Homotopy rigidity of nearby Lagrangian cocores

TL;DR

This paper proves a homotopy rigidity result for nearby Lagrangian cocores in Weinstein sectors, showing under certain conditions they are smoothly isotopic to standard cocores, using advanced Floer theory techniques.

Contribution

It establishes a homotopy rigidity theorem for nearby Lagrangian cocores in Weinstein sectors, a novel result in symplectic topology.

Findings

Nearby Lagrangian cocores are null-homotopic to standard cocores under certain conditions.

In many cases, they are smoothly isotopic to cocores in the complement of missed cocores.

The proof employs the spectral wrapped Donaldson-Fukaya category with advanced coefficients.

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.