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

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.