I’m writing my undergrad thesis on coverings and fundamental groups, with a focus on proving the theorem for the existence of the universal covering. For the last chapter, I wanted to provide a few examples, nothing too complicated, of some application of the theory. I’ve read that the Borsuk Ulam theorem for n=2 can be proved with covering theory but Hatcher actually doesn’t really use it much. Is there some theorem/example of calculation in which the theory of coverings/existence of a universal covering is useful/necessary?
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$\begingroup$ Computing invariants such as homotopy groups (including fundamental groups). $\endgroup$– ConCommented Aug 11 at 9:55
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$\begingroup$ But are there examples of computation of fundamental groups using the universal covering that aren’t bouquet of circles or the projective space? $\endgroup$– ccnptrCommented Aug 11 at 10:12
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$\begingroup$ @ccnptr Whenever you have a group $G$ acting on a space $X$ in a "nice" way, then the quotient projection is $X \to X / G$ is a covering space with deck transformation group $G$; this is often useful to construct spaces with prescribed fundamental group and certain other properties (for instance, if $X$ is manifold so is $X / G$, and if $X$ is contractible the result is a $K(G, 1)$). $\endgroup$– Ben SteffanCommented Aug 11 at 12:07
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$\begingroup$ Do you know about the Galois correspondence of coverings of a space and subgroups of the fundamental group? $\endgroup$– Anton OdinaCommented Aug 11 at 12:12
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2$\begingroup$ Here's another two things: First, you shouldn't underestimate the importance of coverings for higher homotopy groups: Since a covering induces an isomorphism on all homotopy groups in degrees $> 1$, you can hope that a covering space has htpy. groups that are easier to understand than the base space (and since homotopy groups are hard to compute whenever this happens there's much rejoicing): For instance, using the universal covering you can show that $\pi_2(S^1 \vee S^2) \cong \bigoplus_{n \in \mathbb{Z}} \mathbb{Z}$. $\endgroup$– Ben SteffanCommented Aug 11 at 12:12
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So you probably know that covering spaces can be used to prove that subgroups of free groups are free, using the fact that covering spaces of graphs are graphs. What is less well-known is that you can also use covering spaces to prove the Kurosh subgroup theorem, that subgroups of a free product $G \ast H$ (let me stick to the case of two groups for simplicity but this implies the case of finitely many groups and the argument generalizes) are free products of free groups and subgroups of $G$ and $H$.
The idea is that $G \ast H$ is the fundamental group of the wedge sum $X \vee Y$ of any two (reasonable) spaces with fundamental groups $G$ and $H$, which you can take to be Eilenberg-MacLane spaces $BG, BH$. Actually it will turn out to be better to take a modified version of the wedge sum where we insert a path between the basepoint of $BG$ and $BH$. Then we can analyze covering spaces of this modified wedge $BG \vee BH$ in terms of coverings of $BG$ and $BH$ with basepoints connected by paths; this works out to being a version of Bass-Serre theory but the upshot in my mind is that you don't need to develop Bass-Serre theory separately, it really follows from a careful investigation of covering spaces. The subgroups of $G$ and $H$ come from nontrivial covers of $BG$ and $BH$ appearing, while the free groups come from nontrivial loops appearing in the resulting "graph of groups." So here we're saying that a covering space of a "graph of groups" is still a graph of groups, where in this case "graph of groups" means a topological graph with the vertices replaced by Eilenberg-MacLane spaces.
You can check out Drawing subgroups of the modular group for an example of this kind of analysis for the modular group $\Gamma = PSL_2(\mathbb{Z})$, which is the free product $C_2 \ast C_3$. Here's an example of a covering of degree $5$ corresponding to a subgroup of index $5$ isomorphic to $C_2 \ast C_3 \ast C_3$:
This analysis implies that finite-index subgroups can only be torsion-free (hence free) if their index is divisible by $6$, which also follows from the fact that the "orbifold Euler characteristic" of $B \Gamma \cong BC_2 \vee BC_3$ is $\frac{1}{2} + \frac{1}{3} - 1 = -\frac{1}{6}$; see this answer for a bit more on this.
answered Aug 11 at 17:27
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Here's some cool things you can do with covering spaces. One is any covering space of a Lie group can be again made into a Lie group. Some naturally arising Lie groups have nontrivial fundamental group, and thus have interesting covers. For example, $SO(n)$ (rigid rotations in $\mathbb{R}^n$ has fundamental group $\mathbb{Z}/2$ for $n\geq 3$ (for $n=3$, you can explicitly show $SO(3)=\mathbb{RP}^3$ by identifying the former with a direction and magnitude of rotation, i.e. the ball of radius $\pi$, but really antipodal points on the boundary sphere should be identified). The double (universal) cover of these groups are called spin groups. This is where "spin" comes from - it's something which is acted on by the spin group and not $SO(n)$ (although they have the same "infinitesimal action"), which is why you need to rotate twice to get to where you started (the identity in the spin group).
Another neat thing is there's a nice relationship with geometry of surfaces. The plane is the universal cover of a torus, and you can consider the torus as being the plane modulo deck transformations of translations, or a fundamental parallelogram with edges identified. This endows the torus with a flat metric (think folding a fundamental domain to get 4 sheets which overlap but slide on top of each other). You can ask whether you can to the same for a genus $g$ surface for $g\geq 2$ (that is, tile the plane with $4g$-gons), and it turns out no. There's a theorem called the Gauss-Bonnet formula which equates the total curvature of a surface with the Euler Characteristic, so since the latter is negative for $g\geq2$, the curvature must be too. However, there is a space with constant negative curvature - the hyperbolic plane, and it turns out that you can tile these with $4g$-gons! Then, deck transformations correspond to "translations along hyperbolic lines" which connect identified sides of a tile. Not purely topological, but a very neat thing that still somewhat relates to covering spaces.
