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.

Definition

A $n$-dimensional manifold $\Sigma$ with:

$$H_k(\Sigma,\mathbb{Q}) =H_k(S^n,\mathbb{Q}) \cong\begin{cases} \mathbb{Q} & ;k=0\text{ or }k=n \\ 1 & ;\text{otherwise} \end{cases}$$

is a rational homology $n$-sphere.

Properties

Corollary

Every homology sphere is a rational homology sphere.

Examples

Example

The $n$-sphere $S^n$ is in particular a rational homology $n$-sphere.

Example

The Klein bottle $K$ has two dimensions, but has the same rational homology as the $1$-sphere. Its integral homology groups are given by: (Hatcher 02, Ex. 2.47.)

$$H_0(K) \cong\mathbb{Z};$$

$$H_1(K) \cong\mathbb{Z}\oplus\mathbb{Z}_2;$$

$$H_2(K) \cong 1.$$

Hence it is not a rational homology sphere, but would be if the requirement to be of same dimension was dropped.

Example

Real projective space $\mathbb{R}P^n$ is a rational homology $n$-sphere for all $n$ odd. Its integral homology groups are given by: (Hatcher 02, Ex. 4.42.)

$$H_k(\mathbb{R}P^n) \cong\begin{cases} \mathbb{Z} & ;k=0\text{ or }k=n\text{ if odd} \\ \mathbb{Z}_2 & ;k\text{ odd},0\lt k\lt n \\ 1 & ;\text{otherwise} \end{cases}$$

$\mathbb{R}P^1\cong S^1$ is the sphere in particular.

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

References

• Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

See also:

• Wikipedia, Rational homology sphere