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.

The Sullivan model of the connected sum of two complex projective planes has two generators in each of degrees two and three, and none elsewhere. Its rational homotopy groups are therefore exactly the displayed two copies of ${\mathbb{Q}}^2$, with all other positive degrees zero.

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 $\mathbb{Q}$.