Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes
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Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes Pascal Zschumme1 Received: 15 November 2019 / Accepted: 16 October 2020 © The Author(s) 2020
Abstract We study the homology of Riemannian manifolds of finite volume that are covered by an r -fold product (H2 )r = H2 × · · · × H2 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 Hr (M; Z). Keywords Homology · Geometric cycles · Locally symmetric spaces · Arithmetic groups · Quaternion algebras · Hyperbolic plane Mathematics Subject Classification 57T99 · 11F75 · 22E40 · 53C35 · 11R52
1 Introduction Let M be a Riemannian manifold of finite volume that is covered by (H2 )r = H2 × · · · × H2 . If r = 1, then M is a hyperbolic surface and its homology is well understood. Otherwise, M can be a complicated object. For √example, let d > 0 be a square-free integer and consider the real quadratic field F = Q( d) with its two distinct embeddings σ1 , σ2 : F → R. Let O F be the ring of integers of F. Then the group SL2 (O F ) acts properly discontinuously on the product H2 × H2 by γ · (z 1 , z 2 ) := (σ1 (γ ) · z 1 , σ2 (γ ) · z 2 ), where σi (γ ) · z i is the action of SL2 (R) on H2 by fractional linear transformations. For any torsion-free subgroup of finite index Γ ⊂ SL2 (O F ), the quotient Γ \(H2 × H2 ) is a Riemannian manifold of finite volume that is covered by H2 × H2 . It is called a Hilbert modular surface and is an irreducible locally symmetric space of higher rank.
The author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—281869850 (RTG 2229).
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Pascal Zschumme [email protected] Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
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Geometriae Dedicata
The homology of such a locally symmetric space is in general hard to compute, and even if one can do so, the geometric meaning of the homology classes is often lost during the computation. We choose a more geometric approach going back to Millson [13], in which one studies homology classes that are the fundamental classes of totally geodesic submanifolds. Promising candidates for such submanifolds are the compact flat totally geodesic submanifolds of dimension equal to the rank of the locally symmetric space. It is known that these submanifolds exist in any nonpositively curved locally symmetric space of finite volume (see [17, Proposition 5.1]). Furthermore, Pettet and Souto proved in [17, Theorem 1.2] that they are non-peripheral, which means they cannot be homotoped outside of every compact subset of the locally symmetric space. This suggests that these submanifolds might contribute to the homology of the locally symmetric space. Avramidi and Nguyen-Phan [2] have investigated
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