Editing Connected sum (section)

== Connected sum at a point ==

A '''connected sum''' of two ''m''-dimensional [[manifold]]s is a manifold formed by deleting a [[ball (mathematics)|ball]] inside each manifold and [[adjunction space|gluing together]] the resulting boundary [[sphere]]s.

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 function|smooth]] [[category (mathematics)|category]], 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 <math>M_1</math> and <math>M_2</math> which gives a unique well-defined connected sum.<ref>Kervaire and Milnor, Groups of Homotopy Spheres I, Annals of Mathematics Vol 77 No 3 May 1963</ref> Choose embeddings <math>i_1 : D_n \rightarrow M_1</math> and <math>i_2 : D_n \rightarrow M_2</math> so that <math>i_1</math> preserves orientation and <math>i_2</math> reverses orientation. Now obtain <math>M_1 \mathbin{\#} M_2</math> from the disjoint sum
:<math>(M_1 - i_1(0)) \sqcup (M_2 - i_2(0))</math>
by identifying <math>i_1(tu)</math> with <math>i_2((1 - t)u)</math> for each [[unit vector]] <math>u \in S^{n-1}</math> and each <math>0 < t < 1</math>. Choose the orientation for <math>M_1 \mathbin{\#} M_2</math> which is compatible with <math>M_1</math> and <math>M_2</math>. The fact that this construction is well-defined depends crucially on the [[disc theorem]], which is not at all obvious. For further details, see Kosinski, ''Differential Manifolds''.<ref>Antoni A. Kosinski, ''Differential Manifolds'', Academic Press (1992), reprinted by Dover Publications (2007).</ref>

The operation of connected sum is denoted by <math>\#</math>.

The operation of connected sum has the sphere <math>S^m</math> as an [[identity element|identity]]; that is, <math>M \mathbin{\#} S^m</math> is homeomorphic (or diffeomorphic) to <math>M</math>.

The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number <math>g</math> of [[torus|tori]] and some number <math>k</math> of [[real projective plane]]s.