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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 $H^1(X;\mathbb{Z})$ 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})$Hom(H_1(X;\mathbb{Z}),\mathbb{Q}/\mathbb{Z})$. 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 $\pi_1(X) \rightarrow H^1(X;\mathbb{Z})$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 : $$c_1(E_{\phi}) = \rho(\phi).$$ 1 # First Chern class of a flat line bundle 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 :

$$c_1(E_{\phi}) = \rho(\phi).$$