Consider the short exact sequence (call it "coefficient sequence'')
$$ 0 \to {\bf{Z}} \to {\bf{R}} \to U(1) \to 0$$
Suppose $X$ is a topological space. The long-exact sequence in cohomology derived from the coefficient sequence above reads
$$ \cdots \to H^{p-1}(X; {\bf{Z}}) \to H^{p-1}(X; {\bf{R}}) \to H^{p-1}(X; U(1)) \to H^{p}(X; {\bf{Z}}) \to \cdots $$
By "the group of components $\overline{H}^{p-2}(X; U(1))$'' one presumably means the quotient group
$$ \overline{H}^{p-2}(X; U(1)) = H^{p-2}(X; U(1))/H^{p-2}(X; U(1))_{0}$$
where $H^{p-2}(X; U(1))_{0}$ denotes the identity component of $H^{p-2}(X; U(1))$.
How does one argue that $\overline{H}^{p-2}(X; U(1)) = H^{p-1}(X; {\bf{Z}})_{\text{tors}}$, the torsion subgroup of $H^{p-1}(X; {\bf{Z}})$?
[This question stems from the discussion around (2.18) of a paper by Moore and Witten [hep-th/9912279].]
If $\beta$ is the map
$$\beta: H^{p-2}(X, {\bf{R}}) \longrightarrow H^{p-2}(X, U(1))$$
I am able to argue that $$H^{p-1}(X,Z)_{\text{tors}} \cong H^{p-2}(X, U(1))/\text{im}(\beta)$$
I guess my question is: how does one show that $\text{im}(\beta)$ is the connected component of the identity, i.e. $H^{p-2}(X, U(1))_{0}$?