Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 114 / 4 / 2 / Solution

Part 1 makes $q:S^d\to N$ a degree-one map of closed oriented $d$-manifolds. The cohomological injectivity of a degree-one map embeds $H^{\ast }(N;\mathbb{Z})$ into $H^{\ast }(S^d;\mathbb{Z})$. Hence $N$ has no cohomology in degrees $0<i<d$; Poincare duality and the fundamental class giveThus $N$ is an integral homology sphere.

The long exact sequence of the pair $(N,N\setminus \{p\})$ has local relative homology $\mathbb{Z}$ only in degree $d$, and the map $H_d(N)\to H_d(N,N\setminus \{p\})$ is an isomorphism because it sends the fundamental class to the local orientation. Exactness now givesfor every $i$. Since $N\setminus \{p\}\cong S^d\setminus X$, the latter has the integral homology of a point.