I've heard that many problems may be simplified when looking at covering spaces, but I haven't been able to find a good list. What are the most common covering spaces one should understand by heart? The only one I really know is $SU(2)$ covering $SO(3)$.

Are there any slick examples of problems that are greatly simplified when working with covering spaces? (Computing fundamental groups, for example)

  • $\begingroup$ Given a space, $X$, there are many covering spaces of $X$. But they are dependent on $X$, so it it does not make sense to ask for covering spaces without giving the space in question. $\endgroup$ – N. Owad Dec 4 '13 at 23:38
  • $\begingroup$ I'm looking for a list of spaces $X$ with useful coverings. $\endgroup$ – user801923 Dec 4 '13 at 23:52
  • $\begingroup$ You should of course know $\mathbb{R} \to S^1$! Actually it seems that this and $\text{SU}(2) \to \text{SO}(3)$ are the only examples Wikipedia gives which is a bit bizarre. $\endgroup$ – Qiaochu Yuan Dec 5 '13 at 0:08
  • 2
    $\begingroup$ I wrote down some examples here (most of the links don't work but you can google the keywords): qchu.tiddlyspace.com/#%5B%5BCovering%20space%5D%5D $\endgroup$ – Qiaochu Yuan Dec 5 '13 at 0:58

Some examples

You can make a lot of covering spaces from one. If $F$ is a discrete space, then the projection $X \times F \to X$ is called a trivial covering. It's the most uninteresting covering, but "locally" a covering is nothing but a trivial cover.

The restriction of a covering to a saturated subset is a still a covering.

If $p_1 : Y_1 \to X_1$ and $p_2 : Y_2 \to X_2$ are covering map, then $$p_1 \times p_2 : Y_1 \times Y_2 \to X_1 \times X_2$$ is a covering space. For example $\mathbb R^n$ is a covering of the $n$-torus $\mathbb T^n = \mathbb S^1 \times \dots \times \mathbb S^1$ (just take $n$ product of the covering $p : \mathbb R \to \mathbb S^1$). Another example of a 2-sheeted covering is given by the canonical map : $p : \mathbb S^n \to \mathbb {RP}^n$.

Complex Analysis

Three useful covering spaces are given by $\mathbb C$, $\mathbb H$ and $\mathbb P^1$ (the Riemann sphere). Why are they useful ? (and what are they covering?) A bit of general theory : If $M$ is a manifold, then $M$ admit a universal covering $\tilde M$ and moreover $$M \cong \tilde M / \pi_1(M)$$ (here $\tilde M / \pi_1(M)$ is the space of orbit by the action of $\pi_1$ on $\tilde M$. See the book of Lee for more details for examples. ) Back to Riemann surface. Pick one, $X$. By the Riemann's Uniformization Theorem, the universal covering of $X$ is given by the sphere, the disk or the complex plane. Then, since every manifolds can be expressed as a quotient of their universal covering spaces, we obtain the following classification of the Riemann surfaces in function of their universal covering $\tilde X$ :

  1. $\tilde X = \mathbb P^1$, and then in fact $X = \mathbb P^1$. We have $g = 0$ (where $g$ is the genus of $X$)
  2. $\tilde X = \mathbb C$ and then $X$ is a complex torus, i.e a quotient $\mathbb C/\Gamma$ where $\Gamma$ is a lattice in $\mathbb C$. And then $g = 1$
  3. Else $\tilde X = \mathbb D$ and then $g \geq 2$. Then $X$ is isomorphic to a quotient of the Poincaré plane $\mathbb H / \Gamma$ where $\Gamma$ is a fuschian group.

Now, assume you have an non-constant holomorphic map $f : Y \to X$ between compact Riemann surfaces. Then, $f$ is "almost" a covering map (!) : their is two discrete subsets $A \subset Y, B \subset X$ such that $$f_{Y \backslash A} : Y \backslash A \to X \backslash B$$ is a covering map. It is obviously false for non-compact Riemann surface : take the inclusion $\Omega \hookrightarrow \mathbb C $ which is not a covering map (not surjective).

Other Areas

There are lot of other applications and example of covering spaces. You can look at the link between covering spaces and graphs, explained in the correspondant section of the book of Hatcher, "Algebraic Topology".

Some nice actions of discrete groups give covering space : $X \to X/G$ on the orbit space. (Example : Take the group of translation by integers $\mathbb Z$ acting on $\mathbb R$ and see if it sounds familiar to you !). Lee does it in its book "Introduction to Topological Manifolds" and you have in corollary the classification of the covering of the torus, or classification of universal covering space of surface.

Finally, a natural generalisation to covering spaces is the notion of fibration, which is roughtly speaking a "covering space without discrete fiber".

For more informations about covering spaces and group actions you can look the book "Topology" of Munkres, or "Introduction to Topological Manifolds" by John Lee. For the covering spaces and the Riemann surfaces, Forster is a good reference, "Lectures on Riemann Surfaces".


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