Non-orientable surfaces in homology cobordisms

We investigate constraints on embeddings of a non-orientable 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 Ozsv'ath–Sazb'o $d$-invariants or Atiyah–Singer $\rho$-invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L(2p,q)\times I$ is the same as the minimal genus of a surface in $L(2p,q)$. We also consider embeddings of non-orientable surfaces in closed $4$-manifolds.