Connected sum

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.

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots.

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 of ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ which gives a unique well defined connected sum. Choose embeddings ${\displaystyle i_{1}:D_{n}\rightarrow M_{1}}$ and ${\displaystyle i_{2}:D_{n}\rightarrow M_{2}}$ so that ${\displaystyle i_{1}}$ preserves orientation and ${\displaystyle i_{2}}$ reverses orientation. Now obtain ${\displaystyle M_{1}\#M_{2}}$ from the disjoint sum

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