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Geometry and Topology Seminar

Friday, March 16, 2018
3:00pm to 5:00pm
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Building 15, Room 104
Shake genus and slice genus
Lisa Piccirillo, Department of Mathematics, University of Texas at Austin,

An important difference between high dimensional smooth manifolds and smooth four-manifolds is the ability to represent any middle dimensional homology class with a smoothly embedded sphere. For four-manifolds this is not always possible even among the simplest cases: four-manifolds $X_0(K)$, called $0$-traces, 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 of any smooth embedded generator of the second homology of $X_0(K)$ and is clearly bounded above by the slice genus of $K$. It is conjectured that the $0$-shake genus can be strictly less than the slice genus. We prove that slice genus is not a $0$-trace invariant, and thereby provide infinitely many examples of knots with $0$-shake genus strictly less than slice genus. This resolves problem 1.41 from the Kirby list. As a corollary we give examples, via the satellite operation, of bijective maps on the smooth concordance group which fix the identity but don't preserve slice genus.

For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at [email protected].