Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes

TL;DR

This paper constructs many independent homology classes in certain Riemannian manifolds covered by products of hyperbolic planes, revealing rich geometric and topological structures.

Contribution

It introduces a geometric method to produce numerous linearly independent homology classes in manifolds covered by hyperbolic plane products, extending previous techniques.

Findings

Existence of arbitrarily many independent homology classes

Construction of flat totally geodesic submanifolds

Application to manifolds covered by hyperbolic plane products

Abstract

We study the homology of Riemannian manifolds of finite volume that are covered by an $r$-fold product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of hyperbolic planes. Using a variation of a method developed by Avramidi and Nguyen-Phan, we show that any such manifold $M$ possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic $r$-dimensional submanifolds whose fundamental classes are linearly independent in the homology group $H_r(M;\mathbb{Z})$.