Geometry & Topology

Shake genus and slice genus

Lisa Piccirillo

Abstract

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: $X0(K)$ obtained by attaching an $0$–framed $2$–handle to the $4$–ball along a knot $K$ in $S3$. The $0$–shake genus of $K$ records the minimal genus among all smooth embedded surfaces representing a generator of the second homology of $X0(K)$ and is clearly bounded above by the slice genus of $K$. We prove that slice genus is not an invariant of $X0(K)$, and thereby provide infinitely many examples of knots with $0$–shake genus strictly less than slice genus. This resolves Problem 1.41 of Kirby’s 1997 problem list. 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 a conference at the Max Planck Institute, Bonn (2016).

Article information

Source
Geom. Topol., Volume 23, Number 5 (2019), 2665-2684.

Dates
Revised: 17 September 2018
Accepted: 1 January 2019
First available in Project Euclid: 22 October 2019

https://projecteuclid.org/euclid.gt/1571709633

Digital Object Identifier
doi:10.2140/gt.2019.23.2665

Mathematical Reviews number (MathSciNet)
MR4019900

Zentralblatt MATH identifier
07121758

Citation

Piccirillo, Lisa. Shake genus and slice genus. Geom. Topol. 23 (2019), no. 5, 2665--2684. doi:10.2140/gt.2019.23.2665. https://projecteuclid.org/euclid.gt/1571709633

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