  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.

Geometric Topology