Four-dimensional analogues of Dehn's lemma: FOUR-DIMENSIONAL ANALOGUES OF DEHN'S LEMMA
Four-dimensional analogues of Dehn's lemma: FOUR-DIMENSIONAL ANALOGUES OF DEHN'S LEMMA
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德恩引理的四维类比： 德恩引理的四维类比
DOI：
10.1112/jlms.12062
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Ruberman, Daniel
中科院分区：
文献类型：
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作者：
Ray, Arunima;Ruberman, Daniel
我们研究了经典三维Dehn引理的某些四维类似物，给出了在光滑和拓扑范畴中这些类似物成立或不成立的例子。特别地，我们证明了，一个本质的2-spere的边界的单连通4-manifold，使零同伦不需要扩展到嵌入的球。然而，如果具有边界为同调球面的交换基本群，则有一个拓扑嵌入球的边界为。此外，我们还给出了这样的例子，其中这种球不会约束任何平滑嵌入的球。类似地，我们构造了不可压缩的环面，其中是一个可收缩的4-流形，使得它可以扩展到一个实心环面的映射，但不能扩展到一个实心环面的任何嵌入。此外，我们在可收缩四维流形的边界上构造了一个不可压缩环面，使得它扩展为一个实心环面的拓扑嵌入，但不是光滑嵌入。作为我们的结果的应用环面，我们解决了一个问题所提出的贡普夫关于延长某些家庭的同胚的3-流形，他最近使用这样的家庭建设无限阶软木塞。
We investigate certain 4‐dimensional analogues of the classical 3‐dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential 2‐spherein the boundary of a simply connected 4‐manifoldsuch thatis null‐homotopic inneed not extend to an embedding of a ball in. However, ifhas abelian fundamental group with boundary a homology sphere, thenbounds a topologically embedded ball in. Additionally, we give examples where such andoes not bound any smoothly embedded ball in. In a similar vein, we construct incompressible toriwhereis a contractible 4‐manifold such thatextends to a map of a solid torus in, but not to any embedding of a solid torus in. Moreover, we construct an incompressible torusin the boundary of a contractible 4‐manifoldsuch thatextends to a topological embedding of a solid torus inbut no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of 3‐manifolds; he has recently used such families to construct infinite order corks.