# Acyclic space

From Topospaces

This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spacesView all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces

This is a variation of contractibility. View other variations of contractibility

## Contents

## Definition

A topological space is said to be **acyclic** if the homology groups in all dimensions are the same as those of a point, for any homology theory. Equivalently, it suffices to say that the singular homology groups are the same as those for a point.

## 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, or equivalently, has a contracting homotopy | (via weakly contractible) | (via weakly contractible) | Weakly contractible space|FULL LIST, MORE INFO |

weakly contractible space | acyclic space that is also simply connected | (obvious) | complement of a point in a homology sphere | |FULL LIST, MORE INFO |

### Weaker properties

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

rationally acyclic space | homology groups over the rationals are the same as those of a point | (direct) | real projective space in even dimension (see homology of real projective space) | |FULL LIST, MORE INFO |

space with finitely generated homology | only finitely many nonzero homology groups, and each is finitely generated | (direct) | any space with nontrivial homology, such as a sphere | Space with Euler characteristic one|FULL LIST, MORE INFO |

space with homology of finite type | all homology groups are finitely generated | (direct) | (via finitely generated homology) | Space with Euler characteristic one|FULL LIST, MORE INFO |

space with free homology | all homology groups are free abelian groups | (direct) | a sphere has free homology but is not acyclic | |FULL LIST, MORE INFO |

space with perfect fundamental group | the fundamental group is perfect. This is equivalent to the first homology group being trivial | (direct) | a 2-sphere has perfect fundamental group but is not acyclic | |FULL LIST, MORE INFO |

space with Euler characteristic one | the Euler characteristic is well-defined and equals one | |FULL LIST, MORE INFO |

## Metaproperties

A product of acyclic spaces is acyclic. The proof of this relies on the Kunneth formula.