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:

  $H_{0}(K)\cong \mathbb {Z}$

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

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

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:

  $H_{k}(\mathbb {R} P^{n})\cong {\begin{cases}\mathbb {Z} &;k=0{\text{ or }}k=n{\text{ if odd}}\\\mathbb {Z} _{2}&;k{\text{ odd}},0<k<n\\1&;{\text{otherwise}}\end{cases}}.$

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

See also

• Rational homotopy sphere

Literature

• Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0

External links

• rational homology sphere at the nLab