Fundamental group

Mathematical group of the homotopy classes of loops in a topological space

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups.

Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point—paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.

Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs".[1] The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.

A loop on a 2-sphere (the surface of a ball) being contracted to a point

More generally, the fundamental group of a bouquet of r circles is the free group on r letters.

The fundamental group of a wedge sum of two path connected spaces X and Y can be computed as the free product of the individual fundamental groups:

This generalizes the above observations since the figure eight is the wedge sum of two circles.

The fundamental group of the plane punctured at n points is also the free group with n generators. The i-th generator is the class of the loop that goes around the i-th puncture without going around any other punctures.

The fundamental group can be defined for discrete structures too. In particular, consider a connected graph G = (V, E), with a designated vertex v0 in V. The loops in G are the cycles that start and end at v0.[4] Let T be a spanning tree of G. Every simple loop in G contains exactly one edge in E \ T; every loop in G is a concatenation of such simple loops. Therefore, the fundamental group of a graph is a free group, in which the number of generators is exactly the number of edges in E \ T. This number equals |E| − |V| + 1.[5]

For example, suppose G has 16 vertices arranged in 4 rows of 4 vertices each, with edges connecting vertices that are adjacent horizontally or vertically. Then G has 24 edges overall, and the number of edges in each spanning tree is 16 − 1 = 15, so the fundamental group of G is the free group with 9 generators.[6] Note that G has 9 "holes", similarly to a bouquet of 9 circles, which has the same fundamental group.

The fundamental group of a genus n orientable surface can be computed in terms of generators and relations as

This includes the torus, being the case of genus 1, whose fundamental group is

This mapping from continuous maps to group homomorphisms is compatible with composition of maps and identity morphisms. In the parlance of category theory, the formation of associating to a topological space its fundamental group is therefore a functor

is a homotopy equivalence and therefore yields an isomorphism of their fundamental groups.

The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of algebraic topology.

The abelianization of the fundamental group can be identified with the first homology group of the space.

In the parlance of category theory, the theorem can be concisely stated by saying that the fundamental group functor takes pushouts (in the category of topological spaces) along inclusions to pushouts (in the category of groups).[11]

is called a covering or E is called a covering space of B if every point b in B admits an open neighborhood U such that there is a homeomorphism between the preimage of U and a disjoint union of copies of U (indexed by some set I),

A covering is called a universal covering if E is, in addition to the preceding condition, simply connected.[13] It is universal in the sense that all other coverings can be constructed by suitably identifying points in E. Knowing a universal covering

The quotient of an action of a (discrete) group G on a simply connected space Y has fundamental group

Let G be a connected, simply connected compact Lie group, for example, the special unitary group SU(n), and let Γ be a finite subgroup of G. Then the homogeneous space X = G/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space G. Among the many variants of this construction, one of the most important is given by locally symmetric spaces X = Γ\G/K, where

In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups).

As an example take G = SL(2, R), K = SO(2) and Γ any torsion-free congruence subgroup of the modular group SL(2, Z).

From the explicit realization, it also follows that the universal covering space of a path connected topological group H is again a path connected topological group G. Moreover, the covering map is a continuous open homomorphism of G onto H with kernel Γ, a closed discrete normal subgroup of G:

Since G is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of G. In particular π1(H) = Γ is an abelian group; this can also easily be seen directly without using covering spaces. The group G is called the universal covering group of H.

As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism

which generalizes the above fact about the universal covering (which amounts to the case where the fiber F is also discrete). If instead F happens to be connected and simply connected, it reduces to an isomorphism

The fundamental group of noncompact Lie groups can be reduced to the compact case, since such a group is homotopic to its maximal compact subgroup.[19] These methods give the following results:[20]

can be shown to be surjective[21] with kernel given by the set I of integer linear combination of coroots. This leads to the computation

When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.

If X is a connected simplicial complex, an edge-path in X is defined to be a chain of vertices connected by edges in X. Two edge-paths are said to be edge-equivalent if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in X. If v is a fixed vertex in X, an edge-loop at v is an edge-path starting and ending at v. The edge-path group E(Xv) is defined to be the set of edge-equivalence classes of edge-loops at v, with product and inverse defined by concatenation and reversal of edge-loops.

The edge-path group is naturally isomorphic to π1(|X|, v), the fundamental group of the geometric realisation |X| of X.[24] Since it depends only on the 2-skeleton X2 of X (that is, the vertices, edges, and triangles of X), the groups π1(|X|,v) and π1(|X2|, v) are isomorphic.

The edge-path group can be described explicitly in terms of generators and relations. If T is a maximal spanning tree in the 1-skeleton of X, then E(Xv) is canonically isomorphic to the group with generators (the oriented edge-paths of X not occurring in T) and relations (the edge-equivalences corresponding to triangles in X). A similar result holds if T is replaced by any simply connected—in particular contractible—subcomplex of X. This often gives a practical way of computing fundamental groups and can be used to show that every finitely presented group arises as the fundamental group of a finite simplicial complex. It is also one of the classical methods used for topological surfaces, which are classified by their fundamental groups.

The universal covering space of a finite connected simplicial complex X can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (w,γ) where w is a vertex of X and γ is an edge-equivalence class of paths from v to w. The k-simplices containing (w,γ) correspond naturally to the k-simplices containing w. Each new vertex u of the k-simplex gives an edge wu and hence, by concatenation, a new path γu from v to u. The points (w,γ) and (u, γu) are the vertices of the "transported" simplex in the universal covering space. The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X.

It is well known that this method can also be used to compute the fundamental group of an arbitrary topological space. This was doubtless known to Eduard Čech and Jean Leray and explicitly appeared as a remark in a paper by André Weil;[25] various other authors such as Lorenzo Calabi, Wu Wen-tsün, and Nodar Berikashvili have also published proofs. In the simplest case of a compact space X with a finite open covering in which all non-empty finite intersections of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the nerve of the covering.

This yields a theory applicable in situation where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a finite field. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties X over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.[35]

The fundamental group of a root system is defined, in analogy to the computation for Lie groups.[36] This allows to define and use the fundamental group of a semisimple linear algebraic group G, which is a useful basic tool in the classification of linear algebraic groups.[37]