Rational homotopy group (${\pi }_n(X)\otimes \mathbb{Q}$)

The rational homotopy group is obtained by tensoring a homotopy group with the rational numbers. It retains the free part and discards torsion.

For an odd-dimensional sphere $S^{2m+1}$, only ${\pi }_{2m+1}(S^{2m+1})\otimes \mathbb{Q}$ is nonzero. For an even-dimensional sphere $S^{2m}$, the nonzero rational homotopy groups occur in degrees $2m$ and $4m-1$, and both are isomorphic to $\mathbb{Q}$.