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. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism. There are subtle problems in the smooth case: not every diffeomorphism between the boundaries of the spheres gives the same composite manifold, even if the orientations are chosen correctly. For example, Milnor showed that two 7-cells can be glued along their boundary so that the result is an exotic sphere homeomorphic but not diffeomorphic to a 7-sphere. However there is a canonical way to choose the gluing which gives a unique well defined connected sum. This uniqueness depends crucially on the disc theorem, which is not at all obvious.
The operation of connected sum is denoted by ; for example denotes the connected sum of and .
The operation of connected sum has the sphere as an identity; that is, is homeomorphic (or diffeomorphic) to .
The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number of tori and some number of real projective planes.
Read more about Connected Sum: Connected Sum Along A Submanifold, Connected Sum Along A Codimension-two Submanifold, Local Operation, Connected Sum of Knots
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