Rational homology sphere

In algebraic topology, a rational homology -sphere is an -dimensional manifold with the same rational homology groups as the -sphere. These serve, among other things, to understand which information the rational homology groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the homology groups of the space.

Definition

A rational homology -sphere is an -dimensional manifold with the same rational homology groups as the -sphere :

Every homology sphere is a rational homology sphere.
Every simply connected rational homology '-sphere with is homeomorphic to the '-sphere.

The -sphere itself is obviously a rational homology -sphere.
The pseudocircle is a rational homotopy -sphere, which is not a homotopy -sphere.
The Klein bottle has two dimensions, but has the same rational homology as the [Sphere|]-sphere as its homology groups are given by:
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The real projective space is a rational homology sphere for odd as its homology groups are given by:
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The five-dimensional Wu manifold is a simply connected rational homology sphere, which is not a homotopy sphere.