Duke Mathematical Journal

Homology of curves and surfaces in closed hyperbolic 3-manifolds

Yi Liu and Vladimir Markovic

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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 π1-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, π1-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented connected compact π1-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

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

Received: 12 November 2013
Revised: 7 December 2014
First available in Project Euclid: 26 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

homology quasi-Fuchsian subsurface


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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