Editing Connected sum (section)

== Connected sum along a codimension-two submanifold ==

Another important special case occurs when the dimension of <math>V</math> is two less than that of the <math>M_i</math>. Then the isomorphism <math>\psi</math> of normal bundles exists whenever their [[Euler class]]es are opposite:

:<math>e\left(N_{M_1} V\right) = -e\left(N_{M_2} V\right).</math>

Furthermore, in this case the [[structure group]] of the normal bundles is the [[circle group]] <math>SO(2)</math>; it follows that the choice of embeddings can be canonically identified with the [[group (mathematics)|group]] of [[homotopy]] classes of maps from <math>V</math> to the [[circle]], which in turn equals the first integral [[cohomology]] group <math>H^1(V)</math>. So the diffeomorphism type of the sum depends on the choice of <math>\psi</math> and a choice of element from <math>H^1(V)</math>.

A connected sum along a codimension-two <math>V</math> can also be carried out in the category of [[symplectic manifold]]s; this elaboration is called the [[symplectic sum]].