Sn, where X is homeomorphic to Sn-1, and if X satisfies a topological condition of local smoothness, then there is a homeomorphism of Sn onto itself which takes X onto an equatorial (n - 1)-sphere of Sn. One wonders whether an analogous statement is true about subsets Y of Sn, where Y is homeomorphic to Sk, for values of k other than k = n - 1. It is a classical fact that knotted smooth spheres occur when k = n - 2. But otherwise there are no known examples of truly knotted spheres Sk in Sn, except when there is some sort of local pathology. Here we shall show that a locally smooth k-sphere embedded in So is indeed topologically unknotted, provided k 5. The case of a locally smooth 1-sphere in S4 is unsolved. A condition will be given which insures that an (n - 2)-sphere X contained smoothly in Sn is unknotted. It is that n > 5 and Sn - X have the homotopy type of S1. The case of a 2-sphere in S4 is unsolved. As for 1-spheres in S3, the problem has been settled by a special result in 3-dimensional topology [1] and by Dehn's lemma [11]. When the k-sphere X contained in Sn is smooth except possibly at one point, and k 5, then it is shown that X is unknotted. The case of knots which may fail to be smooth at two points cannot be handled by this method. This odd state of affairs appears related to the difficulties in the isotopy conjecture, that a homeomorphism of degree one of Sn on itself should be isotopic to the identity map. The proof of these results is rather complicated. It is necessary to study in detail homotopy properties of the complement of a smooth knot. A delicate application of the engulfing theorem [12] is made. The proof is completed by applying a result about the union of open cones [13]. Comparion with the piecewise-linear unknotting theorem of E. C. Zeeman [15] shows this. Zeeman's theorem, and ours, have an analogous appearance and a region of overlap. Furthermore, both are proved by somewhat similar piecewise-linear methods. Zeeman's result is much stronger than ours when applied to piecewise-linear embeddings of