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

Pascal Zschumme

Published 2019-11-05Version 1

We study the homology of Riemannian manifolds of finite volume that are covered by a product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of the real hyperbolic plane. Using a variation of a method developed by Avramidi and Nyguen-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 real homology group $H_r(M;\mathbb{R})$.