[Paper Review] Topologically Distinct Lagrangian and Symplectic Fillings

Chang Cao, Nathaniel Gallup+2 more|arXiv (Cornell University)|Jul 30, 2013

Geometric and Algebraic Topology45 references3 citations

TL;DR

This paper constructs infinitely many Legendrian links in $S^3$ with arbitrarily many topologically distinct exact Lagrangian fillings, demonstrating that the topology of such fillings is not determined solely by the Thurston-Bennequin invariant. Using a combination of Lagrangian handle attachment, spinning constructions, and branched covers, the authors establish new examples of symplectic fillings and prove that Legendrian submanifolds in higher dimensions can have arbitrarily large Chekanov or generating family homology invariants.

ABSTRACT

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-manifolds with topologically distinct symplectic fillings, and is generalized to higher dimensions.

Motivation & Objective

• To show that the topology of Lagrangian fillings of Legendrian links is not uniquely determined by the Thurston-Bennequin invariant, contradicting a natural extension of Chantraine’s theorem.
• To resolve Boileau and Fourrier’s question about the existence of links in $S^3$ that bound non-homeomorphic algebraic curves in $\mathbb{C}^2$.
• To construct contact 3-manifolds with strong symplectic fillings that are topologically distinct despite having the same Euler characteristic.
• To extend the construction to higher-dimensional Legendrian submanifolds with arbitrarily large Chekanov and generating family homology invariants.

Proposed method

• Construct a Legendrian link $\Lambda_N$ in $S^3$ with $N$ non-homeomorphic exact Lagrangian fillings via iterative handle attachment and isotopy in the symplectization.
• Use the Künneth theorem and homology computations to show that the first homology ranks of the fillings differ, proving topological distinctness.
• Apply a double branched cover construction over the symplectic fillings to produce strong symplectic fillings of transverse links with distinct $H_3$-rank.
• Employ a spinning construction to elevate 2-dimensional fillings to higher-dimensional Legendrian submanifolds, preserving topological distinctness of fillings.
• Utilize compatibility between generating families and Lagrangian fillings to show that distinct fillings induce distinct generating family homology invariants.
• Leverage results from [5], [36], and [39] to relate relative cohomology of fillings to invariants in Legendrian contact homology and generating family homology.

Experimental results

Research questions

• RQ1 Is the topology of a Lagrangian filling of an oriented Legendrian link in $S^3$ completely determined by its Thurston-Bennequin invariant?
• RQ2 Do there exist links in $S^3$ that bound non-homeomorphic complex algebraic curves in $\mathbb{C}^2$ intersecting $B^4$ transversally?
• RQ3 Can contact 3-manifolds admit strong symplectic fillings that are topologically distinct despite having the same Euler characteristic?
• RQ4 Can Legendrian submanifolds in higher dimensions have arbitrarily large Chekanov or generating family homology invariants?

Key findings

• For every integer $N>1$, there exists a Legendrian link $\Lambda \subset (S^3, \xi_0)$ with $p(N)$ non-homeomorphic exact orientable Lagrangian fillings.
• The fillings constructed have distinct first Betti numbers: $\operatorname{rank}H_1(L_k) = 2k$, proving topological distinctness.
• The construction yields links in $S^3$ that bound $p(N)$ non-singular complex algebraic curves in $\mathbb{C}^2$ intersecting $B^4$ in pairwise non-homeomorphic pieces.
• Contact 3-manifolds exist with $N$ strong symplectic fillings having the same Euler characteristic but $\operatorname{rank}H_3(X_k, Y) = k-1$, showing they are not related by symplectic blow-ups.
• For every $n>1$ and $N>1$, there exists a connected Legendrian submanifold in $(S^{2n+1}, \xi_0)$ with $N$ non-homeomorphic connected exact Lagrangian fillings.
• The Chekanov number and GF number of Legendrian submanifolds in $\mathbb{R}^{2n+1}$ can be made arbitrarily large, with examples constructed via compatible fillings and generating families.