Universal Cover

The universal cover of a connected topological space  is a simply connected space  with a map  that is a covering map. If  is simply connected, i.e., has a trivial fundamental group, then it is its own universal cover. For instance, the sphere  is its own universal cover. The universal cover is always unique and, under very mild assumptions, always exists. In fact, the universal cover of a topological space  exists iff the space  is connected, locally pathwise-connected, and semilocally simply connected.

Any property of  can be lifted to its universal cover, as long as it is defined locally. Sometimes, the universal covers with special structures can be classified. For example, a Riemannian metric on  defines a metric on its universal cover. If the metric is flat, then its universal cover is Euclidean space. Another example is the complex structure of a Riemann surface , which also lifts to its universal cover. By the uniformization theorem, the only possible universal covers for  are the open unit disk, the complex plane , or the Riemann sphere .

The above left diagram shows the universal cover of the torus, i.e., the plane. A fundamental domain, shaded orange, can be identified with the torus. The real projective plane is the set of lines through the origin, and its universal cover is the sphere, shown in the right figure above. The only nontrivial deck transformation is the antipodal map.

The compact Riemann surfaces with genuses  are -holed tori, and their universal covers are the unit disk. The figure above shows a hyperbolic regular octagon in the disk. With the colored edges identified, it is a fundamental domain for the double torus. Each hole has two loops, and cutting along each loop yields two edges per loop, or eight edges in total. Each loop is also shown in a different color, and arrows are drawn to provide instructions for lining them up. The fundamental domain is in gray and can be identified with the double torus illustrated below. The above animation shows some translations of the fundamental domain by deck transformations, which form a Fuchsian group. They tile the disk by analogy with the square tiling the plane for the square torus.

Although it is difficult to visualize a hyperbolic regular octagon in the disk as a cut-up double torus, the illustration above attempts to portray this. It is unfortunate that no hyperbolic compact manifold with constant negative curvature, can be embedded in . As a result, this picture is not isometric to the hyperbolic regular octagon. However, the generators for the fundamental group are drawn in the same colors, and are examples of so-called cuts of a Riemann surface.

Roughly speaking, the universal cover of a space is obtained by the following procedure. First, the space is cut open to make a simply connected space with edges, which then becomes a fundamental domain, as the double torus is cut to become a hyperbolic octagon or the square torus is cut open to become a square. Then a copy of the fundamental domain is added across an edge. The rule for adding a copy across an edge is that every point has to look the same as the original space, at least nearby. So the copies of the fundamental domain line up along edges which are identified in the original space, but more edges may also line up. Copies of the fundamental domain are added to the resulting space recursively, as long as there remains any edges. The result is a covering map with possibly infinitely many copies of a fundamental domain which is simply connected.

Any other covering map of  is in turn covered by the universal cover of , . In this sense, the universal cover is the largest possible cover. In rigorous language, the universal cover has a universal property. If  is a covering map, then there exists a covering map  such that the composition of  and  is the projection from the universal cover to .