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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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 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|