what does it mean that something fibers?

Question

For example, in an article I have found that "compact abelian group which fibers over the circle $S^1$ [...]" and surely I have heard that phrase before.

What does it mean?

Answer 1

There are various kinds of maps $f: E \to B$ called fiber bundles. They occur in topology and algebraic geometry (I'm guessing on that last one). Saying that $X$ fibers over $B$ usually means that there exists a bundle $f:X \to B$.

Topologically, a fiber bundle $f: E \to B$ with fiber $F$ is a continuous map of spaces with the following long-winded property: to each point $b \in B$ there is an open set $U$ about $b$ such that $f^{-1} (U) \cong U \times F$ canonically through projections. Some people might impose additional conditions on connectedness, smoothness, the structure of $F$, etc. This all depends on the needs and goals. For example, you may want everything to be a topological group, which sounds like your situation.

The standard first examples are (a) product projections $p: B \times F \to B$ and (b) the exponential cover $p: \mathbb{R} \to S^1$ by $p(t) = e^{2\pi i t}$.