Motivation of the fundamental theorem of covering spaces

Question

The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "Grothendieck's Galois theory" states that is is even true for $X$ a connected scheme, if we replace $\pi_1(X)$ be its étale analogue, covering spaces by finite étale covers, and left $\pi_1(X)$-sets by finite continuous left $\pi_1(X)$-sets.

Certainly these are nice statements, because they both show that two a priori different stories ($G$-sets vs. coverings) turn out to be the same. However, I keep wondering:

QUESTION: What is the actual motivation people (could) came up with these statements? This could be answered by: What are some applications of these theorems?

A more down-to-earth formulation used in algebraic topology courses is the "local" version using posets instead of categories (isomorphism classes of covering spaces $\cong$ conjugacy classes of subgroups of $\pi_1(X)$), as discussed in Hatcher's book Theorem 1.38. I skimmed through this section of Hatcher's book but I can't find any concrete motivation except "here's a nice theorem / classification result: ...".

Answer 1

Many results in algebraic topology are proved using an argument along the following lines. Suppose such and such holds. Then there is a subgroup of the fundamental group with the following properties. Pass to the corresponding covering space and do something. Contradiction. For example, Miles Reid has a nice argument along these lines which shows that the etale fundamental group of a Godeaux surface has size at most 5. See theorem 0.2.1 of his paper Godeaux and Campedelli surfaces: https://homepages.warwick.ac.uk/~masda/surf/more/Godeaux.pdf

Answer 2

The key motivation for the étale version is that it gives us a definition of the étale fundamental group. In other words, one can show that there exists a unique profinite group satisfying that theorem, and define $\pi_1(X)$ to be that group.

Then you know you have the right definition because it's the unique group that satisfies the same property as the original topological fundamental group.

Furthermore, it gives you a roadmap for proving things about the étale fundamental group. If you want to show that, for example, $\pi_1(X) \to \pi_1(Y)$ is surjective for a map of varieties $X \to Y$, you figure out an equivalent property of $\pi_1$-sets (i.e. that for every finite set with a transitive continuous action of $\pi_1(Y)$, the induced action of $\pi_1(X)$ is transitive) and then the corresponding property for étale covers (i.e. that every finite étale cover of $Y$ that is connected pulls back to a connected cover of $X$), which you can check by geometric means, say, in the case $X \to Y$ is an open immersion of normal varieties.

The motivation for the property in the topological setting is a little different. There, the definition in terms of paths is usually easiest to work with. Instead, this theorem is often useful for understanding covering spaces, once you already understand the fundamental group. For example, it gives you an algebraic way to compute how many covering spaces of a particular degree a manifold has, or to find covering spaces with a large automorphism group.

But I think even before Grothendieck people understood there was an analogy between the topological fundamental group and the Galois group, and so another motivation is that it gives a precise statement of the analogy - the topological fundamental group classifies coverings in the same way the Galois group classifies field extensions.

Answer 3

A more topological reason why this result is interesting (although I agree with Will Sawin's answer about Galois theory) is simply that it lets you compute $\pi_1$'s !

Here's an example: it's very easy to prove that higher dimensional spheres $S^n, n\geq 2$ have $\pi_1(S^n)= 1$, and it follows immediately from that and the fact that there is only one group of order $2$, that $\pi_1(\mathbb RP^n)\cong \mathbb Z/2$.

With a bit more work, one can prove using covering theory that $\pi_1(S^1)\cong \mathbb Z$, which you could prove using a groupoid version of van Kampen's theorem, but not using the classical version.

In other words, this kind of result + the knowledge that you can recover $\pi_1$ from the category of $\pi_1$-sets tells you that if you understand the geometry of $X$, you can compute $\pi_1(X)$.

It also tells you a second thing, that might be obvious in hindsight, that the covering theory of a (nice) space $X$ is entirely homotopical. The definition of a covering space uses the topology of $X$, and it is not obvious at first sight that this should only depend on the homotopy type of $X$, but this result tells you that.

A final reason I'll give, is that it gives you an early view of a fundamental insight, which is that one way to describe systems parametrized by $X$ is as things over $X$ (or conversely, that one way to understand things over $X$ is as things parametrized by $X$). This insight is present in many more places in algebraic topology and geometry (vector bundles being another very famous instance)