I want to understand the following sentence:
Let X a compact (complex) manifold which has a non-zero cohomology class $\alpha \in H^1(X,\mathbb Z)$. Let $\pi: \bar X\to X$ be the corresponding infinite cyclic covering.
What does this mean? It seems that an infinite cyclic covering is a cover with fiber $\mathbb Z$.
But why does such a covering exist, and how is it related to the cohomology class?