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 $H^*(N;\mathbb Z)$ into $H^*(S^d;\mathbb Z)$. Hence $N$ has no cohomology in degrees $0<i<d$; <Poincare duality> and the fundamental class give
$$
H_i(N;\mathbb Z)\cong H_i(S^d;\mathbb Z).
$$
Thus $N$ 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 gives
$$
\widetilde H_i(N\setminus\{p\};\mathbb Z)=0
$$
for every $i$. Since $N\setminus\{p\}\cong S^d\setminus X$, the latter has the integral homology of a point.

Solution

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

A degree-one map $f:M\to N$ between closed connected oriented manifolds of the same dimension induces an injection $f^*:H^*(N;R)\to H^*(M;R)$. The wrong-way map defined using <Poincare duality> is a left inverse by the projection formula and $f_*[M]=[N]$.

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 Incoming links: Cohomological injectivity of a degree-one map