Paper detail

Homology of curves and surfaces in closed hyperbolic 3-manifolds

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $π_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $π_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques developed earlier by Kahn and Markovic about good pants constructions, but we only distill geometric and topological ingredients from their papers so no hard analysis is involved in this paper.