Covering Space

Definition

A covering space of a topological space $X$ is a continuous surjection $p:E\to X$ such that for every point $x\in X$ there exists an open neighbourhood $U\subseteq X$ with
$p^{-1}(U)\cong {\coprod }_{j\in J}{U}_j,$
and for each component $U_j$ the restriction $p{\mid }_{U_j}:U_j\to U$ is a homeomorphism. Such a $U$ is called evenly covered by $p$.

Equivalently, one often writes $p^{-1}(U)\cong {\coprod }_{j\in J}U$, i.e. a disjoint union of copies of $U$.

Properties

• Local homeomorphism: $p$ is a local homeomorphism with discrete fibres $p^{-1}(x)$.
• Fibre-bundle viewpoint: a covering space is a Fibre Bundle with discrete fibre.
• Path lifting: any path $\gamma :I\to X$ with a chosen lift of its starting point lifts uniquely to a path $\tilde{\gamma }:I\to E$ with $p\circ \tilde{\gamma }=\gamma$.
• Homotopy lifting: homotopies in $X$ lift uniquely relative to a chosen lift of the base.
• Classification (classical): for path-connected, locally path-connected, semilocally simply connected $X$, connected covering spaces correspond to subgroups of ${\pi }_1(X,x)$; deck transformations identify with normalisers, and regular covers correspond to normal subgroups.¹

Examples

• Universal cover of the circle:
  $p:\mathbb{R}\to S^1,t\mapsto e^{2\pi it}.$
  Every sufficiently small arc $U\subseteq S^1$ is evenly covered: $p^{-1}(U)$ is a disjoint union of open intervals, each mapped homeomorphically onto $U$.
• $n$-fold covering of the circle:
  $q:S^1\to S^1,z\mapsto z^n.$
  Each point has neighbourhoods whose preimages are $n$ disjoint arcs mapping homeomorphically.
• Trivial covering: ${\mathrm{pr}}_1:X\times F\to X$ with $F$ a discrete set.

Cover vs covering space

• A cover of X is a family of subsets (often open) whose union is $X$.
• A covering space of $X$ is a single space $E$ with $p:E\to X$ that locally looks like a disjoint union of copies of $X$.

Related Concepts

Cover (Topology)
Fibre Bundle
Topological Fibration
Hurewicz Fibration
Serre Fibration
Fundamental Group
Deck Transformation

References

²³

Footnotes

1. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf ↩

2. https://ncatlab.org/nlab/show/covering+space

3. https://topology.jdabbs.com/ (supplementary notes)