### All these fundamental groups!

Tuesday, April 02nd, 2013 | Author:

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

#### A list of some things that have a fundamental group flavor:

• The usual fundamental group $\pi_1(M,m_0)$ of a topological space $M$ with basepoint $m_0 \in M$.
• The simplicial fundamental group $\pi_1(N,n_0)$ of a simplicial set $N$ with basepoint $n_0 \in N_0$.
• The fundamental group $\pi_1(A,a_0)$ of an object $A$ with basepoint $a_0 \in A$ in a Quillen model category.
• The absolute Galois group $Gal(k^{sep}/k)$ of a field $k$ with separable hull $k^{sep}$.
• The fundamental group of a linear algebraic group: root lattice modulo weight lattice.
• The étale fundamental group $\pi_1^{et}(X)$ of a scheme $X$.
• The fundamental group scheme $\pi_1(X,x_0)$ of a scheme $X$ over a perfect field $k$ with rational basepoint $x_0 \in X(k)$.
• The motivic fundamental group $\pi_1(X,x_0)_M$ of a scheme $X$ over a perfect field $k$ with rational basepoint $x_0 \in X(k)$.
• The Tannakian fundamental group object $\pi_1(\omega)$ of a fiber functor $\omega$ of a Tannakian category.
• The polynomial fundamental group presheaf $\pi_1^{poly}(X,x_0)$ of a scheme $X$ over a field $k$, with rational basepoint $x_0 \in X(k)$.
• The A¹-fundamental group (pre)sheaf $\pi_1^{\mathbb{A}^1}(X,x_0)$ of a scheme $X$ over a field $k$, with rational basepoint $x_0 \in X(k)$.

#### Related concepts and observations:

• For an object $M$, the set of objects of some type over $M$, i.e. the set of certain maps to $M$, is often classified by something like a fundamental group. In the case of the topological fundamental group, the covering spaces are in 1:1 correspondence with subgroups of the fundamental group. In particular, there is one covering space which has a trivial fundamental group itself (it is then called simply connected) and the automorphisms of this covering map form precisely the fundamental group of the base space. A similar covering space theory holds for the absolute Galois group and field extensions instead of coverings.
• There are operations that convert one notion of fundamental group to another. For example the singular set of a topological space is a simplicial set that has as simplicial fundamental group exactly the topological fundamental group of the original space (and the geometric realization goes the other way around).
• For many fundamental-group concepts, there are higher homotopy groups $\pi_n$ and long exact sequences that connect the information of $\pi_1$ with the $\pi_n$. For some fundamental-group concepts, there are no "higher groups" (known so far).
• One may often neglect basepoints, if the resulting groups are isomorphic anyway (under some connectedness assumption). This, however, can be dangerous, as one gets an action of the fundamental group at one basepoint on each other homotopy group, and this action might be non-trivial. Another danger comes from fundamental groups that are not just groups, but also carry some other structure (that of an algebraic variety, or a mixed Hodge structure, for example) -- these additional structures may then depend on the basepoint. If one wants to forget about basepoints, there is the safe way of working with fundamental groupoids instead.
• There are more delicate relations between the various fundamental groups. For a connected projective algebraic variety $X$ over $\mathbb{C}$ one may use an embedding $X \to \mathbb{P}^N$ to put the submanifold topology of the complex manifold $\mathbb{P}^N(\mathbb{C})$ on $X(\mathbb{C})$, call this $X(\mathbb{C})^{an}$ and compare $\pi_1^{et}(X)$ with $\pi_1(X(\mathbb{C})^{an})$. It turns out that the étale fundamental group is the profinite completion of the topological fundamental group. This may be explained by the fact that the étale topology only allows finite covers, so for example $exp : \mathbb{R} \to S^1$ is a topological covering that doesn't come from any étale covering of $\mathbb{C}^\times$.
• Some of the fundamental groups on the list are special cases of others. For example, both the topological and the simplicial fundamental group can be defined in the context of Quillen model categories. This also clarifies the relation between them, since the singular set functor and the geometric realization form what is called a "Quillen adjunction", which axiomatizes this relationship. On the other hand, the Galois group of a field is just the étale fundamental group of the spectrum of that field -- which has no direct connection to model categories at all.
• Well, actually it is a bit more complicated: the precise relation between étale fundamental groups and Galois groups with respect to basepoints is described in this MO answer by Minhyong Kim.
• Some of these fundamental groups are really not very fundamental-group-like: The algebraic fundamental group of a reductive linear algebraic group, as explained on MO by Brian Conrad here, is not the étale fundamental group and its relation to it (and other fundamental groups) can be very loose.

### Systematically:

I claim that there are two or three major approaches to fundamental groups. The first is via Quillen model categories, which captures the intution of loops at some basepoint modulo basepointed homotopy, and which directly gives us higher homotopy groups and long exact sequences. The second is via Tannakian categories, which captures the intuition of being an automorphism group, and which may directly give us some extra structure on the fundamental group.
Grothendieck's Galois Theory of Topoi captures the intuition of monodromy, but essentially is a variation of the Tannakian point of view (in my eyes).

