Rationally acyclic space

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 among manifolds
We list some examples of compact connected manifolds:

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.