Rationally acyclic space

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

View all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces

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

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.