# Homology of compact orientable surfaces

From Topospaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology and the topological space/family is compact orientable surface

Get more specific information about compact orientable surface | Get more computations of homology

## Contents

## Statement

Suppose is a nonnegative integer. We denote by the compact orientable surface of genus , i.e., is the connected sum of copies of the 2-torus (and it is the 2-sphere when ). In other words, , , , and so on.

### Unreduced version over the integers

We have:

In other words, the zeroth and second homology groups are both free of rank one, and the first homology group is , i.e., the free abelian group of rank .

### Reduced version over the integers

We have:

### Unreduced version with coefficients in

With coefficients in a module over a ring , we have:

### Reduced version with coefficients in

## Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant | General description | Description of value for genus surface | Comment |
---|---|---|---|

Betti numbers | The Betti number is the rank of the torsion-free part of the homology group. | , , all higher are zero | |

Poincare polynomial | Generating polynomial for Betti numbers | ||

Euler characteristic | In particular, this means that the Euler characteristic is negative for , and also that the Euler characteristic and genus can be computed in terms of each other -- given the genus, we can find the Euler characteristic, and vice versa. |

## Facts used in homology computation

- Homology of spheres: This tackles the case.
- Homology of torus: This tackles the case.
- Homology of connected sum: This tackles the inductive procedure of taking connected sums.