Let $X$ be a topological space. I would like to show that $H^n_c(X,G) \cong H^{n+1}_c(X \times \mathbb{R},G)$ for every abelian group $G$ and for every $n \in \mathbb{N}$.
This is an exercise from Hatcher book, but I 've got really no idea of how to do it.
Note that $X\times\mathbb R = A\cup B$, where $$ A = X\times (-\infty ,1) \qquad\text{and}\qquad B = X\times (-1,\infty ). $$ The product space has a cofinal system of compact sets of the form $K\times I$, where $K\subset X$ compact and $I = \lbrack -a,a\rbrack$ for some $a > 0$. For a pair of such sets $(K,I)$, let $$ C=A - K\times I\qquad\text{and}\qquad D=B-K\times I $$ Then trivially $X\times\mathbb R - K\times I = C\cup D$. Consider the following segment from the relative MVS of $(A\cup B,C\cup D)$: $$ H^{n-1}(A,C)\oplus H^{n-1}(B,D)\rightarrow H^{n-1}(A\cap B,C\cap D)\rightarrow H^n(X\times\mathbb R,X\times\mathbb R-K\times I)\rightarrow H^n(A,C)\oplus H^n(B,D) $$ It is not difficult to see that $(A,C)\simeq (X\times\mathbb R,X\times\mathbb R )\simeq (B,D)$ and $(A\cap B,C\cap D)\simeq (X,X- K)$. (Try drawing them!) Hence $H^j(A,C)$ and $H^j(B,D)$ vanish for all $j$, so $$H^n(X\times\mathbb R,X\times\mathbb R-K\times I)\cong H^{n-1}(A\cap B,C\cap D) \cong H^{n-1}(X,X- K).$$ Taking limits, this gives us the desired isomorphism. (Strictly speaking, we first have to argue that the isomorphisms assemble into isomorphisms of directed systems before passing to limits. This seems reasonable since all isomorphisms given above are induced by inclusions or coboundary/connecting maps.)