# Compact orientable surface

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

## Contents

## Classification

`Further information: classification of compact orientable surfaces`

These are classified by the nonnegative integers. In other words, there is a correspondence:

Nonnegative integers Homeomorphism classes of compact orientable surfaces

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

The surface of genus is sometimes denoted , or .

Pictorially, the surface of genus can be embedded in with as many *holes* as the genus.

## Particular cases

Value of genus | 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 are given as follows: and are both , and is isomorphic to .

In particular, the Betti numbers are , the Poincare polynomial is , and the Euler characteristic is .

We see from this that the surfaces of genus 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 .

### 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 and a surface with genus is a surface with genus . If the Euler characteristics of the surfaces are and respectively, the Euler characteristic of the connected sum is .

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 is a surface of genus , . Then, it turns out that for any finite group of order , there exists a regular covering map with base and degree such that the group of deck transformations for the covering map is . The covering space for this map must also be a compact orientable surface, and have genus for some . are related as follows:

or, upon simplification:

The justification is as follows: and 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.