## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 19, Number 1 (2015), 439-494.

Dates
Accepted: 25 May 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858685

Digital Object Identifier
doi:10.2140/gt.2015.19.439

Mathematical Reviews number (MathSciNet)
MR3318756

Zentralblatt MATH identifier
1311.57019

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R40: Embeddings 57R58: Floer homology

#### Citation

Levine, Adam; Ruberman, Daniel; Strle, Sašo. Nonorientable surfaces in homology cobordisms. Geom. Topol. 19 (2015), no. 1, 439--494. doi:10.2140/gt.2015.19.439. https://projecteuclid.org/euclid.gt/1510858685

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