An invariant of link cobordisms from symplectic 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.