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Concept
Connected sum
Summary
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 , 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 and which gives a unique well defined connected sum. Choose embeddings and so that preserves orientation and reverses orientation. Now obtain from the disjoint sum
by identifying with for each unit vector and each . Choose the orientation for which is compatible with and . The fact that this construction is well-defined depends crucially on the disc theorem, which is not at all obvious. For further details, see
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 .
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Ontological neighbourhood