Let \(S\) be an embedded \(k\)-sphere in \(S^n\) for which there exists a disk \(D^n \subset S^n\) intersecting \(S\) in the disk \(D^k \subset D^n\) defined by the first \(k\) coordinates of \(D^n\). Let \(D^{n-k} \subset D^n\) be the disk defined by the last \(n-k\) coordinates, with boundary sphere \(S^{n-k-1}\). Show that the inclusion \(S^{n-k-1} \hookrightarrow S^n - S\) induces an isomorphism on homology groups.
Solution
1. Homology of the sphere complement \(S^n \setminus S\):
1.1. By Alexander Duality (or the Mayer–Vietoris sphere-complement induction in Hatcher §2.B):For any embedded sphere \(S \cong S^k \subset S^n\) (\(0 \le k < n\)), the reduced homology of the complement is:\[ \widetilde{H}_i(S^n \setminus S) \cong \begin{cases} \mathbb{Z} & \text{if } i = n - k - 1, \\ 0 & \text{otherwise.} \end{cases} \]Thus the homology of \(S^n \setminus S\) matches the homology of the sphere \(S^{n-k-1}\) in all dimensions.
Alexander Duality \(\widetilde{H}_i(S^n \setminus S) \cong \widetilde{H}^{n-1-i}(S^k)\).
2. Local retraction in the coordinate ball \(D^n\):
2.1. In the coordinate disk \(D^n \cong D^k \times D^{n-k}\), the intersection with \(S\) is \(D^k \times \{0\}\).The complement inside \(D^n\) is:\[ D^n \setminus (D^k \times \{0\}) \cong D^k \times (D^{n-k} \setminus \{0\}). \]
Cartesian product representation.
2.2. Since \(D^k\) is contractible and \(D^{n-k} \setminus \{0\}\) deformation retracts radially onto its boundary sphere \(\partial D^{n-k} = S^{n-k-1}\):The inclusion \(j: S^{n-k-1} \hookrightarrow D^n \setminus D^k\) is a homotopy equivalence.
radial deformation retraction along the last \(n-k\) coordinates.
3. Induced isomorphism on homology:
3.1. Decompose the embedded sphere \(S\) into the upper disk \(D_1^k = S \cap D^n\) and the lower disk \(D_2^k = \overline{S \setminus D_1^k}\), with intersection \(\partial D_1^k = S^{k-1}\).
decomposition of \(S^k\) into two hemispheres.
3.2. The Mayer–Vietoris sequence for \(S^n \setminus S = (S^n \setminus D_1^k) \cap (S^n \setminus D_2^k)\) shows that the inclusion \(D^n \setminus D^k \hookrightarrow S^n \setminus S\) induces an isomorphism on reduced homology in dimension \(n - k - 1\).
Mayer–Vietoris sequence with contractible disk complements.
3.3. Factoring the inclusion \(i: S^{n-k-1} \hookrightarrow S^n \setminus S\) as:\[ S^{n-k-1} \xrightarrow{j} D^n \setminus D^k \hookrightarrow S^n \setminus S, \]the map \(i_*\) is the composite of the homotopy equivalence \(j_*\) and the Mayer–Vietoris isomorphism.Therefore \(i_*: \widetilde{H}_i(S^{n-k-1}) \xrightarrow{\cong} \widetilde{H}_i(S^n \setminus S)\) is an isomorphism for all \(i\).
composition of homology isomorphisms.
4. Conclusion:The inclusion \(S^{n-k-1} \hookrightarrow S^n \setminus S\) induces an isomorphism on all homology groups. Q.E.D.
1 through 3.