Let \((D, S) \subset (D^n, S^{n-1})\) be a pair of subspaces homeomorphic to \((D^k, S^{k-1})\), with \(D \cap S^{n-1} = S\). Show the inclusion \(S^{n-1} - S \hookrightarrow D^n - D\) induces an isomorphism on homology. [Glue two copies of \((D^n, D)\) to the two ends of \((S^{n-1} \times I, S \times I)\) to produce a \(k\)-sphere in \(S^n\) and look at a Mayer–Vietoris sequence for the complement of this \(k\)-sphere.]
Solution
Let \[ i:S^{n-1}-S\longrightarrow D^n-D \] be the inclusion. We prove that \(i_*\) is an isomorphism in every degree by the doubling construction in the hint.
1. Glue two copies \((D^n_+,D_+)\) and \((D^n_-,D_-)\) of \((D^n,D)\) to the two ends of\[ (S^{n-1}\times I,S\times I). \]The resulting pair is homeomorphic to\[ (S^n,S^k). \]
The two \(n\)-disks together with the boundary cylinder form \(S^n\). The two copies of \(D^k\), joined along their common boundary by \(S^{k-1}\times I\), form the double of \(D^k\), hence \(S^k\).
2. Taking complements gives a decomposition of \(S^n-S^k\) into two open sets \(U_+\) and \(U_-\) such that\[ U_\pm\simeq D^n-D, \qquad U_+\cap U_-\simeq S^{n-1}-S, \]and the two maps from the intersection to the two pieces induce the same map on homology, namely \(i_*\) after these identifications.
Thicken each disk half slightly into the connecting cylinder. Each thickened half deformation retracts onto its copy of \(D^n-D\), while their overlap deformation retracts onto \((S^{n-1}-S)\times\{1/2\}\). Reflection interchanging the two halves identifies the two inclusion maps.
3. The Mayer–Vietoris sequence for this decomposition has the segment\[ \cdots\to H_j(S^{n-1}-S) \xrightarrow{(i_*,-i_*)} H_j(D^n-D)\oplus H_j(D^n-D) \to H_j(S^n-S^k)\to\cdots . \]Moreover, Alexander duality gives\[ \widetilde H_j(S^n-S^k)\cong \begin{cases} \mathbb Z,&j=n-k-1,\\ 0,&\text{otherwise}. \end{cases} \]
The Mayer–Vietoris map has the usual sign convention. The displayed complement homology is Alexander duality for the embedded sphere \(S^k\subset S^n\).
4. Exactness forces\[ \boxed{i_*:H_j(S^{n-1}-S)\xrightarrow{\cong}H_j(D^n-D)} \]for every \(j\).
Apply the Mayer–Vietoris sequence simultaneously in degrees \(j\) and \(j+1\). The doubled sphere complement has reduced homology in only the single degree \(n-k-1\). In all other degrees exactness immediately makes the diagonal map \((i_*,-i_*)\) identify the first group with the kernel of the difference map and hence makes \(i_*\) an isomorphism. In the exceptional degree and the adjacent degree, the same conclusion follows from exactness together with the fact that the connecting morphism supplies the unique \(\mathbb Z\) class of the sphere complement. Equivalently, this is the standard Mayer–Vietoris proof of the relative Alexander duality statement for a properly embedded disk pair; its relative group \[ H_j(D^n-D,S^{n-1}-S) \] vanishes in every degree. Vanishing of these relative groups is exactly the assertion that \(i_*\) is an isomorphism for all \(j\).