# Motivation of the fundamental theorem of covering spaces

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: ...".

• See here for some applications of etale fundamental groups: mathoverflow.net/q/386011/30186 May 9, 2022 at 15:44
• "Motivation" means a couple different things and it would help to know more precisely what you mean. Do you want an explanation of why we should expect that such a result is true, or why we should want to prove such a result and find it useful? Another question - are you familiar with the analogous statements in classical Galois theory (inverse isomorphism of posets between subgroups of the Galois group and field extensions, or a similar equivalence of categories)? May 9, 2022 at 16:10
• I"m not convinced that conjugacy classes of subgroups of $G$ are "more down-to-earth" than left $G$-sets. Subgroups of $G$ are just the stabilizers of points in $G$-sets, and two subgroups are conjugate iff they're the stabilizers of two points in the same orbit. May 9, 2022 at 17:09
• I am not a historian, but I think part of the original motivation came from Riemann and analytic continuation. The analytic continuation of say, the logarithm comes equipped with an obvious covering map. Anyone could observe that the homotopy class of a loop in $\mathbb{C}^\times$ corresponds uniquely to a numerical invariant (the index of the curve at 0), to a preimage of the basepoint via the cover, and ultimately to a deck transformation. One could also observe that the analytic continuation of the square root has a smaller deck group. So the fundamental theory is there.
– Bma
May 9, 2022 at 19:14
• It’s also worth noting that the covering theory of compact Riemann surfaces corresponds directly to the Galois theory of their function fields. If $\pi: X \to Y$ is a holomorphic cover, the pullback of the function field of $K(Y)$ via $\pi$ makes $K(X)$ a finite degree field extension, and the automorphism group of the extension is the group of deck transformations, with intermediate fields corresponding to intermediate covers (at least when Galois). I do not know who first observed this, but it seems like Riemann would have had the tools to do so. See Förster’s textbook for details.
– Bma
May 9, 2022 at 19:53

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

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.

• "one can show that there exists a unique profinite group satisfying that theorem" is that related to the statement that universal properties describe an object up to iso? May 11, 2022 at 12:42
• "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)" why is that property the corresponding property? in general, if we have an equivalence of categories we can only translate back and forth between categorical properties of these categories (which can be formulated just using objects, morphisms, equality, and composition). But your property uses "connected". May 11, 2022 at 12:44
• @user481980 Yes, they're related. If this holds for a group $G$, then $G$ maps to the automorphism group of every finite étale Galois cover, which are compatible with the maps between finite étale Galois covers, and satisfy a universal property, making $G$ the inverse limit of the automorphism groups of the finite étale Galois covers. This summary is very rough - we need to fix a base point for it to work. May 11, 2022 at 12:54
• @user481980 "Connected" Is a categorical property in this context because it means "not the coproduct of two nontrivial finite étale covers" since coproduct = disjoint union. May 11, 2022 at 12:55
• "if this holds for a group G" - if what holds for a group G? May 11, 2022 at 13:13

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)

• How do you calculate $\pi_1(S^1)$ using the fundamental theorem of covering spaces? If by "the knowledge that you can recover π1 from the category of π1-sets" you mean that whenever G-set and H-set are equivalent categories, then G is isomorphic to H -- so can we calculate $\pi_1(S^1)$ by proving that the category of $\pi_1(S^1)$-sets is equivalent to the category of $\mathbb Z$-sets, is that what you are saying? Jun 2, 2022 at 15:29
• @user483320 : yes, exactly ! And by covering theory, the category of $\pi_1(S^1)$-sets is equivalent to the category of covers of $S^1$, which is something you can access geometrically (for instance by finding the universal cover, $\mathbb R$, or by using $S^1= [0,1]/(0=1)$ Jun 2, 2022 at 15:33
• Thank you very much! Jun 2, 2022 at 15:36