Homotopy rigidity of nearby Lagrangian cocores

TL;DR

The paper establishes a homotopy rigidity phenomenon for nearby Lagrangian cocores in Weinstein sectors: under $n\ge 2k+2$ and subcritical-index bounds, a natural composed map from $L/\partial_\infty L$ into the core's $n-k$-skeleton is null-homotopic, forcing strong topological constraints. The authors develop a spectral wrapped Donaldson--Fukaya category with coefficients in the Thom spectrum $MO\langle k+1\rangle$, showing that nearby cocores correspond to isomorphic objects after a suitable handle attachment and Maslov-data choices; a fold map to the core, together with obstruction-theoretic and bordism arguments, yields the null-homotopy. As a consequence, in many subcritical settings nearby cocores are smoothly unknotted rel boundary, and in turn correspond to unique smooth isotopy classes of certain exact Lagrangian fillings of Legendrian unknots in the subcritical boundary. The results combine higher-connectivity bordism theory with modern spectral Floer techniques to yield new topological rigidity statements for Lagrangian embeddings in Weinstein sectors, with broad implications for unknotting and filling problems in symplectic topology.

Abstract

An exact Lagrangian submanifold $L \subset X^{2n}$ in a Weinstein sector is called a nearby Lagrangian cocore if it avoids all Lagrangian cocores and is equal to a shifted Lagrangian cocore at infinity. Let $k$ be the dimension of the core of the subcritical part of $X$. For $n \geq 2k+2$ we prove that that the inclusion of $L$ followed by the retract to the Lagrangian core of $X$ and the quotient by the $(n-k-1)$-skeleton of the core, is null-homotopic. As a consequence, in many examples, a nearby Lagrangian cocore is smoothly isotopic (rel boundary) to a Lagrangian cocore in the complement of the missed Lagrangian cocores. The proof uses the spectral wrapped Donaldson-Fukaya category with coefficients in the ring spectrum representing the bordism group of higher connective covers of the orthogonal group.

Figures (2)

• Figure 1: The projection of the Weinstein handle $H^\delta_k$ in the $(x_i,y_i)$-plane together with its Liouville vector field.
• Figure 2: The projection of the Weinstein half-handle $\varPi^\delta_k$ in the $(x_i,y_n)$-plane together with its Liouville vector field.

Theorems & Definitions (144)

• Definition 1.1: Nearby Lagrangian cocore
• Theorem 1.3
• Remark 1.4
• Theorem 1.6: \ref{['thm:main']}
• Remark 1.7
• Theorem 1.8: \ref{['thm:main_conc']}
• Remark 1.9
• Lemma 1.10: \ref{['lem:closures_quiv']}
• Remark 1.11
• Lemma 1.12: \ref{['thm:oc_fund_classes']}