4
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ The idea, roughly, is to use trivializations of the bundle $\tilde{M} \times \mathbb{K}^d$ over small open sets. The changes of trivialization are of the form $n(\gamma_1)^{-1} n(\gamma_2)$ (or something like that). $\endgroup$ – Phillip Andreae Feb 21 at 21:42
  • 1
    $\begingroup$ Could you clarify exactly what kind of data you want to write down to find the trivializations? And is there something you want to do with this, or are you just trying to make sure you understand this well? (which would be admirable!) $\endgroup$ – Phillip Andreae Feb 21 at 21:46
  • $\begingroup$ Hi @PhillipAndreae. I tried to use the trivializations of $\widetilde{M}\times \mathbb{K}^d$ as you said, but I didn't see exactly how to use them. Maybe the problem is that I can't see how the fibers of $E_\eta$ should be. Do you know a nice way to look at it? $\endgroup$ – Leonardo Schultz Feb 21 at 22:22
  • $\begingroup$ About your second question, I just want to really undestand that. I couldn't find any thing about this construction with more details. $\endgroup$ – Leonardo Schultz Feb 21 at 22:27
2
$\begingroup$

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 :)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.