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

A local orientation of a manifold at $x\in M$ is a generator ofAn $R$-orientation is a locally coherent choice of such generators. An R-fundamental class is a class $[M{]}_R\in H_d(M;R)$ whose image in every one of these local homology groups is a generator. Those images vary coherently under the restriction maps between small coordinate balls, so an $R$-fundamental class determines an $R$-orientation.

For a closed $R$-oriented $d$-manifold, Poincare duality says that cap product with its fundamental class is an isomorphismfor every $k$.

Choose a generator $[S^d]\in H_d(S^d;\mathbb{Z})\cong \mathbb{Z}$ from the homology of a sphere. For every $x\in S^d$, the long exact sequence of $(S^d,S^d\setminus \{x\})$, together with the contractibility of $S^d\setminus \{x\}\cong {\mathbb{R}}^d$, shows thatThe chosen generator is therefore a local generator at every point and is a $\mathbb{Z}$-fundamental class.

Put $N=S^d/X$ and let $p=q(X)$. The restrictionis a homeomorphism. At every point of $N\setminus \{p\}$, transport the local orientation of $S^d$ through this homeomorphism. The localization of the global class $q_{\ast }[S^d]$ is therefore a generator at one, and hence every, point of that connected open set. The localizations of a global homology class form a section of the orientation local system; because the manifold $N$ is connected, this generator extends across $p$. Thus $q_{\ast }[S^d]$ is a fundamental class for an orientation of $N$.

Choose $[N]=q_{\ast }[S^d]$. By the degree of a map between oriented manifolds,so $\boxed{\mathrm{deg}q=1}$.

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.