# Topological finiteness of etale fundamental group for arithmetic schemes

By comparison theorem and finite CW complex structure we know every complex variety's etale fundamental group is topological finitely generated (see here). It seems natural to ask similiar things for arithmetic schemes (like Bass conjecture for algebraic $$K$$-groups).

Question: assume $$X$$ is a irreducible (hence connected) scheme finite type over $$\Bbb Z$$ and the structure morphism $$X \rightarrow \operatorname{Spec} \Bbb Z$$ has dense image (to avoid the wild part), then is $$\pi_1^{et}(X)$$ topologically finitely generated? What about higher homotopy groups (defined by Artin and Mazur)? If there are counterexamples, could we avoid them by more restricted condition like smoothness or restriction on $$X_{\Bbb C}$$?

Example: $$\operatorname{Spec} \Bbb Z$$ (by Minkowski theorem and duality theorem, see What are the higher homotopy groups of Spec Z ?)

Q1: Is this question related to some conjectures or false for some known counterexamples? What cases are known at present?

Q2: Is it true for $$X=\operatorname{Spec} O_{K,S}$$? Here $$K$$ is a number field and $$S$$ consists of finite many finite places of $$K$$. In other words, is $$\operatorname{Gal}(K_S/K)$$ topologically finitely generated where $$K_S$$ is the maximal extension of $$K$$ unramified outside S? It has topologically finite generated abelization by class field theory, and is type $$(F)$$ (defined in Serre's Galois cohomology) i.e has only finite many open subgroups with given index by Minkowski theorem. I see the claim in a note (last paragraph in 1.2) but without proof. More generally, as Bass conjecture is true for $$\text{dim} \leq 1$$ proven by Quillen maybe the question is at least true for $$\text{dim}X \leq 1$$ ?

Q3: Could we make some reductions? For example, we can assume $$X$$ is reduced hence integral as the etale site does not change (are higher homotopy groups also unchanged?). Could we reduce to the case $$X$$ is affine? Could we reduce to the normal case? and even smooth case by some purity results? If it's true for $$X$$, is it true for $$X \times \Bbb A^1$$ ? For instance, is it true for $$X=\Bbb{Z}[x_1,\dots,x_n]$$ ?

Q4: There is a motivic spectral sequence connecting motivic cohomology and algebraic $$K$$ groups in the complex variety case (like Atiyah–Hirzebruch spectral sequence), so could we reduce the problem to finiteness of some cohomology groups? (see Finiteness of etale cohomology for arithmetic schemes)

Edit: As the comment points out, Shinya Harada and Toshiro Hiranouchi proved that smallness of fundamental groups for arithmetic schemes in this paper. And there is also a generalization to algebraic stack proper etale over the base arithmetic scheme, see this paper.

• Just a comment on Q2: the fact that $\pi_1$ contains only finitely many finite index subgroups of bounded index is true for any arithmetic scheme; see arxiv.org/pdf/0803.2096.pdf – Ariyan Javanpeykar May 19 '18 at 9:28
• The accepted answer to mathoverflow.net/questions/65996/… claims the finite generation of $\pi_1$ is not known already for an open subset of ${\rm Spec}(Z)$. – Piotr Achinger May 19 '18 at 9:28
• Contrary to the finite generation of the étale fundamental groups of open subschemes of curves over finite fields, it is unknown for $O_{K,S}$ for $K$ any number field, see [Neukirch-Schmidt-Wingberg] (they say that they even don't have a conjecture if it is true or false!). For the abelianisation, one can use class field theory, and for more general schemes, there is the Katz-Lang finiteness theorem. – TKe May 19 '18 at 15:09
• For the abelianised tame fundamental group, there is Schmidt Theorem 3.1 mathi.uni-heidelberg.de/~schmidt/papers/tame.pdf – TKe May 19 '18 at 15:14
• The topological finite generation of $G_{K,S}$ is an old conjecture of Shafarevich, though as TKe writes not everyone expects it to be true. – Daniel Litt May 19 '18 at 19:58