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

The cap product is the chain operationobtained by evaluating the cochain on the front $k$-face of a singular simplex and retaining its back $(p-k)$-face, with the standard sign convention. It descends to homology and cohomology.

A fundamental class $[M]\in H_d(M;\mathbb{Z})$ restricts at every $x\in M$ to the local generator of $H_d(M,M\setminus \{x\};\mathbb{Z})$ selected by the orientation. Poincare duality states thatis an isomorphism for every $k$ and coefficient ring $R$.

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The small simplex theorem says that the inclusion of the subcomplex generated by singular simplices whose images lie wholly in $U$ or wholly in $V$,is a chain-homotopy equivalence. Repeated barycentric subdivision supplies the inverse up to chain homotopy.

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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.