Classification of surfaces

This article describes the classification of connected surfaces, i.e., connected two-dimensional manifolds. Note that any two-dimensional manifold is a disjoint union of connected two-dimensional manifolds, so this also gives a classification of (possibly disconnected) two-dimensional manifolds.

Classification of compact orientable surfaces
We first discuss compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds. These are classified by the nonnegative integers. In other words, there is a correspondence:

Nonnegative integers $$\leftrightarrow$$ Homeomorphism classes of compact orientable surfaces

The correspondence, in the forward direction, is as follows: given a nonnegative integer $$g$$, the corresponding compact orientable surface, called the surface of genus $$g$$, is defined as a connected sum of $$g$$ copies of the 2-torus. Two special cases are of note: for $$g = 0$$, we take the corresponding surface to be the 2-sphere, and for $$g = 1$$, we take the corresponding surface to be the 2-torus. After that, each time we increment $$g$$ by $$1$$, we take the connected sum with a new 2-torus.

Pictorially, the surface of genus $$g$$ can be embedded in $$\R^3$$ with as many holes as the genus.

The number $$g$$ can be recovered from the homeomorphism class of the surface (in fact, even from the homotopy type), for instance, it can be recovered by knowing the Euler characteristic of the surface, because the Euler characteristic is $$2 - 2g$$.

Classification of compact non-orientable surfaces
There is a bijection:

Positive integers $$\leftrightarrow$$ Homeomorphism classes of compact non-orientable surfaces

The correspondence, in the forward direction, is as follows: given a positive integer $$k$$, the corresponding compact non-orientable surface is a connected sum of $$k$$ copies of the real projective plane.