Rational homotopy sphere

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

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

Properties

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

Examples

• The ${\displaystyle n}$-sphere ${\displaystyle S^{n}}$ itself is obviously a rational homotopy ${\displaystyle n}$-sphere.
• The Poincaré homology sphere is a rational homology ${\displaystyle 3}$-sphere in particular.
• The real projective space ${\displaystyle \mathbb {R} P^{n}}$ is a rational homotopy sphere for all ${\displaystyle n>0}$. The fiber bundle ${\displaystyle S^{0}\rightarrow S^{n}\rightarrow \mathbb {R} P^{n}}$^[1] yields with the long exact sequence of homotopy groups^[2] that ${\displaystyle \pi _{k}(\mathbb {R} P^{n})\cong \pi _{k}(S^{n})}$ for ${\displaystyle k>1}$ and ${\displaystyle n>0}$ as well as ${\displaystyle \pi _{1}(\mathbb {R} P^{1})=\mathbb {Z} }$ and ${\displaystyle \pi _{1}(\mathbb {R} P^{n})=\mathbb {Z} _{2}}$ for ${\displaystyle n>1}$,^[3] which vanishes after rationalization. ${\displaystyle \mathbb {R} P^{1}\cong S^{1}}$ is the sphere in particular.

See also

• Rational homology sphere

Literature

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

External links

• rational homotopy sphere at the nLab