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

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} ${\mathcal{E}}_0^{LS}(X,L)$. 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 $L\ne \varnothing$, our new invariant, KLS lasagna homotopy type, is stronger than the Khovanov-Rozansky ${\mathfrak{gl}}_2$ skein lasagna modules or KR lasagna modules.

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