Covering Spaces

Path lifting, the degree, and computing the fundamental group of the circle.

The idea

A covering space of $B$ unrolls $B$: it is a larger space lying above $B$ in which a loop of $B$ need not close up, so that a loop's failure to shrink becomes a measurable displacement rather than a claim to be checked against every possible deformation. Covering spaces are the standard tool for computing fundamental groups.

A covering map is a continuous surjection $p \colon E \to B$ such that every point of $B$ has an open neighbourhood $U$ whose preimage $p^{-1}(U)$ is a disjoint union of open sheets, each carried homeomorphically onto $U$ by $p$. Locally, $E$ consists of identical copies of $B$ stacked above it; globally it can be a different space altogether. The model case wraps the real line around the circle: $p \colon \mathbb{R} \to S^{1}, \qquad p(t) = (\cos 2\pi t,\ \sin 2\pi t).$ As the figure shows, the line then sits over the circle as an infinite spiral, and the fibre $p^{-1}(x_{0})$ over a point $x_{0}$ of the circle is a copy of the integers, one point per turn.

The key property is lifting: given a path in $B$ and a starting point in the fiber over its initial point, exactly one path in $E$ lies over it. The uniqueness is what makes lifts informative. A loop in $B$ lifts to a path in $E$ that need not close up, and where the lift ends is forced by the loop rather than chosen — so the endpoint records genuine information about the loop, such as how many times it winds.

Ways to work on it

• Walkthrough. Covering maps, unique path lifting, and _1(S^1) ℤ by degree.
• Practice. Lift a loop on the circle and read off its winding number.
• Hardest. Count the sheets of a cover via the index of the image subgroup.

Not sure where to start? Take the ten-question placement test.