A refinement of Rasmussen's s-invariant

TL;DR

This paper refines Rasmussen's s-invariant using spectrum-level Khovanov homology, demonstrating that certain cohomology operations, like Steenrod squares, provide stronger bounds on slice genus than the original invariant.

Contribution

It introduces a spectrum-level refinement of Rasmussen's s-invariant that incorporates stable cohomology operations, enhancing the original bound's strength.

Findings

Cohomology operations commute with cobordism maps

Refinement provides stronger slice genus bounds for certain operations

Steenrod square Sq^2 yields a strictly stronger bound than s

Abstract

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq^2 our refinement is strictly stronger than s.