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

Abstract

We study the homology of Riemannian manifolds of finite volume that are covered by an $r$-fold product $({\mathbb{H}}^2{)}^r={\mathbb{H}}^2\times \dots \times {\mathbb{H}}^2$ 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 $H_r(M;\mathbb{Z})$.

arXiv comment21 pages, 2 figures; updated to match the journal version

References in corpus (1)

• Flat cycles in the homology of Γ\SL(m,R)/SO(m)