Connected Sum Along A Codimension-two Submanifold
Another important special case occurs when the dimension of is two less than that of the . Then the isomorphism of normal bundles exists whenever their Euler classes are opposite:
Furthermore, in this case the structure group of the normal bundles is the circle group ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from to the circle, which in turn equals the first integral cohomology group . So the diffeomorphism type of the sum depends on the choice of and a choice of element from .
A connected sum along a codimension-two can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum.
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