Apanasov, Boris,Ivansic, Dubravko.2013-08-162013-08-161998http://hdl.handle.net/11244/5715We 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 $M = S\sp4-\{\rm tori \cup Klein\ bottles\}.$ If M is a codimension-1 complement, we show that the universal cover of N is $\IR\sp{n+1}$ 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 ${\ge}3$ 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 $4\pi\sp2/3.$ This is "half" the set of possible values for volumes, which is the integral multiples of $4\pi\sp2/3$ due to the Gauss-Bonnet formula Vol$(M) = 4\pi\sp2/3\cdot\chi(M).$vi, 90 leaves :Four-manifolds (Topology)Mathematics.Geometry, Hyperbolic.Volume (Cubic content)Thesis