A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?


Let $X$ be a nice space (eg a smooth manifold, or more generally a CW complex). The topological Picard group $Pic(X)$ is the set of isomorphism classes of $1$-dimensional complex vector bundles on $X$. The set $Pic(X)$ is an abelian group with group operation the fiberwise tensor product, and the first Chern class map


$$c\_1 : Pic(X) \longrightarrow H^2(X;\mathbb{Z})$$


is an isomorphism of abelian groups.


Now make the assumption that $H\_1(X;\mathbb{Z})$ is a finite abelian group. One nice construction of elements of $Pic(X)$ is as follows. Consider $\phi \in Hom(H\_1(X;\mathbb{Z}),\mathbb{Q}/\mathbb{Z})$. Let $\tilde{X}$ be the universal cover, so $\pi\_1(X)$ acts on $\tilde{X}$ and $X = \tilde{X} / \pi\_1(X)$. Let $\psi : \pi\_1(X) \rightarrow \mathbb{Q}/\mathbb{Z}$ be the composition of $\phi$ with the natural map $\pi\_1(X) \rightarrow H\_1(X;\mathbb{Z})$. Define an action of $\pi\_1(X)$ on $\tilde{X} \times \mathbb{C}$ by the formula


$$g(p,z) = (g(p),e^{2 \pi i \psi(g)}z) \quad \quad \text{for $g \in \pi\_1(X)$ and $(p,z) \in \tilde{X} \times \mathbb{C}$}.$$


Observe that this makes sense since $\psi(g) \in \mathbb{Q} /\mathbb{Z}$. Define $E\_\phi = (\tilde{X} \times \mathbb{C}) / \pi\_1(X)$. The projection onto the first factor induces a map $E\_{\phi} \rightarrow X$ which is easily seen to be a complex line bundle. The line bundle $E\_{\phi}$ is known as the flat line bundle on $X$ with monodromy $\phi$.


Now, the universal coefficient theorem says that we have a short exact sequence


$$0 \longrightarrow Ext(H\_1(X;\mathbb{Z}),\mathbb{Z}) \longrightarrow H^2(X;\mathbb{Z}) \longrightarrow Hom(H\_2(X;\mathbb{Z}),\mathbb{Z}) \longrightarrow 0.$$


Since $H\_1(X;\mathbb{Z})$ is a finite abelian group, there is a natural isomorphism $\rho : Hom(H\_1(X;\mathbb{Z}),\mathbb{Q}/\mathbb{Z}) \rightarrow Ext(H\_1(X;\mathbb{Z}),\mathbb{Z}) $. We can finally state the fact for which I am looking for a reference :


$$c\_1(E\_{\phi}) = \rho(\phi).$$



First Chern class of a flat line bundle
I noticed that someone voted this up today. Since this might indicate that someone else is interested in the answer, I thought I'd remark that Oscar Randal-Williams and I worked out a proof of this when I visited him earlier this year. A version of this proof can be found in Section 2.2 of my paper


The Picard group of the moduli space of curves with level structures,
to appear in Duke Math. J.


which is available on my webpage [here](http://www.nd.edu/~andyp/papers/).


(marked community wiki since it feels weird to get reputation for answering my own question)


