Nonorientable surfaces in homology cobordisms

Levine, Adam¹ ; Ruberman, Daniel² ; Strle, Sašo³

¹ Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08540, USA
² Department of Mathematics, Brandeis University, MS 050, Waltham, MA 02454, USA
³ Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia

Geometry & topology, Tome 19 (2015) no. 1, pp. 439-494.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.439

Classification : 57M27, 57R40, 57R58
Keywords: nonorientable surfaces, $4$–manifold, Heegaard Floer homology, Dedekind sums

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