Connection Between Étale Fundamental Group and Topological Fundamental Group

Question

I have some problems with the understanding of the theory of fundamental groups. I hope everything I write down here is understandable and correctly written.

Let $X$ be a (smooth) complex algebraic variety. Realize the variety as a complex manifold $X(\mathbb{C})$. This question is concerned about the connection between the étale fundamental group $$\pi_1^{\text{ét}}(X,x)$$ and the topological fundamental group $$\pi_1(X(\mathbb{C}),x).$$ More precisely, I would like to understand how I can describe the étale fundamental group by using the classical fundamental group - which should be, if I'm not mistaken; $$ \pi_1^{\text{ét}}(X,x)=\widehat{\pi_1(X(\mathbb{C}),x)}, $$ where the right hand side denotes the profinite completion of the topological fundamental group.

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There are a few facts I have proven. Let me begin by mentioning them.

Proven Results

Statement 1. If we let $X$ denote a topological space with base point $x$, $\mathbf{Cov}(X)$ denote the category of covering spaces over $X$, and $\pi_1(X,x)-\mathbf{Sets}$ be the category of sets equipped by a group action by the fundamental group. Then we have the following equivalence of categories $$ \mathbf{Cov}(X)\simeq \pi_1(X,x)-\mathbf{Sets} $$ Also, I have proven the following two statements:

Statement 2. If $\mathbf{G}$ is a group considered as a category and $P_{\mathbf{G}}:\mathbf{G}-\mathbf{Sets}\to\mathbf{Sets}$ is the forgetful functor, then $$ \mathbf{G}\cong \operatorname{Aut}(P_{\mathbf{G}}), $$ where $\operatorname{Aut}(P_{\mathbf{G}})$ is the automorphism group of $P_{\mathbf{G}}$.

Statement 3. If $\mathbf{\hat{G}}$ denotes the profinite completion of $\mathbf{G}$ and $\hat{P}_{\mathbf{G}}:\mathbf{G}-\mathbf{FinSets}\to\mathbf{FinSets}$ is the forgetful functor from the category of profinite completion actions on finite sets to the category of finite sets, then $$ \mathbf{\hat{G}}\cong\operatorname{Aut}(\hat{P}_{\mathbf{G}}). $$ Lastly, we also have the following theorem called The Riemann Existence Theorem, which can be found in Lectures on Étale Cohomology by James Milne (see page 28):

Riemann Existence Theorem. Let $X$ be a nonsingular variety over $\mathbb{C}$. The functor sending a finite étale covering $(Y,\pi)$ of $X$ to the finite covering space $(Y(\mathbb{C}),\pi)$ of $X(\mathbb{C})$ is an equivalence of categories.

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A Quest to Prove the Isomorphism

I had a discussion with a professor (not too long ago) over Zoom, he told me that the above is enough to find the connection between the fundamental groups. But I am not exactly sure how.

In the notes, this is how professor Milne gave the proof

From the Riemann Existence Theorem it follows that the étale universal covering space $\widetilde{X}=(X_i)_{i\in I}$ of $X$ has the property that every finite topological covering space of $X(\mathbb{C})$ is a quotient of some $X_i(\mathbb{C})$. Let $x$ any element of $X(\mathbb{C})$. Then

$$\pi_1^{\text{ét}}(X,x)=\varprojlim_i\operatorname{Aut}_X(X_i)=\varprojlim_i\operatorname{Aut}_{X(\mathbb{C})}(X_i(\mathbb{C})) =\widehat{\pi_1(X(\mathbb{C}),x)}.$$

I cannot see how anything I have proven comes into the picture in the above quotation.

Questions

Question 1. To use the results I have proven I guess I would like to put $\mathbf{G}=\pi_1(X(\mathbb{C}),x)$ in Statement 2 and Statement 3? This gives us $$ \pi_1(X(\mathbb{C}),x)\cong\operatorname{Aut}(P_{\mathbf{G}}) \\ \widehat{\pi_1(X(\mathbb{C}),x)}\cong \operatorname{Aut}(\hat{P}_{\mathbf{G}}) $$ where do I go from where?

Question 2. To make The Riemann Existence Theorem more compact, let $\mathbf{FinÉtCov}(X)$ denote the category of finite étale coverings of $X$ and $\mathbf{FinCov}(X(\mathbb{C}))$ denote the finite covers of the topological space. Then the theorem says $$ (1)\text{ }\text{ }\text{ }\mathbf{FinÉtCov}(X)\simeq\mathbf{FinCov}(X(\mathbb{C})) $$ As given in Statement 1, I have the following equivalence $$(2)\text{ }\text{ }\text{ }\mathbf{Cov}(X(\mathbb{C}))\simeq \pi_1(X(\mathbb{C}),x)-\mathbf{Sets}.$$ The two equivalence just given are not exactly the same, but "almost". Can I use the above equivalence somehow?

