Nonorientable surfaces in homology cobordisms

Abstract

We investigate constraints on embeddings of a nonorientable surface in a $4$–manifold with the homology of $M\times I$, 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 $\rho$–invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L\left(2k,q\right)\times I$ is the same as the minimal genus of a surface in $L\left(2k,q\right)$. We also consider embeddings of nonorientable surfaces in closed $4$–manifolds.