Editing Surface (topology) (section)

==Connected sums==
The [[connected sum]] of two surfaces ''M'' and ''N'', denoted ''M'' # ''N'', is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The [[Euler characteristic]] <math>\chi</math> of {{nowrap|''M'' # ''N''}} is the sum of the Euler characteristics of the summands, minus two:

:<math>\chi(M \mathbin{\#} N) = \chi(M) + \chi(N) - 2.\,</math>

The sphere '''S''' is an [[identity element]] for the connected sum, meaning that {{nowrap|1='''S''' # ''M'' = ''M''}}. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from ''M'' upon gluing.

Connected summation with the torus '''T''' is also described as attaching a "handle" to the other summand ''M''. If ''M'' is orientable, then so is {{nowrap|'''T''' # ''M''}}. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.

The connected sum of two real projective planes, {{nowrap|'''P''' # '''P'''}}, is the [[Klein bottle]] '''K'''. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus; in a formula, {{nowrap|1='''P''' # '''K''' = '''P''' # '''T'''}}. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.