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

Use the path-loop fibrationand its rational Serre spectral sequence. The total space is contractible, while $H^{\ast }(S^4;\mathbb{Q})$ is $\mathbb{Q}$ in degrees $0$ and $4$ and zero otherwise. The only possible nonzero differential isConvergence to the cohomology of a point first forces $d_4:H^3(\Omega S^4;\mathbb{Q})\to H^4(S^4;\mathbb{Q})$ to be an isomorphism, and then inductively forces an isomorphism from each nonzero vertical group to the group three degrees below it in the other column. Therefore

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Apply the rational Serre spectral sequence to the path-loop fibration of $K(\mathbb{Z},3)$. Its fiber iswhose rational cohomology ring is $\mathbb{Q}[c]$ with $|c|=2$. Since the path space is contractible, $c$ must transgress to a nonzero class $x\in H^3(K(\mathbb{Z},3);\mathbb{Q})$. Multiplicativity givesOver $\mathbb{Q}$ these differentials pair and kill every positive-degree class except $x$, while graded commutativity gives $x^2=0$. Hence

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The loop-space shift of homotopy groups givesThus $\Omega S^4$ is $2$-connected and its first nonzero homotopy group isThe Hurewicz theorem now givesand every positive integral homology group below degree $3$ vanishes. The smallest requested degree is therefore $i=3$.

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The rational cohomology from part i has one generator in every degree divisible by $3$. Its connected graded-commutative Hopf algebra structure isas 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 $0\le i\le 6$,This also agrees with the rational homotopy groups of a sphere and the loop-space shift of homotopy groups.

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By the loop-space shift of homotopy groups,Part iv therefore gives

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