# Genus two surface

From Topospaces

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

## Contents

## Definition

This topological space, denoted or , is defined in the following equivalent ways:

- It is the connected sum of two copies of the 2-torus.
- It is the compact orientable surface of genus .

## Topological space properties

Property | Satisfied? | Is the property a homotopy-invariant property of topological spaces? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|---|

manifold | Yes | No | By definition as a connected sum of manifolds | satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc. |

path-connected space | Yes | Yes | By definition as a connected sum of connected manifolds. | satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous) |

simply connected space | No | Yes | Fundamental group is nontrivial, see homotopy of compact orientable surfaces | dissatisfies: simply connected manifold |

rationally acyclic space | No | Yes | The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See homology of spheres | dissatisfies: acyclic space, weakly contractible space, contractible space |

space with Euler characteristic zero | No | Yes | The Euler characteristic is -2, see homology of compact orientable surfaces | |

space with Euler characteristic one | No | Yes | The Euler characteristic is -2, see homology of compact orientable surfaces | |

compact space | Yes | No | connected sum of compact manifolds | satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness |

## Algebraic topology

### Homology

`Further information: homology of compact orientable surfaces`

The homology groups over the integers are as follows:

More generally, with coefficients in a module , the homology groups are:

The reduced homology looks the same except that the zeroth homology groups/modules are now zero.

### Cohomology

`Further information: cohomology of compact orientable surfaces`

The cohomology groups over the integers are as follows:

More generally, with coefficients in a module , the cohomology groups are:

### Homology-based invariants

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

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 | -2 |

### Homotopy groups

`Further information: homotopy of compact orientable surfaces`