On compact hyperbolic manifolds of Euler characteristic two

TL;DR

This paper proves the nonexistence of certain compact arithmetic hyperbolic manifolds with Euler characteristic two in dimensions greater than four, impacting the understanding of hyperbolic rational homology spheres.

Contribution

It establishes a nonexistence result for compact arithmetic hyperbolic manifolds with Euler characteristic two in dimensions greater than four.

Findings

No compact arithmetic hyperbolic n-manifold with Euler characteristic two for n>4

Implication for hyperbolic rational homology spheres in even dimensions

Clarifies limitations on the topology of hyperbolic manifolds

Abstract

We prove that for n>4 there is no compact arithmetic hyperbolic n-manifold whose Euler characteristic has absolute value equal to 2. In particular, this shows the nonexistence of arithmetically defined hyperbolic rational homology n-sphere with n even different than 4.