nLab rational homology sphere

Idea

A rational homology sphere is a topological space which need not be homeomorphic to an n-sphere, but which has the same rational homology as an $n$-sphere.

Properties

Corollary

Every homology sphere is a rational homology sphere.

Examples

Example

The Wu manifold $W=SU(3)/SO(3)$ is a simply connected rational homology $5$-sphere (with non-trivial homology groups $H_0(W)\cong\mathbb{Z}$, $H_2(W)\cong\mathbb{Z}_2$ and $H_5(W)\cong\mathbb{Z}$), but isn’t a homotopy $5$-sphere.

Related concepts

• homology sphere
• homotopy sphere
• rational homotopy sphere