# 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 :

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

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I'm hoping someone answers with a proof, and then you can give the referees this MO question as a reference. –  Dylan Wilson Feb 20 '11 at 20:32
@Mohan : (some more details) Flat just means that you can find a connection whose curvature vanishes. The Chern-Weil homomorphism will then be zero, but this only gives you an element of de Rham cohomology, so it doesn't see torsion phenomena. –  Andy Putman Feb 20 '11 at 21:36
@Andy:Thanks for clearing up my confusion. –  Mohan Ramachandran Feb 20 '11 at 21:39
You could just redefine $Pic(X)$ to be $H^1(X;\mathcal{C}^\times)$ (cohomology of the sheaf of continuous functions into $\mathbb{\C}^\times$). Then your construction is just the map on $H^1$ induced by $\mathbb{Q}/\mathbb{Z}\to \mathcal{C}^\times$, and the boundary map $H^1(X;\mathcal{C}^\times)\to H^2(X;\mathbb{Z})$ is $c_1$. –  Charles Rezk Feb 20 '11 at 23:20
Not a reference, but one proof goes a little like this: Think about $R/Z$ instead of $Q/Z$. Then $K(Z,2)=BU(1)=B(R/Z)$. Giving $R/Z$ the discrete topology gives a homomorphism of topological groups $Id:(R/Z)^d\to R/Z$ and hence a map on classifying spaces $K(R/Z,1)\to K(Z,2)$. This induces the Bockstein $H^1(-;R/Z)\to H^2(-;Z)$ whose image is the set of complex line bundles whose $c_2$ is sent to 0 in $H^2(-;R)$. Then to see the relation to your construction you just have to consider the "universal" case of the flat bundle over $K(R/Z,1)$. –  Paul Feb 21 '11 at 0:31

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.

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

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Haha, thanks! :) –  Chris Gerig Dec 29 '11 at 7:13

It's probably not exactly what you want (in particular, they're dealing with real bundles and the Stiefel Whitney classes), but something sort of close is discussed in the appendix to

MR2003827 (2004h:53116) Ho, Nan-Kuo(3-TRNT); Liu, Chiu-Chu Melissa(1-HRV) Connected components of the space of surface group representations. Int. Math. Res. Not. 2003, no. 44, 2359–2372. 53D30 (22F05 57N05)

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