Compact orientable surface

This article is about compact orientable surfaces, i.e., two-dimensional compact connected orientable manifolds.

Classification
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.

The surface of genus $$g$$ is sometimes denoted $$\Sigma_g$$, $$S_g$$ or $$M_g$$.

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

Homology
The homology groups of the surface with genus $$g$$ are given as follows: $$H_0$$ and $$H_2$$ are both $$\mathbb{Z}$$, and $$H_1$$ is isomorphic to $$\mathbb{Z}^{2g}$$.

In particular, the Betti numbers are $$b_0 = 1, b_1 = 2g, b_2 = 1$$, the Poincare polynomial is $$1 + 2gx + x^2$$, and the Euler characteristic is $$2 - 2g$$.

We see from this that the surfaces of genus $$g$$ are all in different homotopy classes and are in fact not even homology-equivalent. We can in fact recover the genus of a compact orientable surface simply from its Euler characteristic, by $$g = 1 - (\chi/2)$$.

Connected sum
The connected sum of a surface with genus $$g_1$$ and a surface with genus $$g_2$$ is a surface with genus $$g_1 + g_2$$. If the Euler characteristics of the surfaces are $$\chi_1$$ and $$\chi_2$$ respectively, the Euler characteristic of the connected sum is $$\chi_1 + \chi_2 - 2$$.

Thus, the set of homeomorphism classes of compact orientable surfaces under connected sum is isomorphic to the monoid of nonnegative integers under addition.

Covering spaces
Suppose $$S_g$$ is a surface of genus $$g$$, $$g > 0$$. Then, it turns out that for any finite group $$N$$ of order $$n$$, there exists a regular covering map with base $$S_g$$ and degree $$d$$ such that the group of deck transformations for the covering map is $$N$$. The covering space for this map must also be a compact orientable surface, and have genus $$h$$ for some $$h$$. $$g,h,n$$ are related as follows:

$$(2 - 2h) = n(2 - 2g)$$

or, upon simplification:

$$h = 1 + n(g - 1)$$

The justification is as follows: $$2 - 2h$$ and $$2 - 2g$$ are respectively the Euler characteristics of the compact orientable surfaces, and Euler characteristic of covering space is product of degree of covering and Euler characteristic of base.