Exotic 4-manifolds and Khovanov-Lipshitz-Sarkar homotopy type

TL;DR

This paper introduces a new stable homotopy invariant for smooth 4-manifolds with boundary links, extending and strengthening previous link homology invariants by assigning a CW complex-based homotopy type.

Contribution

It defines the KLS lasagna homotopy type as a new diffeomorphism invariant that generalizes and enhances existing link homology invariants for 4-manifolds.

Findings

The invariant assigns a stable homotopy type to smooth structures.

It is not weaker than existing KR lasagna modules.

It is stronger than Khovanov-Rozansky skein modules for non-empty links.

Abstract

We introduce a new diffeomorphism invariant of smooth compact oriented 4-manifolds $X$ with a framed oriented 1-link $L$ in the boundary, where $L$ may be the empty set, and call it {\it Khovanov-Lipshitz-Sarkar skein lasagna homotopy type} or {\it KLS lasagna homotopy type} $\mathcal E^{{LS}}_0(X,L)$. Our invariant assigns to a smooth structure a stable homotopy type of a CW complex. Our new invariant is not weaker than KR lasagna module, which were defined by Morrison, Walker and Wedrich. For a pair $(X,L)$ such that $L\neq\emptyset$, our new invariant, KLS lasagna homotopy type, is stronger than the Khovanov-Rozansky $\mathfrak{gl}_2$ skein lasagna modules or KR lasagna modules.