Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Question

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting homomorphism $$ H^j(X;\mathbb C^\times) \to H^{j+1}(X;\mathbb Z). $$ What is a good way of seeing that the image of this map is contained in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$?

Answer 1

Unless I'm being silly, the torsion subgroup is contained in the kernel of the map $H^{j+1}(X,\Bbb Z)\to H^{j+1}(X,\Bbb C)$, so you're done by exactness of the long exact sequence. (For reasonable spaces, e.g., manifolds or simplicial complexes, the sheaf cohomology and singular cohomology agree.)