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

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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