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A connected sum of two m-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation.

Connected sum with a torus is equivalent to adding a handle. Every surface is the connected sum of either the sphere, projective plane, or Klein bottle with zero or more tori. For example, the connected sum of two tori is the two-handled torus.

Examples: - the connected sum of two spheres is a sphere: (in fact the sphere is the identity element for this operation).

Two oriented knots (or links) can be summed by placing them side by side and joining them by straight bars so that orientation is preserved in the sum. The knot sum is also known as composition (Adams 1994) or connected sum (Rolfsen 1976, p. 40).

The fact that the connected sum is associative and commutative follows naturally from the fact that it is actually a pushout (if you're careful, it does make a pushout in the smooth category).

Definition 1.3 (Connected Sum). The connected sum of two surfaces is the space obtained when an open disk is removed from each surface, and the resulting boundaries are mapped together via a continuous function. We write A#B for the connected sum A and B.

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.