Compact orientable surface

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

Classification

Further information: classification of compact orientable surfaces

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 $Σ_{g}$ , $S_{g}$ or $M_{g}$ .

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

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 $Z$ , and $H_{1}$ is isomorphic to $Z^{2 g}$ .

In particular, the Betti numbers are $b_{0} = 1, b_{1} = 2 g, b_{2} = 1$ , the Poincare polynomial is $1 + 2 g x + x^{2}$ , and the Euler characteristic is $2 - 2 g$ .

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 $g = 1 - (χ / 2)$ .

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 $χ_{1}$ and $χ_{2}$ respectively, the Euler characteristic of the connected sum is $χ_{1} + χ_{2} - 2$ .

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:

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

or, upon simplification:

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

The justification is as follows: $2 - 2 h$ and $2 - 2 g$ 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.