Editing Low-dimensional topology (section)

===Classification of surfaces===
The ''[[classification theorem]] of closed surfaces'' states that any [[connected (topology)|connected]] [[closed manifold|closed]] surface is homeomorphic to some member of one of these three families:

# the sphere;
# the [[connected sum]] of ''g'' [[torus|tori]], for <math>g \geq 1</math>;
# the connected sum of ''k'' [[real projective plane]]s, for <math>k \geq 1</math>.

The surfaces in the first two families are [[orientability|orientable]]. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number ''g'' of tori involved is called the ''genus'' of the surface. The sphere and the torus have [[Euler characteristic]]s 2 and 0, respectively, and in general the Euler characteristic of the connected sum of ''g'' tori is {{nowrap|2 &minus; 2''g''}}.

The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of ''k'' of them is {{nowrap|2 &minus; ''k''}}.