Adjunction inequality for spatially refined $s$-invariants

TL;DR

This paper establishes an adjunction inequality for a specific spatially refined version of Rasmussen's s-invariant in a particular 4-manifold, highlighting limitations of such inequalities for general refinements.

Contribution

It introduces an adjunction inequality for the s-invariant's spatial refinement in $k\overline{\mathbb{CP}^2}$, revealing its specific applicability.

Findings

Adjunction inequality holds for the $s$-version of the $Sq^1$-refinement in $k\overline{\mathbb{CP}^2}$.

The inequality does not extend to general spatial refinements of $s$-invariants.

The result clarifies the scope of spatial refinements in relation to adjunction inequalities.

Abstract

We note an adjunction inequality in $k\overline{\mathbb{CP}^2}$ for the $s$-version of the $Sq^1$-refinement of Rasmussen's $s$-invariant. This does not hold for general spatial refinements of $s$-invariants.