Knotted surfaces in 4‐manifolds and stabilizations
Knotted surfaces in 4‐manifolds and stabilizations
复制标题
4 歧管中的打结表面和稳定装置
DOI：
10.1112/jtopol/jtv039
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发表时间：
2015
影响因子：
1.1
通讯作者：
Nathan Sunukjian
中科院分区：
文献类型：
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作者：
R. Baykur;Nathan Sunukjian
在本文中，我们研究了4-流形中奇异纽结曲面的稳定等价性，这些曲面是拓扑同位素的，但不是光滑同位素的。我们证明了在同一同调类中的任何一对嵌入曲面在通过在周围4流形中的柄加法使它们稳定后成为光滑的同位素，而且，在许多有利的情况下，可以假设它们以标准的方式（局部和无结地）附着。特别是，任何具有环状基本群互补的奇异纽结表面对在相同数量的标准稳定化之后变得平滑同位素，类似于C.T.C.沃尔关于单连通4流形稳定等价的著名结果。此外，我们表明，所有建设的外来knottings的表面，我们知道，这显示了各种各样的技术和想法，产生表面，成为顺利同位素后，一个单一的稳定。
In this paper, we study stable equivalence of exotically knotted surfaces in 4‐manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4‐manifold, which can, moreover, be assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations, analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply connected 4‐manifolds. We, moreover, show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.