Rational homology sphere

Template:Short description 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 $\mathrm{\Sigma }$ with the same rational homology groups as the $n$-sphere $S^n$:

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

Properties

• Every (integral) homology sphere is a rational homology sphere.
• Every simply connected rational homology $n$-sphere with $n\le 4$ 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 $ℝP^n$ is a rational homology sphere for $n$ odd as its (integral) homology groups are given by:

  $H_k(ℝP^n)\cong \{\begin{array}{cc}\mathbb{Z}& ;k=0\text{ or }k=n\text{ if odd}\\ {\mathbb{Z}}_2& ;k\text{ odd},0<k<n\\ 1& ;\text{otherwise}\end{array}.$

$ℝP^1\cong S^1$ is the sphere in particular.

• The five-dimensional Wu manifold $W=\mathrm{SU}(3)/\mathrm{SO}(3)$ 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

• Template:Citation

External links

• rational homology sphere at the nLab