I am quite sure I cannot do it like this, but something similar to $$\mathbf{FinCov}(X(\mathbb{C}))\simeq\pi_1(X(\mathbb{C}),x)-\mathbf{Sets}?$$ I don't think I can use the above because $(1)$ is about finite covers, when $(2)$ is not.

After having done the above, then I could replace the fundamental group, with the automorphism group of the forgetful functor. But this seems just like it is making things much more complicated for me. I'm just trying to use all of the results I have proved.

´Something which bothers me is that I want to prove that two groups are isomorphic. But right now, everything I am doing is at a category-theoretical level - which makes me confused. I want to do things at a group-theoretic level.

Question 3. At the end of the day, I would like someone to help me put everything together. Maybe someone can give me a guiding hand and tell me how this relates to what Milne did? Maybe there are more work to do before I can prove the isomorphism?

I would be really happy if someone could help me get a better understanding of this problem.

Best wishes,

Joel

Answer 1

At some point in the argument you need to introduce a definition of the etale fundamental group. Here is one that will work fine: $\pi_1^{et}(X, x)$ is the automorphism group of the fiber functor

$$P^{et} : \text{FinEtCov}(X) \to \text{Set}$$

given by taking the fiber at the point $x$. The Riemann existence theorem produces an equivalence of categories $\text{FinEtCov}(X) \cong \text{FinCov}(X(\mathbb{C}))$ (this is the "actual hard part" of the argument and everything else is "just abstract nonsense"); importantly, this equivalence is compatible with taking fibers in the sense that there is also an analytic fiber functor

$$P^{an} : \text{FinCov}(X(\mathbb{C})) \to \text{Set}$$

given by taking the fiber at the point $x$ and the equivalence sends the etale fiber functor to the analytic fiber functor. So we get

$$\pi_1^{et}(X, x) \cong \text{Aut}(P^{et}) \cong \text{Aut}(P^{an}).$$

Now how do we compute $\text{Aut}(P^{an})$? We start from the fact that the category of covering spaces of $X(\mathbb{C})$ is equivalent to the category of actions of the topological fundamental group $\pi_1(X(\mathbb{C}), x)$ on sets. It follows that $\text{FinCov}(X(\mathbb{C}))$ is equivalent to the category of actions of $\pi_1(X(\mathbb{C}), x)$ on finite sets. Now we need the following:

Theorem: Let $G$ be a group and let $P : G\text{-FinSet} \to \text{Set}$ be the forgetful functor from the category of finite $G$-sets to sets. Then $\text{Aut}(P)$ is naturally isomorphic to the profinite completion $\widehat{G}$.

Proof. Here is a sketch. If we replaced "finite sets" with "sets" above, the automorphism group of the forgetful functor would just be $G$, and this would be because the forgetful functor in this case is representable by $G$ itself (being acted on by left multiplication).

If $G$ is infinite the forgetful functor is no longer representable. However, it can be written as a filtered colimit of representable functors; it is the filtered colimit over the functors represented by the finite quotients $G/G_i$ of $G$ (this is the key step). Together with the Yoneda lemma, it follows that

$$\text{Hom}(P, P) \cong \text{Hom}(\text{colim}_i G/G_i, P) \cong \lim_i \text{Hom}(G/G_i, P) \cong \lim_i P(G/G_i) \cong \lim_i G/G_i$$

and although this is only an identification as sets, as written, you can show that it is also an identification as groups. $\Box$

So we get that $\pi_1^{et}(X, x)$ is the profinite completion $\widehat{\pi_1(X(\mathbb{C}), x)}$ as desired. This isomorphism is, properly understood, exactly equivalent to the Riemann existence theorem, once you internalize the idea that a group $G$ is completely determined by the pair consisting of the category of $G$-sets and its forgetful functor to sets, and similarly a profinite group $G$ is completely determined by the pair consisting of the category of continuous finite $G$-sets and its forgetful functor to (finite) sets.

You can see more examples of these kinds of profinite gadgets appearing as endomorphisms of forgetful functors in my blog post Operations, pro-objects, and Grothendieck's Galois theory which may or may not be helpful reading.