[Paper Review] Cobordisms of sutured manifolds

András Juhász|arXiv (Cornell University)|Oct 22, 2009

Geometric and Algebraic Topology18 references16 citations

TL;DR

This paper introduces a new notion of cobordism between sutured manifolds and proves that such cobordisms induce maps on sutured Floer homology, unifying the closed 3-manifold cobordism map and the contact gluing map. The construction establishes sutured Floer homology as a TQFT-like theory and proves that 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 Matic. 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.

Motivation & Objective

• To resolve the open problem of whether cobordisms of links induce homomorphisms on link Floer homology.
• To define a natural notion of cobordism between sutured manifolds that generalizes both 3-manifold cobordisms and contact gluing.
• To show that sutured Floer homology with these cobordism maps satisfies an Atiyah-style TQFT axiomatic framework.
• To demonstrate that the complementary cobordism of a decorated link cobordism induces the desired map on link Floer homology.
• To provide an alternative definition of the contact gluing map using only the contact element and handle maps.

Proposed method

• Introduce a new definition of cobordism between sutured manifolds that respects the sutured structure and boundary conditions.
• Construct a functorial map on sutured Floer homology associated to each such cobordism, using holomorphic curve techniques from Heegaard Floer theory.
• Show that the cobordism map restricts to the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory.
• Prove that the cobordism map also recovers the contact gluing map of Honda, Kazez, and Matic via the contact element and handle attachment maps.
• Establish functoriality and composition laws to verify the TQFT-like structure of sutured Floer homology under these cobordisms.
• Use Weinstein cobordisms to show that the contact element is preserved under sutured manifold cobordisms of Weinstein type.

Experimental results

Research questions

• RQ1 Can a well-defined cobordism relation between sutured manifolds be defined such that it induces a map on sutured Floer homology?
• RQ2 Does this cobordism map generalize both the closed 3-manifold cobordism map and the contact gluing map in Heegaard Floer theory?
• RQ3 Can sutured Floer homology with these cobordism maps be viewed as a TQFT in the sense of Atiyah?
• RQ4 Does the complementary cobordism of a decorated link cobordism induce a well-defined map on link Floer homology?
• RQ5 Can the contact gluing map be redefined using only the contact element and handle maps, without auxiliary data?

Key findings

• A natural notion of cobordism between sutured manifolds is defined, and each such cobordism induces a well-defined map on sutured Floer homology.
• The induced cobordism map generalizes both the hat version of the closed 3-manifold cobordism map and the contact gluing map of Honda, Kazez, and Matic.
• Sutured Floer homology with these cobordism maps satisfies the axioms of a TQFT in the sense of Atiyah, including functoriality and composition laws.
• The theory applied to the complementary cobordism of a decorated link cobordism yields the desired homomorphism on link Floer homology, confirming that link Floer homology categorifies the multi-variable Alexander polynomial.
• An alternative definition of the contact gluing map is provided, relying solely on the contact element and handle maps, without additional structures.
• A Weinstein sutured manifold cobordism preserves the contact element in sutured Floer homology, establishing a geometric consistency condition.