Rational homotopy sphere

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 $\Sigma$ with the same rational homotopy groups as the $n$ -sphere $S^{n}$ :

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