Rational homology sphere

In algebraic topology, a rational homology $n$ -sphere is an $n$ -dimensional manifold with the same rational homology groups as the $n$ -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 (integral) homology groups of the space.

Definition

A rational homology $n$ -sphere is an $n$ -dimensional manifold $\Sigma$ with the same rational homology groups as the $n$ -sphere $S^{n}$ :

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

Properties

Every (integral) homology sphere is a rational homology sphere.
Every simply connected rational homology $n$ -sphere with $n\leq 4$ is homeomorphic to the $n$ -sphere.

Examples

The $n$ -sphere $S^{n}$ itself is obviously a rational homology $n$ -sphere.
The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy $1$ -sphere, which is not a homotopy $1$ -sphere.
The Klein bottle has two dimensions, but has the same rational homology as the $1$ -sphere as its (integral) homology groups are given by:^[1]
$H_{0}(K)\cong \mathbb {Z}$

$H_{1}(K)\cong \mathbb {Z} \oplus \mathbb {Z} _{2}$

$H_{2}(K)\cong 1$

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

The real projective space $\mathbb {R} P^{n}$ is a rational homology sphere for $n$ odd as its (integral) homology groups are given by:^[2]^[3]
$H_{k}(\mathbb {R} P^{n})\cong {\begin{cases}\mathbb {Z} &;k=0{\text{ or }}k=n{\text{ if odd}}\\\mathbb {Z} _{2}&;k{\text{ odd}},0<k<n\\1&;{\text{otherwise}}\end{cases}}.$

\mathbb {R} P^{1}\cong S^{1}

is the sphere in particular.

The five-dimensional Wu manifold $W=\operatorname {SU} (3)/\operatorname {SO} (3)$ is a simply connected rational homology sphere (with non-trivial homology groups $H_{0}(W)\cong \mathbb {Z}$ , $H_{2}(W)\cong \mathbb {Z} _{2}$ und $H_{5}(W)\cong \mathbb {Z}$ ), which is not a homotopy sphere.