Nonorientable surfaces in homology cobordisms

We investigate constraints on embeddings of a nonorientable surface in a 4-manifold with the homology of M I , where M is a rational homology 3-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsvth-Szab d -invariants or Atiyah-Singerinvariants of M . One consequence is that the minimal genus of a smoothly embedded surface in L.2k; q/ I is the same as the minimal genus of a surface in L.2k; q/. We also consider embeddings of nonorientable surfaces in closed 4-manifolds.