Rational homology sphere

In algebraic topology, a rational homology ${\displaystyle n}$-sphere is an ${\displaystyle n}$-dimensional manifold with the same rational homology groups as the ${\displaystyle n}$-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 ${\displaystyle n}$-sphere is an ${\displaystyle n}$-dimensional manifold ${\displaystyle \Sigma }$ with the same rational homology groups as the ${\displaystyle n}$-sphere ${\displaystyle S^{n}}$:

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

Properties

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

Examples

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

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

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

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

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 ${\displaystyle \mathbb {R} P^{n}}$ is a rational homology sphere for ${\displaystyle n}$ odd as its (integral) homology groups are given by:^[2][3]

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

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

• The five-dimensional Wu manifold ${\displaystyle W=\operatorname {SU} (3)/\operatorname {SO} (3)}$ is a simply connected rational homology sphere (with non-trivial homology groups ${\displaystyle H_{0}(W)\cong \mathbb {Z} }$, ${\displaystyle H_{2}(W)\cong \mathbb {Z} _{2}}$ und ${\displaystyle H_{5}(W)\cong \mathbb {Z} }$), 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