Rational homotopy sphere

Template:Short description In algebraic topology, a rational homotopy $n$-sphere is an $n$-dimensional manifold with the same rational homotopy groups as the $n$-sphere. These serve, among other things, to understand which information the rational homotopy groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homotopy groups of the space.

Definition

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

${\pi }_k(\mathrm{\Sigma })\otimes \mathbb{Q}={\pi }_k(S^n)\otimes \mathbb{Q}\cong \{\begin{array}{cc}\mathbb{Z}& ;k=n\text{ if }n\text{ even}\\ \mathbb{Z}& ;k=n,2n-1\text{ if }n\text{ odd}\\ 1& ;\text{otherwise}\end{array}.$

Properties

• Every (integral) homotopy sphere is a rational homotopy sphere.

Examples

• The $n$-sphere $S^n$ itself is obviously a rational homotopy $n$-sphere.
• The Poincaré homology sphere is a rational homology $3$-sphere in particular.
• The real projective space $ℝP^n$ is a rational homotopy sphere for all $n>0$. The fiber bundle $S^0\to S^n\to ℝP^n$ yields with the long exact sequence of homotopy groups that ${\pi }_k(ℝP^n)\cong {\pi }_k(S^n)$ for $k>1$ and $n>0$ as well as ${\pi }_1(ℝP^1)=\mathbb{Z}$ and ${\pi }_1(ℝP^n)={\mathbb{Z}}_2$ for $n>1$, which vanishes after rationalization. $ℝP^1\cong S^1$ is the sphere in particular.

See also

• Rational homology sphere

Literature

• Template:Citation

External links

• rational homotopy sphere at the nLab