Finding Irreducible Submanifolds of 3‐Manifolds

If M is a compact 3-manifold, then Kneser's theorem [2] tells us that M is a connected sum of prime manifolds. For many problems, this reduces the study of compact 3-manifolds to that of compact prime manifolds. When M is non-compact, one might hope that M is a finite or an infinite connected sum of prime manifolds. However, in [4], Scott gives an example of a 3-manifold M which is not a finite or an infinite connected sum of primes. Since his example is simply connected, we see that the problem is a geometric one rather than one of pathology of the fundamental group of the manifold. If M is a compact 3-manifold with indecomposable fundamental group, then M is the connected sum of a prime manifold JV together with a simply connected manifold. ./Vis irreducible except when TV is a 2-sphere bundle over the circle, in which case nx(M) = Z, and so in either case there is an irreducible manifold with the same fundamental group as M. In [4], Scott proves the following somewhat weaker result when the 3-manifold is not compact.