nLab rational homotopy sphere

Idea

A rational homotopy sphere is a topological space which need not be homeomorphic to an n-sphere, but which has the same rational homotopy type as an $n$-sphere, hence whose rationalization is weakly homotopy equivalent to a rational n-sphere.

Definition

A $n$-dimensional manifold $\Sigma$ with:

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

is a rational homotopy $n$-sphere.

Properties

Corollary

Every homotopy sphere is a rational homotopy sphere.

Examples

Example

The $n$-sphere $S^n$ is in particular a rational homology $n$-sphere.

Example

Real projective space $\mathbb{R}P^n$ is a rational homotopy $n$-sphere for all $n\gt 0$. The fiber bundle $S^0\hookrightarrow S^n\twoheadrightarrow\mathbb{R}P^n$ (Hatcher 02, Ex. 4.44.) yields with the long exact sequence of homotopy groups (Hatcher 02, Thrm. 4.41.) that $\pi_k(\mathbb{R}P^n)\cong\pi_k(S^n)$ for $k\gt 1$ and $n\gt 0$ as well as $\pi_1(\mathbb{R}P^1)=\mathbb{Z}$ and $\pi_1(\mathbb{R}P^n)=\mathbb{Z}_2$ for $n\gt 1$, which vanishes after rationalization. $\mathbb{R}P^1\cong S^1$ is the sphere in particular.

Related concepts

• homotopy sphere
• homology sphere
• rational homology sphere

References

• Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

See also:

• Wikipedia, Rational homotopy sphere