0.3. The ultimate duality theorem 7 For a compact pair (A, B), there is a fundamental relationship between the cohomology with compact supports of A B and the unrestricted co- homology of that pair (Spanier, 1966, p. 321): Theorem 0.3.3. For any compact Hausdorff pair (A, B), ˇ q c (A B G) ∼ ˇ q (A, B G). Corollary 0.3.4. A compact, connected, n-dimensional ∂-manifold M is G-orientable if and only if Hn(M, ∂M G) ∼ G. Corollary 0.3.5. If A is a locally compact Hausdorff space and B is a closed subset of A, then ˇ q c (A B G) ∼ ˇ q c (A, B G). Proof. Let A+,B+ denote the one-point compactifications of A, B, respec- tively, with ∞ the ideal point. Then ˇ q c (A G) ∼ ˇ q (A+, {∞} G) ∼ ˇ q (A+ G) and similarly for B. The Five Lemma and another application of Theorem 0.3.3 yield ˇ q c (A, B G) ∼ ˇ q (A+,B+ G) ∼ ˇ q c (A B : G). Corollary 0.3.6. Every open subset of a G-orientable manifold is G-orient- able. Proof. If U is a connected open subset of the G-orientable manifold M n , then G ∼ H0(U G) ∼ ˇ n c (M, M U G) by duality, and ˇ n c (M, M U G) ∼ Hc n (U G) by 0.3.5. Corollary 0.3.7. If S is a locally compact Hausdorff space and q ≥ 2, then ˇ q c (S × R G) ∼ ˇ q−1 c (S G). Proof. Let S+ denote the one-point compactification of S, with ∞ the ideal point. A Mayer-Vietoris argument shows that ˇ q (Suspension of S+ G) ∼ ˇ q−1 (S+ G) for q ≥ 2. Thus, ˇ q c (S × R G) ∼ ˇ q (Suspension of S+, Suspension of ∞ G) ∼ ˇ q (Suspension of S+ G) ∼ Hq−1(S+ G) ∼ ˇ q−1 (S+, {∞} G) ∼ ˇ q−1 c (S G). Corollary 0.3.8. A manifold M is G-orientable if and only if M × R is G-orientable.

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