A hyperbolic 4-manifold

Abstract

There is a regular 4 4 -dimensional polyhedron with 120 dodecahedra as 3 3 -dimensional faces. (Coxeter calls it the " 120 120 -cell".) The group of symmetries of this polyhedron is the Coxeter group with diagram: \[ [ u n k ] [unk] \] For each pair of opposite 3 3 -dimensional faces of this polyhedron there is a unique reflection in its symmetry group which interchanges them. The result of identifying opposite faces by these reflections is a hyperbolic manifold M 4 {M^4} .