We also consider the question of when a finite volume hyperbolic (n+1)-manifold M may be embedded as a complement of a closed codimension-k submanifold A inside a closed (n+1)-manifold N. We show that if this is possible, then every flat manifold E corresponding to an end of M is either an S\sp0- or an S\sp1-bundle, giving that k can only be 1 or 2. We give a criterion in terms of the fundamental group for when a flat manifold is an S\sp1-bundle, which is used to classify flat 3-manifolds from this viewpoint and construct higher-dimensional examples of flat manifolds that are not S\sp1-bundles. Furthermore, we show that there are at most finitely many 4-manifolds M so that M=S\sp4−{tori∪Klein bottles}. If M is a codimension-1 complement, we show that the universal cover of N is \IR\spn+1 and, with an additional assumption, that there are only finitely many choices for N in dimensions n=2,3. We also give a criterion in terms of the fundamental group that detects when a flat manifold is an S\sp0-bundle over another flat manifold with the same holonomy group. It is used to classify flat 3-manifolds from this viewpoint.

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 4π\sp2/3. This is "half" the set of possible values for volumes, which is the integral multiples of 4π\sp2/3 due to the Gauss-Bonnet formula Vol(M)=4π\sp2/3⋅χ(M).