Exact Lagrangian immersions with one double point revisited

TL;DR

This paper revisits the classification of exact Lagrangian immersions with one double point, establishing conditions under which the manifold is homotopy equivalent to a sphere and bounding a parallelizable manifold, using solutions to perturbed Cauchy-Riemann equations.

Contribution

It provides a new, simplified proof of classification results for exact Lagrangian immersions with one double point, extending previous work to odd dimensions.

Findings

If the Maslov grading is not 1, the manifold is homotopy equivalent to a sphere.

Under additional conditions, the manifold bounds a parallelizable (n+1)-manifold.

The proof uses solutions to perturbed Cauchy-Riemann equations with boundary on the immersion.

Abstract

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling of K obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image f(K) of the immersion. This leads to a new and simplified proof of some of the main results of arXiv:1111.5932, which treated Lagrangian immersions in the case n=2k by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.