1997/05/08 by Dubravko Ivanšić, Ivanšić, Dubravko
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/9705217
openalex publication_date 1997/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the volume function for hyperbolic manifolds of dimension ≥ 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π2/3. This is ``half'' the set of possible values for volumes, which is the integral multiples of 4π2/3 due to the Gauss-Bonnet formula.