Show that if \(M\) is a compact contractible \(n\)-manifold then \(\partial M\) is a homology \((n-1)\)-sphere, that is, \(H_i(\partial M; \mathbb{Z}) \approx H_i(S^{n-1}; \mathbb{Z})\) for all \(i\).
Solution
1. \(M\) contractible implies \(H_i(M)=0\) for \(i>0\), \(H_0=\ZZ\).
contractible.
2. Lefschetz duality: \(H_i(M,\partial M)\cong H^{n-i}(M)=0\) for \(i<n\).
duality for compact \(n\)-manifold.
3. Long exact sequence of pair \((M,\partial M)\): \(\cdots\to H_i(\partial M)\to H_i(M)\to H_i(M,\partial M)\to\cdots\).
LES.
4. For \(i<n-1\), \(H_i(M,\partial M)=0\) and \(H_i(M)=0\), so \(H_i(\partial M)=0\).
2 and 3.
5. For \(i=n-1\), \(0\to H_{n-1}(\partial M)\to0\to\ZZ\to H_{n-2}(\partial M)\to0\) gives \(H_{n-1}(\partial M)\cong\ZZ\).
3 with \(H_n(M,\partial M)\cong\ZZ\), \(H_n(M)=0\).
6. Hence \(H_i(\partial M)\cong H_i(S^{n-1})\).
4 and 5.
7. Q.E.D.
6.