On cobordism groups of Lagrangian immersions

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.