On cobordism groups of Lagrangian immersions

We compute the cobordism group ${\mathrm{\Omega }}^{\mathrm{lag}}(M)$ 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 ${\mathrm{\Omega }}^{\mathrm{lag}}(M)$ when $M$ is a closed surface and give a partial description of ${\mathrm{\Omega }}^{\mathrm{lag}}(M)$ when $M$ is a monotone manifold.