#### Volume 19, issue 1 (2015)

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Nonorientable surfaces in homology cobordisms

### Appendix: Ira M Gessel

Geometry & Topology 19 (2015) 439–494
##### 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 $\rho$–invariants of $M$. One consequence is that the minimal genus of a smoothly embedded surface in $L\left(2k,q\right)×I$ 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.

##### Keywords
nonorientable surfaces, $4$–manifold, Heegaard Floer homology, Dedekind sums
##### Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57R40, 57R58