The rational cohomology from part i has one generator in every degree divisible by $3$. Its connected graded-commutative Hopf algebra structure is
$$
H^*(\Omega S^4;\mathbb Q)
\cong\Lambda(x_3)\otimes\mathbb Q[y_6]
$$
as a graded vector-space-compatible algebra: the odd class has square zero and the degree-six class supplies the even multiples. The rational Hurewicz and Hopf-algebra correspondence for a connected loop space identifies the indecomposable generators with the duals of its <rational homotopy groups>. Hence, for $0\leq i\leq6$,
$$
\pi_i(\Omega S^4)\otimes\mathbb Q\cong
\begin{cases}
\mathbb Q,&i=3,6,\\
0,&i=0,1,2,4,5.
\end{cases}
$$
This also agrees with the <rational homotopy groups of a sphere> and the <loop-space shift of homotopy groups>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 165 / 3 / iv / Solution

{title2=$\pi_n(X)\otimes\mathbb Q$}
{wiki}

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

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 Incoming links: Rational homotopy group