Lisa Piccirillo - Shake genus and slice genus

June 21, 2018 - This talk was part of the 2018 RTG mini-conference Low-dimensional topology and its interactions with symplectic geometry

An important difference between high dimensional smooth manifolds and smooth 4-manifolds is 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 the Kirby 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.

Видео Lisa Piccirillo - Shake genus and slice genus канала princetonmathematics