The connected sum of -manifolds and is formed by deleting the interiors of -balls in and attaching the resulting punctured manifolds to each other by a homeomorphism , so

is required to be interior to and bicollared in to ensure that the connected sum is a manifold.

Topologically, if and are pathwise-connected, then the connected sum is independent of the choice of locations on and where the connection is glued.

The illustrations above show the connected sums of two tori (top figure) and of two pairs of multi-handled tori.