For an \(n\)-manifold \(M\) and a compact subspace \(A \subset M\), show that \(H_n(M, M - A; R)\) is isomorphic to the group \(\Gamma_R(A)\) of sections of the covering space \(M_R \to M\) over \(A\), that is, maps \(A \to M_R\) whose composition with \(M_R \to M\) is the identity.
Solution
Write \[ H_n(M\mid A;R)=H_n(M,M-A;R). \] For each \(x\in A\), inclusion of pairs gives a homomorphism \[ H_n(M\mid A;R)\longrightarrow H_n(M\mid x;R). \] Thus every class \(\alpha\in H_n(M\mid A;R)\) determines a function \[ x\longmapsto \alpha_x, \] where \(\alpha_x\) is its image in the local homology group at \(x\).
By the local constancy statement built into the orientation covering \(M_R\to M\), this function is a section of \(M_R\) over \(A\). Hence there is a homomorphism \[ \Phi:H_n(M,M-A;R)\longrightarrow\Gamma_R(A), \qquad \Phi(\alpha)(x)=\alpha_x. \]
Hatcher’s Lemma 3.27(a) says precisely that if \(A\) is compact and \(x\mapsto\alpha_x\) is a section of \(M_R\to M\) over \(A\), then there exists a unique class \[ \alpha_A\in H_n(M\mid A;R) \] whose image in every \(H_n(M\mid x;R)\) is \(\alpha_x\). Therefore every section lies in the image of \(\Phi\), and uniqueness says that \(\Phi\) is injective.
Consequently \[ \boxed{H_n(M,M-A;R)\cong\Gamma_R(A).} \] The isomorphism is the natural one sending a relative homology class to its family of local orientation classes along \(A\).