Connected Sum Along A Submanifold
Let and be two smooth, oriented manifolds of equal dimension and a smooth, closed, oriented manifold, embedded as a submanifold into both and . Suppose furthermore that there exists an isomorphism of normal bundles
that reverses the orientation on each fiber. Then induces an orientation-preserving diffeomorphism
where each normal bundle is diffeomorphically identified with a neighborhood of in, and the map
is the orientation-reversing diffeomorphic involution
on normal vectors. The connected sum of and along is then the space
obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted
Its diffeomorphism type depends on the choice of the two embeddings of and on the choice of .
Loosely speaking, each normal fiber of the submanifold contains a single point of, and the connected sum along is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along is often called the fiber sum.
The special case of a point recovers the connected sum of the preceding section.
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