On the topology of Lagrangian fillings of the standard Legendrian sphere

TL;DR

This paper investigates the topology of Lagrangian fillings of the standard Legendrian sphere, showing they are homology balls and, in certain cases, diffeomorphic to the n-ball, revealing new topological constraints.

Contribution

It establishes that all exact Maslov zero Lagrangian fillings are homology balls and that real fillings are diffeomorphic to the n-ball for dimensions n ≥ 6.

Findings

All exact Maslov zero Lagrangian fillings are homology balls.

Real Lagrangian fillings are diffeomorphic to the n-ball for n ≥ 6.

Provides topological classification constraints for Lagrangian fillings.

Abstract

In this paper we study the uniqueness of Lagrangian fillings of the standard Legendrian sphere $\mathcal{L}_0$ in the standard contact sphere $(S^{2n-1}, \xi_{\text st})$. We show that every exact Maslov zero Lagrangian filling $L$ of $\mathcal{L}_0$ in a Liouville filling of $(S^{2n-1}, \xi_{\text st})$ is a homology ball. If we restrict ourselves to real Lagrangian fillings, then $L$ is diffeomorphic to the $n$-ball for $n \geq 6$.