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

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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 $\mathbb{Z}$.

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.