# Rationally acyclic space

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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 spaces

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## 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 $\mathbb{Q}$ 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 $\mathbb{Z}$.

## 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