On cobordism groups of Lagrangian immersions

TL;DR

This paper computes the cobordism group of Lagrangian immersions into symplectic manifolds using stable homotopy theory, extending previous results and providing explicit calculations for surfaces and partial results for monotone manifolds.

Contribution

It generalizes Eliashberg's result to non-exact symplectic manifolds by expressing the cobordism group in terms of a Thom spectrum's stable homotopy group.

Findings

Computed cobordism groups for closed surfaces

Provided partial descriptions for monotone manifolds

Extended the understanding of Lagrangian immersion cobordism groups

Abstract

We compute the cobordism group $\Omega^{\operatorname{lag}}(M)$ of Lagrangian immersions into a symplectic manifold $(M, \omega)$ in terms of a stable homotopy group of a Thom spectrum constructed from $M$. This generalizes a result of Eliashberg in the case of exact symplectic manifolds. As applications, we compute $\Omega^{\operatorname{lag}}(M)$ when $M$ is a closed surface and give a partial description of $\Omega^{\operatorname{lag}}(M)$ when $M$ is a monotone manifold.