Cobordisms of sutured manifolds and the functoriality of link Floer homology

Journal article

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.

Authors: Juhász, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 299
• Pages: 940–1038
• Publication date: 2016-01-01
• Acceptance date: 2016-06-08
• DOI: 10.1016/j.aim.2016.06.005
• EISSN: 1090-2082
• ISSN: 0001-8708
• Keywords: sutured manifold; Heegaard Floer homology; cobordism