An invariant of link cobordisms from Khovanov homology

Abstract

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy invariant of smooth surfaces in 4D; a first step in proving the conjectured isomorphism of symplectic Khovanov homology and Khovanov homology. The maps themselves are defined using a generalisation of Seidel's relative invariant of exact Lefschetz fibrations to exact Morse-Bott-Lefschetz fibrations with non-compact singular loci.