Concave spin fillings of contact 3-manifolds

Every contact 3-manifold is known to admit a weak concave symplectic filling, and every spin 3-manifold is known to be the boundary of a 4-manifold such that the spin structure on the boundary is the restriction of a spin structure on the interior. We show that any spin contact 3-manifold admits a weak concave symplectic filling by a spin 4-manifold.;Given a contact 3-manifold, Eliashberg demonstrated that a filling constructed by Akbulut and Ozbagci admits a symplectic form so that the 4-manifold is a weak concave symplectic filling. The topological manifold is built as a Lefschetz fibration over D2 with an extra 2-handle. Our result follows by applying relations in the spin mapping class group of the fiber of the fibration.