I'm reading this notes, and I have some questions about the contruction on page 18.
Let $M$ be a connected manifold and $E\rightarrow M$ a flat vector bundle over $M$. Consider $\{(U_\alpha, \phi_{U_\alpha})\}$ a flat atlas of $E\rightarrow M$, i.e the transition functions $g_{U_\alpha, U_\beta}$ associated to the trivializations are locally constants.
There is a canonical way to associate a representation $\rho:\pi_{1}(M, m_0)\rightarrow \mathrm{GL}(d, \mathbb{K})$ to this flat vector bundle as follows:
- Fix an open set $U$ containing $m_0$, such that $U$ is in the atlas.
- For any loop $\gamma: [0, 1] → M$, $γ(0) = γ(1) = m_0$ there are a subdivision $0 = t_0 ≤ t_1 ≤ · · · ≤ t_N ≤ t_{N+1} = 1$ of $[0, 1]$ and open sets $U_0, \dots, U_N$ in the atlas, such that $U_0 = U_N = U$ and, for all $i=0, \dots, N$, $\gamma([t_i, t_{i+1}])\subset U_i$.
then, define:
$$
\rho(\gamma)=g_{U_N ,U_{N−1}}(γ(t_N))\dots g_{U_2,U_1}(γ(t_2))g_{U_1,U_0}(γ(t_1))
$$
The question is about the proof that the following map is sujective:
$$
\frac{\{\mbox{flat bundles of rank $d$ over $M$}\}}{\mbox{isomorphism}}\rightarrow \frac{\mathrm{Hom}(\pi_{1}(M, m_0), \mathrm{GL}(d, \mathbb{K}))}{\mbox{conjugacy}}
$$
i.e, given a linear representation $\eta$ of $\pi_{1}(M, m_0)$, I need to find a flat vector bundle $E_\eta$, such that the associeted representation $\rho$ described later is equal to $\eta$.
Here is the construction:
Let $\eta : π_1(M, m_0)→\mathrm{GL}(d,\mathbb{K})$ be a morphism, then the trivial bundle $\widetilde{M} × \mathbb{K}^d$ over the universal cover $\widetilde{M}$ of $M$ has an obvious flat structure together with an action of $π_1(M, m_0)$ preserving the flat structure:
$$
γ · (\tilde m, v) = (γ\tilde m, \eta(γ)v)\quad ∀γ ∈ π_1(M, m_0), \tilde m \in M, v \in \mathbb{K}^{d}.
$$
Here $ \tilde m→ γ\tilde m$ is the natural action of $π_1(M, m_0)$ on the universal cover $\widetilde{M}$. The quotient of the trivial bundle $\widetilde{M} × \mathbb{K}^d$ by this action is denoted $E_\eta$ and
it is naturally a flat bundle over the base $M$.
I'm trying to find the trivizalizations of $E_\eta$. Does any one knows how to find it? Any hints?
Best Answer
This is a standart construction on principal bundles called "associated bundle". There is a wikipedia page and you can find it in Kobayashi's Foundations of Differential Geometry vol 1.
Consider $\widetilde{M}$ as the principal bundle $\pi_{1}(M) \circlearrowleft \widetilde{M} \to^{\pi} M$ with the oposite action you defined $\widetilde{m}g := g^{-1}\widetilde{m}$. Given $\eta : \pi_{1}(M) \to GL(K,d)$, for each $u \in \widetilde{M}$ and $v \in \mathbb{K}^{d}$ denote $(u,v]_{\eta}$ the class of $(u,v)$ in $E_{\eta}=(\widetilde{M} \times \mathbb{K}^{d})/\sim$. Denote by $\pi_{\eta} : E_{\eta} \to M$ the bundle projection. Each point $u \in \widetilde{M}$ acts as a linear map $u : \mathbb{K}^{d} \to (E_{\eta})_{\pi(u)}$ by $u(v)= (u,v]_{\eta}$.
Now given a local section $s : U \subseteq M \to \widetilde{M}$ you can associate it to a $E_{\eta}$ trivialization $\phi_{s} : \pi_{\eta}^{-1}(U) \to U \times \mathbb{K}^{d}$ by \begin{align} \phi_{s}(w) = (\pi_{\eta}(w),s(\pi_{\eta}(w))^{-1}w) \end{align} where $s(\pi_{\eta}(w)) \in \pi^{-1}(\pi_{\eta}(w))$. Now, it's a known fact (and relatively easy to see) of principal bundles that sections are in bijection with trivializations (of the principal bundle). So if you construct a bundle atlas for $\widetilde{M}$ you take the associated sections and you have an atlas for $E_{\eta}$.
As a universal cover, the trivializations of $\widetilde{M}$ take the form $\psi : \pi^{-1}[U] \to U \times \pi_{1}(M)$ with $U \times \pi_{1}(M) = \bigsqcup_{g \in \pi_{1}(M)} U$ with $U$ connected.
Hope this helps :)