Show that \(H_c^n(X \times \mathbb{R}; G) \approx H_c^{n-1}(X; G)\) for all \(n\).
Solution
Under the standing locally compact Hausdorff hypotheses for compactly supported cohomology, \[ H_c^q(Y;G)\cong\widetilde H^q(Y^+;G), \] where \(Y^+\) denotes the one-point compactification.
One-point compactification converts products into smash products: \[ (X\times\mathbb R)^+\cong X^+\wedge\mathbb R^+. \] Since \(\mathbb R^+\cong S^1\), this is the reduced suspension \[ X^+\wedge S^1\cong\Sigma X^+. \] Hence the suspension isomorphism gives \[ H_c^n(X\times\mathbb R;G) \cong \widetilde H^n(\Sigma X^+;G) \cong \widetilde H^{n-1}(X^+;G) \cong H_c^{n-1}(X;G). \] Thus \[ \boxed{H_c^n(X\times\mathbb R;G)\cong H_c^{n-1}(X;G)} \] for all \(n\).