## Duke Mathematical Journal

### Homology of curves and surfaces in closed hyperbolic $3$-manifolds

#### Abstract

Among other things, we prove the following two topological statements about closed hyperbolic $3$-manifolds. First, every rational second homology class of a closed hyperbolic $3$-manifold has a positive integral multiple represented by an oriented connected closed $\pi_{1}$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $\pi_{1}$-injectively immersed oriented closed $1$-submanifold in a closed hyperbolic $3$-manifold has an equidegree finite cover which bounds an oriented connected compact $\pi_{1}$-injectively immersed quasi-Fuchsian subsurface. In, we exploit techniques developed by Kahn and Markovic but we only distill geometric and topological ingredients from those papers, so no hard analysis is involved in this article.

#### Article information

Source
Duke Math. J., Volume 164, Number 14 (2015), 2723-2808.

Dates
Revised: 7 December 2014
First available in Project Euclid: 26 October 2015

https://projecteuclid.org/euclid.dmj/1445865571

Digital Object Identifier
doi:10.1215/00127094-3167744

Mathematical Reviews number (MathSciNet)
MR3417184

Zentralblatt MATH identifier
1334.57033

#### Citation

Liu, Yi; Markovic, Vladimir. Homology of curves and surfaces in closed hyperbolic $3$ -manifolds. Duke Math. J. 164 (2015), no. 14, 2723--2808. doi:10.1215/00127094-3167744. https://projecteuclid.org/euclid.dmj/1445865571

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