Relationship between monodromy representations and isomorphism of flat vector bundles

Question

This question is somehow related to this one.

Let $M$ be a smooth (compact, if you wish) connected manifold.

Then, it is well known that there is an equivalence between the isomorphism classes of pairs $(E,\nabla)$, where $E\to M$ is a complex vector bundle of rank $r$ and $\nabla$ a flat connection on $E$, and the isomorphism classes of complex linear representations of dimension $r$ of the fundamental group $\pi_1(M)$ of $M$.

Here, $(E,\nabla)\simeq (F,\nabla')$ iff there exists an isomorphism of differentiable vector bundles $f\colon E\to F$ over $M$ such that $f^*\nabla'=\nabla$.

Question. Are there necessary and/or sufficient conditions such that given two representations $\rho,\rho'\colon\pi_1(M)\to\operatorname{Gl}(r,\mathbb C)$, the associated complex vector bundles $E_\rho,E_{\rho'}\to M$ are isomorphic?

Of course, in the question we are forgetting the flat connections naturally given by these two representations, and we are asking the two complex vector bundles to be isomorphic just as complex vector bundles.

More specifically, can we for instance give an answer in terms of connected components of the space of the representations - or, if you want, in terms of (some well-chosen notion of) homotopies between the representations?

Answer 1

I have two comments:

(1). It is a trivial remark, but one should note that representations $\rho, \rho'$ which are in the same component of $Hom(\pi,G)$ (in your case, $G=GL(r, {\mathbb C}))$ yield flat bundles $E_{\rho}, E_{\rho'}$ which are isomorphic as vector bundles. Here and below $\pi=\pi_1(M)$.

Proof. It suffices to work with principal $G$-bundles. Let $\rho_t$ be a path of representations between $\rho, \rho'$. Then $\rho_t$ determines a (flat) principal $G$-bundle $P$ over $M\times [0,1]$, whose restrictions $P_0, P_1$ to $M\times 0, M\times 1$ are isomorphic to the principal $G$-bundles associated with $\rho, \rho'$. I claim that this bundle is just the product $P_0\times I$. Indeed, construct a section of the bundle $Hom(P, P_0\times I)$ starting with the identity on $M\times 0$ and then extend it to the rest of $M\times I$ (path-lifting property for Serre fibrations). QED.

This gives a sufficient, but, of course, far from necessary, condition for an isomorphism of vector bundles. In particular, you get finiteness of the number of isomorphism classes of vector bundles. As a simple example consider the case $r=1$ and $H_1(M)$ torsion-free. Then $Hom(\pi, {\mathbb C}^\times)= ({\mathbb C}^\times)^n$ is connected (here $n=rank(H_1(M))$). In particular, all flat line bundles in this case are trivial.

(2). A far less trivial is the result of Deligne and Sullivan "Fibres vectoriels complexes a groupe structural discret", C. R. Acad. Sci. Paris 281 (1975), 1081-1083. They prove that for every finite cell complex $M$ and every $\rho: \pi=\pi_1(M)\to GL(r, {\mathbb C})$ there exists a finite cover $\tilde{M}\to M$ so that the pull-back of the associated flat bundle $E_\rho$ to $\tilde{M}$ is trivial (as a bundle). Their proof is constructive and (from what I can tell by reading Math Review of their paper since I lost my copy) gives a sufficient condition for triviality of $E_\rho$: Let $A$ denote the subring of ${\mathbb C}$ generated by the matrix entries of $\rho(\pi)$. Suppose that there are two maximal ideals $m_1, m_2$ in $A$ so that:

a. $A/m_i$ have distinct characteristics and

b. $\rho(\pi)$ maps trivially to $GL(r, A/m_i), i=1,2$.

Then $E_\rho$ is trivial as a vector bundle.

From this one can extract a sufficient condition for an isomorphism of bundles $E_\rho, E_{\rho'}$ (by considering the flat bundle $E_\rho^*\otimes E_{\rho'}$).

If I remember correctly, a proof of this theorem by Deligne and Sullivan was redone by Eric
Friedlander, in "Étale homotopy of simplicial schemes". Annals of Mathematics Studies, 104. Princeton University Press, 1982. I cannot tell if it is easier to read than the original.

Answer 2

Here's an answer for line bundles. A flat connection on the trivial bundle is the same as a closed one-form $\eta$. The corresponding local system is the sheaf of logarithmic antiderivatives of $\eta$. What is the representation attached to a closed one-form? A loop $\gamma$ acts on a vector $v$ via multiplication by $\exp(\int_\gamma \eta)$. So a $\pi_1$ character gives rise to the trivial line bundle (with possibly non-trivial flat connection) iff there is a closed one-form $\eta$ on $X$ such that the character is given by integrating as above, and two arbitrary $\pi_1$ characters give rise to isomorphic line bundles (with possibly different flat connection) iff their quotient gives rise to the trivial line bundle.

Answer 3

Clearly one must have

$$c_k(E_\rho)=c_k(E_{\rho'} ), $$

for any $k\leq r$. When $r=1$ this is a necessary and sufficient condition. This raises an even more concrete question, namely how does one compute the Chern classes of $E_\rho$ which are obviously torsion classes.