Cobordisms of sutured manifolds and the functoriality of link Floer homology

TL;DR

This paper proves that cobordisms of links induce well-defined maps on link Floer homology, establishing a functorial framework that generalizes previous constructions and confirms link Floer homology as a categorification of the multi-variable Alexander polynomial.

Contribution

It introduces a natural notion of cobordism between sutured manifolds, showing it induces maps on sutured Floer homology, thus establishing a TQFT structure and extending functoriality to link Floer homology.

Findings

Cobordisms induce maps on sutured Floer homology.

Sutured Floer homology forms a TQFT with cobordism maps.

Link Floer homology categorifies the multi-variable Alexander polynomial.

Abstract

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Mati\'c. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element.