from them. For example, using Poincare duality and the Hurewicz theorem, we have the following homology groups: where the last two groups are isomorphic to the group homology and cohomology of , respectively; that is,From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov tower there is a canonical mapIf we take the pushforward of the fundamental class into we get an element . It turns out the group together with the group homology class gives a complete algebraic description of the homotopy type of . Connected sums One important topological operation is the connected sum of two 3-manifolds . In fact, from general theorems in topology, we find for a three manifold with a connected sum decomposition the invariants above for can be computed from the . In particularMoreover, a 3-manifold which cannot be described as a connected sum of two 3-manifolds is called prime. Second homotopy groups For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it turns out there is a nice description of the second fundamental group as a -module. For the special case of having each is infinite but not cyclic, if we take based embeddings of a 2-sphere where then the second fundamental group has the presentationgiving a straightforward computation of this group. Important examples of 3-manifolds Euclidean 3-space Euclidean 3-space is the most important example of a 3-manifold, as all others are defined in relation to it. This is just the standard 3-dimensional vector space over the real numbers. 3-sphere A 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions. Many examples of 3-manifolds can be constructed by taking quotients of the 3-sphere by a finite group acting freely on via a map , so . Real projective 3-space Real projective 3-space, or RP3, is the topological space of lines passing through the origin 0 in R4. It is a compact, smooth manifold of dimension 3, and is a special case Gr(1, R4) of a Grassmannian space. RP3 is (diffeomorphic to) SO(3), hence admits a group