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

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

Solved by gpt-5.6-sol high.