摇属和切片属

IF 2 1区 数学
Lisa Piccirillo
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引用次数: 14

摘要

高维光滑流形与光滑4-流形的一个重要区别是,在4-流形中,不可能总是用光滑嵌入球来表示每一个中维同调类。即使在最简单的4-流形中也是如此:$X_0(K)$是通过将$0$框架的2-手柄沿着$S^3$中的结$K$附加到4-球上得到的。$K$的$0$-抖格记录了表示$X_0(K)$第二同调的一个发生器的所有光滑嵌入曲面中的最小格,并且被$K$的切片格清楚地限定在上面。我们证明了片格不是$X_0(K)$的不变量,从而给出了$0$-摇格严格小于片格的无穷多个结的例子。这解决了[Kir97]的问题1.41。作为推论,我们证明了Rasmussen的$s$不变量不是$0$迹不变量,并通过卫星运算给出了光滑协调群上的双射映射的例子,这些映射固定了恒等但不保留片属。这些推论解决了[4MKC16]中的一些问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Shake genus and slice genus
An important difference between high dimensional smooth manifolds and smooth 4-manifolds that in a 4-manifold it is not always possible to represent every middle dimensional homology class with a smoothly embedded sphere. This is true even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an $0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X_0(K)$, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41 of [Kir97]. As corollaries we show that Rasmussen's $s$ invariant is not a $0$-trace invariant and we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but do not preserve slice genus. These corollaries resolve some questions from [4MKC16].
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来源期刊
Geometry & Topology
Geometry & Topology 数学-数学
自引率
5.00%
发文量
34
期刊介绍: Geometry and Topology is a fully refereed journal covering all of geometry and topology, broadly understood. G&T is published in electronic and print formats by Mathematical Sciences Publishers. The purpose of Geometry & Topology is the advancement of mathematics. Editors evaluate submitted papers strictly on the basis of scientific merit, without regard to authors" nationality, country of residence, institutional affiliation, sex, ethnic origin, or political views.
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