If \(M\) is a noncompact \(R\)-orientable \(n\)-manifold with boundary \(\partial M\) having a collar neighborhood in \(M\), show that there are Poincaré duality isomorphisms \(H_c^k(M; R) \approx H_{n-k}(M, \partial M; R)\) for all \(k\), using the five-lemma and the long exact sequence relating compactly supported cohomology of \(M\), \(M \setminus \partial M\), and \(\partial M\).
Solution
Because \(\partial M\) has a collar, compactly supported cohomology has the long exact sequence \[ \cdots\to H_c^{k-1}(\partial M;R) \to H_c^k(M,\partial M;R) \to H_c^k(M;R) \to H_c^k(\partial M;R)\to\cdots. \] The homology long exact sequence of the pair is \[ \cdots\to H_{n-k}(\partial M;R) \to H_{n-k}(M;R) \to H_{n-k}(M,\partial M;R) \to H_{n-k-1}(\partial M;R)\to\cdots. \] Cap product with the fundamental classes gives the vertical maps in Hatcher’s diagram, and Exercise 34’s cap-product/naturality argument shows the diagram commutes up to sign.
Now \(M-\partial M\) is an \(R\)-orientable noncompact manifold without boundary. By excision through the collar, \[ H_c^k(M,\partial M;R)\cong H_c^k(M-\partial M;R), \] and the already-proved noncompact Poincaré duality theorem gives \[ H_c^k(M,\partial M;R)\xrightarrow{\cong}H_{n-k}(M;R). \] Similarly, \(\partial M\) is an \(R\)-orientable \((n-1)\)-manifold without boundary, so \[ H_c^{k-1}(\partial M;R)\xrightarrow{\cong}H_{n-k}(\partial M;R) \] and \[ H_c^k(\partial M;R)\xrightarrow{\cong}H_{n-k-1}(\partial M;R). \] Thus in the commutative diagram all vertical maps except the middle map \[ D_M:H_c^k(M;R)\longrightarrow H_{n-k}(M,\partial M;R) \] are isomorphisms. The five-lemma applied to the surrounding five terms implies that \(D_M\) is also an isomorphism. Hence \[ \boxed{H_c^k(M;R)\cong H_{n-k}(M,\partial M;R)} \] for every \(k\).