Prime 3-manifold

Definition

A compact connected orientable manifold $M$ of dimension 3 is termed a prime 3-manifold if the only possible way of expressing $M$ as a connected sum of manifolds, both of dimension 3, is as $M\#S^{3}$. In other words, one of the summands of the connected sum must be homeomorphic to $M$ and the other summand must be homeomorphic to the 3-sphere $S^{3}$.