Covering Spaces and Vector Bundles

Question

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?

Answer 1

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.

Answer 2

The vanishing of $c_1(E)$ is not sufficient. Take the case $\deg(f)=2$. Then $Y$ is a $\mathbb{Z}/2$-bundle over $X$, which gives via the homomorphism $\mathbb{Z}/2\rightarrow \mathbb{C}^*$ a complex line bundle $L$ with $L\otimes L\cong \mathbb{1}$; your $E$ is $\mathbb{1}\oplus L$. Conversely, any vector bundle $\mathbb{1}\oplus L$ on $X$ with $L\otimes L\cong \mathbb{1}$ comes from a 2-sheeted covering. If $M$ is any nontrivial line bundle on $X$, then $M\oplus M^*$ is a rank 2 vector bundle with $c_1=0$ which does not come from a covering.