Khovanov-Lipshitz-Sarkar homotopy type for links in thickened higher genus surfaces

TL;DR

This paper extends the Khovanov-Lipshitz-Sarkar stable homotopy type and Steenrod square to links in thickened surfaces of genus greater than one, providing a new, stronger invariant for such links in 3-manifolds.

Contribution

It introduces the first meaningful stable homotopy type for links in 3-manifolds beyond the 3-sphere, specifically for thickened surfaces of higher genus.

Findings

Defines the stable homotopy type and Steenrod square for links in higher genus surfaces.

Shows these invariants are stronger than homotopical Khovanov homology.

Highlights unique features in the torus case.

Abstract

We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov-Lipshitz-Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus$>1$ are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. It is the first meaningful Khovanov-Lipshitz-Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.