# Rationally 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

## Contents

## Definition

A topological space is termed **rationally acyclic** if its homology groups with *rational coefficients* in all dimensions, are equal to those of a point. In other words, the zeroth homology group is and all higher homology groups are zero.

Equivalently the homology groups in the usual sense, are all torsion groups, except the zeroth group which is just .

## Examples

### Examples among manifolds

We list some examples of compact connected manifolds:

Manifold | Dimension | Does it satisfy some stronger property than being rationally acyclic? |
---|---|---|

one-point space | 0 | contractible space |

real projective plane | 2 | |

real projective four-dimensional space | 4 |

## Relation with other properties

### Stronger properties

- Contractible space
- Weakly contractible space
- Acyclic space: Every rationally acyclic space need not be acyclic; for instance, real projective space in even dimensions is rationally acyclic but not acyclic.