# Space with Euler characteristic one

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

## Contents

## Definition

Suppose is a topological space that is a space with finitely generated homology, i.e., it has only finitely many nontrivial homology groups and all of them are finitely generated. We say that is a **space with Euler characteristic one** if the Euler characteristic of equals , i.e., .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

contractible space | homotopy-equivalent to a point | (via acyclic) | (via acyclic) | Acyclic space|FULL LIST, MORE INFO |

weakly contractible space | weakly homotopy-equivalent to a point | (via acyclic) | (via acyclic) | Acyclic space|FULL LIST, MORE INFO |

acyclic space | all the homology groups are zero, except for the zeroth homology group | (via rationally acyclic) | (via rationally acyclic) | |FULL LIST, MORE INFO |

space with finitely generated homology that is also a rationally acyclic space |
all the homology groups over the rationals, except the zeroth homology group, are zero. Equivalently, all integral homology groups except the zeroth one are torsion |
rationally acyclic and finitely generated homology implies Euler characteristic one | Euler characteristic one not implies rationally acyclic |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

space with finitely generated homology | finitely many nonzero homology groups, and they are all finitely generated | |FULL LIST, MORE INFO | ||

space with homology of finite type | all homology groups are finitely generated | |FULL LIST, MORE INFO |