Braid loops with infinite monodromy on the Legendrian contact DGA

Abstract

We present the first examples of elements in the fundamental group of the space of Legendrian links in false(S3,ξstfalse)$(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer‐theoretic techniques. These new families include the first‐known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer‐theoretic proof that Legendrian false(n,mfalse)$(n,m)$ torus links have infinitely many Lagrangian fillings if n⩾3,m⩾6$n\geqslant 3,m\geqslant 6$ or false(n,mfalse)=false(4,4false),false(4,5false)$(n,m)=(4,4),(4,5)$. In addition, for any given higher genus, we construct a Weinstein 4‐manifold homotopic to the 2‐sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.

On topologically distinct infinite families of exact Lagrangian fillings

“…exact Lagrangian cobordisms with empty negative ends) for some given Legendrian submanifolds has become one of the central questions in contact and symplectic topology. First it has been positively answered by Casals-Gao [5], and then later it has been investigated using different methods in the works of An-Bae-Lee [1,2], Casals-Zaslow [7], Gao-Shen-Weng [16,17], Casals-Ng [6], Capovilla-Searle [4] and the author [19].…”

“…Remark 3.2. Legendrians in I n and the corresponding fillings have Maslov numbers 0, this follows from the discussion in [6,19]. Now we take Λ T ∈ T n and Λ I ∈ I n .…”

“…Then we consider the class of Legendrian links H with an infinite number of diffeomorphic, but Hamiltonian non-isotopic exact Lagrangian fillings constructed by Casals-Ng in [6]. Note that in this class there are Legendrian knots.…”

“…with Maslov numbers 0. From the discussion in [6] we know that Λ I admits an infinite number of diffeomorphic, but Hamiltonian non-isotopic exact Lagrangian fillings that we denote by {L I m | m ≥ 0}. Without loss of generality we assume that L T 1 , .…”

“…Remark 3.3. Note that from the construction of L I m from [6,19] we know that L I m is given by the concatenation of one given exact Lagrangian filling L, concatenated m times exact Lagrangian concordance C and another fixed exact Lagrangian cobordism L which does not depend on m, i.e.…”