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.
1 авг 2024
