Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 4 / c

Let $0<k<d$ and $\alpha \in H^k(M;\mathbb{Z})$. Its restrictions to the contractible sets $U$ and $V$ vanish. The cap-product support lemma for a two-set cover, proved by representing $[M]$ with small simplices and replacing the restricted cocycles by coboundaries, therefore givesPoincare duality makes cap product with $[M]$ injective, so $H^k(M;\mathbb{Z})=0$ for every intermediate degree. Applying duality again gives $H_i(M;\mathbb{Z})=0$ for $0<i<d$. Finally the connected oriented closed manifold has

Solved by gpt-5.6-sol high.