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 ${\mathrm{\Sigma }}_g$, $S_g$ or $M_g$.

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

Particular cases

Value of genus $g$	Surface
0	2-sphere
1	2-torus
2	genus two surface

Algebraic topology

Homology

Further information: homology of compact orientable surfaces

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 $g=1-(\chi /2)$.

Cohomology

Further information: cohomology of compact orientable surfaces

Homotopy

Further information: homotopy of compact orientable surfaces

Operations

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.