Monotone Lagrangians in $\mathbb{CP}^n$ of minimal Maslov number $n+1$

TL;DR

This paper proves that monotone Lagrangians in complex projective space with minimal Maslov number n+1 are topologically equivalent to real projective space, using Floer homology and quantum cohomology techniques.

Contribution

It establishes a topological classification of certain monotone Lagrangians in CP^n, showing they are homeomorphic to a double quotient of a sphere and homotopy equivalent to RP^n.

Findings

Monotone Lagrangians with minimal Maslov number n+1 are homeomorphic to a double quotient of a sphere.

Such Lagrangians are homotopy equivalent to real projective space RP^n.

The proof utilizes Zapolsky's pearl complex and quantum cohomology actions.

Abstract

We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathbb{RP}^n$. To prove this we use Zapolsky's canonical pearl complex for $L$ with coefficients in $\mathbb{Z}$, and various twisted versions thereof, where the twisting is determined by connected covers of $L$. The main tool is the action of the quantum cohomology of $\mathbb{CP}^n$ on the resulting Floer homologies.