TITLE: Covering Spaces and Vector Bundles
QUESTION [12 upvotes]: Suppose $f: Y \rightarrow X$ is a covering map between compact Hausdorff spaces $X$ and $Y$.  Then $f$ induces a algebra homomorphism $f^*:C(X) \rightarrow C(Y)$ and gives $C(Y)$ the structure of a finitely generated projective module over $C(X)$.  Hence $C(Y)$ can be viewed as the sections of a complex vector bundle $E \rightarrow X$.
I am interested in the converse.  Given a complex vector bundle $E \rightarrow X$, when is $E$ defined from a covering map as above?  In particular, is this the case if the first Chern character $c_1(E) \in \check{H}^2(X ; \mathbb{Z})$ vanishes?  Is there always a complex line bundle $L \rightarrow X$ such that $E \otimes L$ is defined by a covering map as above?

REPLY [10 votes]: The category of flat vector bundles is equivalent to the category of local systems, see for instance this MO-question, which in turn are equivalent to representations of the fundamental group, see this MO-question. Via these correspondences, the vector bundle associated to the covering corresponds to the representation of the fundamental group of $X$ on the fibre of $f:Y\to X$ via deck transformations. In particular, the vector bundles associated to coverings are flat. 
A flat vector bundle has trivial characteristic classes, but triviality of characteristic classes is not sufficient for flatness, as the answer of abx shows.