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.
Read more about this topic: Connected Sum
Famous quotes containing the words connected and/or sum:
“Nothing fortuitous happens in a child’s world. There are no accidents. Everything is connected with everything else and everything can be explained by everything else.... For a young child everything that happens is a necessity.”
—John Berger (b. 1926)
“but Overall is beyond me: is the sum of these events
I cannot draw, the ledger I cannot keep, the accounting
beyond the account:”
—Archie Randolph Ammons (b. 1926)