On the immersion classes of nearby Lagrangians

TL;DR

This paper demonstrates that the transfer map on Floer homotopy types for exact Lagrangian embeddings is an equivalence, offering a new obstruction to representing Lagrangian immersions as embeddings, with concrete sphere examples.

Contribution

It introduces a novel Floer homotopy type transfer map equivalence as an obstruction, sensitive to information beyond Floer cochains.

Findings

Transfer map on Floer homotopy types is an equivalence for exact Lagrangians.

Obstruction distinguishes Lagrangian immersions from embeddings beyond Floer cochains.

Concrete computation provided for spheres.

Abstract

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.