[Paper Review] In simply-connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Hiro TANAKA|ArXiv.org|Apr 24, 2014

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper proves that in simply connected cotangent bundles $T^*Q$, any compact exact Lagrangian cobordism between compact exact Lagrangians is an h-cobordism. The result follows from the Abouzaid-Kragh theorem, which implies that such Lagrangians are homotopy equivalent to $Q$, and via a double-cobordism argument using van Kampen and Mayer-Vietoris techniques, establishing that the cobordism induces isomorphisms on homotopy and homology groups.

ABSTRACT

We show that if Q is simply connected, every exact Lagrangian cobordism between compact, exact Lagrangians in the cotangent bundle of Q is an h-cobordism. The result is an exercise in basic algebraic topology once one invokes the Abouzaid-Kragh theorem.

Motivation & Objective

• To establish that exact Lagrangian cobordisms in $T^*Q$ are h-cobordisms when $Q$ is simply connected.
• To provide a new approach to the Nearby Lagrangian Conjecture by analyzing cobordism classes in $T^*Q$.
• To support the idea that Fukaya category equivalence classes may classify exact Lagrangians via cobordism invariants.
• To demonstrate that non-compact cobordisms are essential in the category of exact Lagrangians, as compact ones are homotopically trivial.
• To extend understanding of symplectic topology by linking cobordism theory with mirror symmetry and Floer-theoretic invariants.

Proposed method

• Use the Abouzaid-Kragh theorem to show that any compact exact Lagrangian in $T^*Q$ is homotopy equivalent to $Q$ when $Q$ is simply connected.
• Construct the double of a cobordism $Y_{01} \circ (Y_{01})^{\text{op}}$ to form a closed Lagrangian $\overline{N}$ in $T^*(Q \times S^1)$.
• Apply the Abouzaid-Kragh theorem to $\overline{N}$, showing the projection $\overline{N} \to Q \times S^1$ is a homotopy equivalence.
• Use a pushout diagram of groupoids and the van Kampen theorem to show $\pi_1(N) = 0$, implying trivial fundamental group.
• Apply the Whitehead and Hurewicz theorems to conclude that $N$ is a homotopy equivalence to $Q \times I$, hence an h-cobordism.
• Use Mayer-Vietoris and van Kampen arguments to show that the original cobordism $Y_{01}$ is simply connected and induces isomorphisms on homology, confirming it is an h-cobordism.

Experimental results

Research questions

• RQ1 Under what conditions is an exact Lagrangian cobordism in $T^*Q$ an h-cobordism?
• RQ2 Can the classification of exact Lagrangians via cobordism classes help resolve the Nearby Lagrangian Conjecture?
• RQ3 How do h-cobordism properties of exact Lagrangian cobordisms relate to their behavior in the Fukaya category?
• RQ4 What role does the fundamental group of $Q$ play in determining the topological type of exact Lagrangian cobordisms?
• RQ5 Can the double-cobordism construction be used to detect homotopy triviality in symplectic cobordisms?

Key findings

• Any compact exact Lagrangian cobordism between compact exact Lagrangians in $T^*Q$ is an h-cobordism when $Q$ is simply connected.
• The double of such a cobordism forms a compact exact Lagrangian in $T^*(Q \times S^1)$, which is homotopy equivalent to $Q \times S^1$ via the Abouzaid-Kragh theorem.
• The fundamental group of the double cobordism $N$ is trivial, as shown by a groupoid pushout argument using van Kampen’s theorem.
• The inclusion of the Lagrangian on either end of the double cobordism is a homotopy equivalence, confirming $N$ is an h-cobordism.
• The original cobordism $Y_{01}$ is shown to be an h-cobordism via Mayer-Vietoris and van Kampen, as it is simply connected and induces isomorphisms on homology.
• When $\dim Q \geq 5$, Corollary 2 implies that any two compact exact Lagrangians related by an exact cobordism are diffeomorphic, due to the h-cobordism theorem.