On a course that ended some time ago, I was handed the following problem:

Problem:Compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}, \overline x)$.

Hint:Find all finite algebraic extensions of $\mathbb{C}((t))$.

It's easy to compute the absolute Galois group of $\mathbb{C}((t))$. Also, I know there are at least two different ways to compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \})$:

by using the comparison with the profinite completion of the topological fundamental group,

by using Hurwitz formula (cf. here).

However, I wonder, if it is possible to formally finish this proof using the above hint.

Question:is it possible to show directly that $\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}$ and $\DeclareMathOperator{\Spec}{Spec} \Spec \mathbb{C}((t))$ have the same étale fundamental group?

The intuition is clear: $\DeclareMathOperator{\Spec}{Spec} \Spec \mathbb{C}((t)) = \Spec \mathbb C[[t]] \setminus \{ (t) \}$ is a "formal" punctured disc and thus should "morally" have the same homotopy type as $\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}$. But is it possible to formalize this intuition (possibly even using some higher machinery from étale homotopy theory)? Does it generalize to higher dimensions?

It seems to me that the answer would be quite informative, not only for this problem.

I am asking this question here, since a similar question posted on Stack Exchange did not get any satisfactory (from my point view) answer.

thinkthe hint was intended as: an etale cover of $\mathbb{A}^1 \setminus \{0\}$ restricts to an etale cover of $\mathrm{Spec}(\mathbb{C}((t))) = \mathbb{D}^1 \setminus \{0\}$; now classify those, and show that each one extends uniquely to $\mathbb{A}^1 \setminus \{0\}$. But this is a disappointing interpretation: as you point out, it is more satisfying to have a general criterion that says that the inclusion $\mathbb{D}^k \setminus\{0\} \subset \mathbb{A}^k \setminus\{0\}$ is an etale homotopy equivalence. $\endgroup$uniquely. Besides, exactly as you mentioned, I would be more satisfied with the "étale homotopy type" statement. $\endgroup$in this casethis is valid, it certainly isn't in general, it'd be nice to know when this sort of reasoning can be applied. $\endgroup$2more comments