Every compact 3-manifold is the connected sum of a unique collection of prime 3-manifolds.
Connected Sum Decomposition
See also
Jaco-Shalen-Johannson Torus DecompositionExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Connected Sum Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConnectedSumDecomposition.html