#### Fundamental groups definable in terms of Quillen model categories

The brief definition of the fundamental group of an object $X$ in a (base-pointed) Quillen model category: Take a fibrant replacement $\tilde{X} \to X$ and then $\pi_1(X) := \pi_1(\tilde{X}) := [S^1,\tilde{X}]$ where the brackets mean base-pointed homotopy, i.e. $Map(S^1,\tilde{X})/\sim$ with $\sim$ the (left)homotopy relation. For that to make sense, you obviously need an object $S^1$ in the model category; this can be constructed once you have some kind of interval, for example if the model category is in fact a simplicial model category. Alternatively, without an interval you can still define the loop space via the path space fibration $\Omega X \to P X \to X$ and then take $\pi_1(X) := \pi_0(\Omega X) := \Omega X / \sim$ where $\sim$ is the (left)homotopy relation.

• The usual fundamental group of a topological space comes from the model structure with homotopy equivalences as weak equivalences and Hurewicz fibrations as fibrations (or any other Quillen equivalent model structure).
• The simplicial fundamental group comes from the Kan model structure, with Kan fibrations as fibrations (and the weak equivalences can be defined without reference to fundamental groups).
• The polynomial fundamental group presheaf is a presheaf of simplicial fundamental groups of singular resolutions.
• The A¹-fundamental group (pre)sheaf is the group (pre)sheaf coming out of the Morel-Voevodsky model category.
• Étale fundamental groups may be defined in terms of an étale homotopy category as well, though this is not widely used.

#### Fundamental groups definable in terms of Tannakian categories

The brief definition of the fundamental group of a fiber functor $\omega$ of a Tannakian category over a field $k$ is just $\underline{\mathcal{A}ut}^{\otimes}(\omega)$, the $\otimes$-automorphisms of the functor (which are invertible natural transformations from $\omega$ to itself). This is, in general, a pro-algebraic group scheme over $k$. If the Tannakian category is $\otimes$-generated by a single object $X$, then $\underline{\mathcal{A}ut}^{\otimes}(\omega) \subset \mathcal{A}ut(\omega(X))$.

• The topological fundamental group of a space $X$ with basepoint $x_0 \in X$ is the Automorphism group of the fiber functor from the covering category. The covering category is the Tannakian category of covering spaces $Y \to X$ with lift $y_0$ of the basepoint, and the fiber functor forgets $X$. If there is a single generator of the Tannakian category, i.e. a universal covering, then the Automorphism group of the fiber functor is just the automorphism group of this single covering (where we're talking about automorphisms of the morphism, not just of the space).
• The absolute Galois group of a field $k$ comes out of the category of all separable algebraic field extensions $L/k$, with the fiber functor $\omega : L/k \mapsto L$, since $\mathcal{A}ut(\omega(k^{sep})) = Gal(k^{sep}/k)$.
• The motivic fundamental group of Tate motives over a scheme $S$ is given by the category of mixed Tate motives $MTM(S)$ and the Betti realization. Here, $MTM(k)$ can be constructed as subcategory of $DM_-$ cut out by a t-structure if $k$ is a number field. For rings of integers of number fields, one can give an ad-hoc modification of $MTM(k)$ that gives the right answer. There are also models from rational homotopy theory (see Deligne-Goncharov, Esnault-Levine).

One might even ask, as Harry Gindi did on MO, whether one could define higher homotopy groups in the Tannakian setting. Obviously, one would need higher Tannakian categories.

#### Grothendieck's Galois Theory

Given a Topos $Sh(C)$ of sheaves on some category C, and a point p, we can look at the functor that assigns to each sheaf its stalk at that point. If we take only locally finite, locally constant sheaves (aka local systems), we get a functor to finite sets
$\rho_p : Loc(C) \to FinSet$
which results in an equivalence of $Loc(C)$ to a category of modules under a group, the corresponding Galois group.

Two more fundamental groups:

• The geometric fundamental group of a scheme $X \to k$ over a field $k$ is given as the kernel of the morphism $\pi_1^{et}(X) \to \pi_1^{et}(k) = Gal(k)$. This morphism can be seen to come from a morphism $Sch/X \to Sch/k$. If $k$ is a number field, the geometric fundamental group is the profinite completion of the usual topological fundamental group of the associated complex space.
• The motivic fundamental group of a scheme $X \to k$ is similarly given as the kernel of the morphism $\pi_1(MTM(X)) \to \pi_1(MTM(k))$. There has been some recent research around this object, especially the special case of motivic $\pi_1$ of $\mathbb{P}^1 \setminus \{0,1,\infty\}$. The reason is a deep connection with multiple Zeta values.

Once you have any definition of a fundamental group, it is interesting whether you can find a sheaf topos such that the local systems correspond 1:1 to the representations of this fundamental group. That would be in the spirit of Caramello's unification of mathematics via topos theory.

Category: English, Mathematics