Nonorientable surfaces in homology cobordisms

Abstract

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 Ozsváth–Szabó d –invariants or Atiyah–Singer ρ– invariants 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.