Fundamental group

In algebraic Topology, the fundamental group or group of Poincaré is a central invariant. The fundamental group of a topological Espace pointed is, by definition, the whole of the classes of Homotopie of laces whose Law of composition interns is the Concaténation. The examination of the fundamental groups makes it possible to prove that two spaces cannot be homeomorphic, or topologically equivalents. The fundamental group makes it possible to classify the coatings of a related space by arcs, except for isomorphism.

A generalization of the fundamental groups is the continuation of the higher Groupes of homotopy.

Intuitive definition through the example of the torus

First of all, we with the idea of the fundamental group familiarize through the example of the two-dimensional Tore (which one can represent as being the surface of a Donut). One fixes on the torus a starting point p .

Starting from this point, one can build lace S, i.e of the closed curves, which leave the point p , walk on the torus and which returns to the starting point. Let us imagine that the laces are made starting from rubber and that it is thus possible to stretch them, to deform them in such a way that the starting point and the point of arrival are always p and that the laces always move on the torus. Such a deformation is called a Homotopie: two laces which can be obtained one from the other by a homotopy are known as homotopiquement equivalent. In fact the laces with deformation close interest us: one thus gathers the laces in classes of homotopy. The fundamental group of the torus is thus the whole of the various classes of homotopy of the laces.

In the figure opposite, the laces has and B is not homotopiquement equivalent: one cannot obtain one by continuously deforming the second without “tearing it” at one time, they represent two elements distinct from the fundamental group. One obtains other classes of homotopy while making turn the laces several times around the hole. As its name indicates it, the fundamental group is not a simple unit, it is provided with a structure of group: the Law of composition interns is that which with two laces associates a third lace obtained by traversing the first then the second at the same speed (there are no problems of definition since the laces start and finish with the same point p ). The neutral element of the fundamental group is the class of homotopy of the lace which remains at the point p . One obtains an element reverses by traversing the laces of a class of homotopy in the contrary direction.

Mathematical definition

That is to say X a topological space, and p a point fixed in X . A continuous arc is a continuous application $\backslash \; gamma:\; \backslash \; to\; X$ and a lace based in p is a continuous arc checking moreover $\backslash \; gamma\; (0)\; =\; \backslash \; gamma\; (1)\; =\; p$.

Two laces $\backslash \; gamma\_0$ and $\backslash \; gamma\_1$ are known as homotopic if there exists a homotopy of worms the other, i.e. an application continues $H:\; ^2\; \backslash \; to\; X$ such as:

• $\backslash \; forall\; T\; \backslash \; in,\; \backslash ,\; H\; (T,\; 0)\; =\; \backslash \; gamma\_0\; (T)$

• $\backslash \; forall\; T\; \backslash \; in,\; \backslash ,\; H\; (T,\; 1)\; =\; \backslash \; gamma\_1\; (T)$
• $\backslash \; forall\; X\; \backslash \; in,\; \backslash ,\; H\; (0,\; X)\; =\; H\; (1,\; X)\; =\; p$

The fact of being homotopic is a Relation of equivalence between laces. One can thus consider the whole of the laces of X quotient unit by homotopy. One will note $$ the class of equivalence of the lace $\backslash \; gamma$ (also called class of homotopy).

One wants to now give a structure of group to this unit. If F and G is two laces of X, their concatenation is the lace H defined by:

Intuitively, it is the lace which traverses F then G (each one at double speed, to manage to traverse the lace in a time unit). One will note $f*g$ concaténé of F and G . One can prove that $f*g$ depends neither on the class of homotopy of F , nor of that of G . Thus, one can define an internal law on the whole of the classes of homotopy of the laces of X , by $*=$.

One can then prove that one obtains then a structure of group: the neutral is the commonplace lace $$ defined by $\backslash \; gamma\; (T)\; =p$ for all T . The reverse of a lace F is simply the same lace, but traversed in the other direction (i.e., defined by $f^\; \{-\; 1\}\; (T)\; =f\; (1-t)$)

The group thus obtained is called fundamental group (or group of Poincaré ) of X based in p , and is noted $\backslash \; pi\_1\; (X,\; p)$.

Examples

• the fundamental group (based in any point) of $\backslash \; mathbb\; R^n$, (or, obviously, of any homeomorphic space with $\backslash \; R^n$), is $\backslash \; \{\backslash \}$: in other words, each lace can be brought back by continuous deformation to the point p . Such spaces whose fundamental groups based in any point are commonplace are known as simply related. This concept reinforces the concept of connexity.
• For $n\; \backslash \; geq\; 2$, the fundamental group of the sphere $\backslash \; mathbb\; S^n$ of Euclidean space $\backslash \; R^\; \{n+1\}$ is also commonplace. In other words, the spheres of higher size or equalizes to 2 are simply related.
• the circle $\backslash \; mathbb\; S^1$ (i.e., the circle of center 0 and ray 1 in $\backslash \; mathbb\; C=\; \backslash \; mathbb\; R^2$) provides a more interesting example. The fundamental group of $\backslash \; mathbb\; S^1$, based for example as in point 1, is infinite monogene, therefore isomorphe  à  $\backslash \; Z$. The representatives of the classes of homotopies are the laces which make the turn of the circle at constant speed, in the positive direction or the negative direction: in other words, they are defined by the $e\_m\; applications:\; T\; \backslash \; mapsto\; e^\; \{2i\; \backslash \; pi\; MT\}$, for $m\; \backslash \; in\; \backslash \; Z$ where m is the number of times that the lace makes the turn of the circle. The composition of the ways being done under the law: $e\_m\; *e\_n=e\_\{m+n\}$.
• One can prove that the fundamental group of the torus with two dimensions $T^2=\; \backslash \; mathbb\; S^1\; \backslash \; times\; \backslash \; mathbb\; S^1$ (cf above) based in a point p is isomorphous with $\backslash \; Z^2$, the classes of homotopy of the generators being those of the laces has and B previously described. More generally, the fundamental group of the torus with N dimensions is isomorphous with $\backslash \; Z^n$.
• the fundamental group can also contain elements of torsion: for example, the fundamental group of the projective Plan $\backslash \; R\; P^2$ is isomorphous with $\backslash \; Z/2\; \backslash \; Z$.
• the fundamental group is not always commutative: For example, the fundamental group based in a point p of the private plan of two points $\backslash \; mathbb\; R^2-\; \backslash \; \{has;\; B\; \backslash \}$, is isomorphous with the free Groupe with two generators $F\_2$. The two generators are here the laces on the basis of p and doing each one the turn of one of the points.

In fact, one can show that for any group G , there exists a topological space of fundamental group G . (One can in makes find CW-complex dimension 2 or even a Variété of dimension 4 if the group is of finished presentation).

Properties

Independence of the fundamental group compared to the starting point

Let us examine the particular case where topological space X is related by arcs. Two fundamental groups based in two points p and Q ($\backslash \; pi\_1\; (X,\; p)$ and $\backslash \; pi\_1\; (X,\; Q)$) are isomorphous. Indeed, there exists a way φ energy of p to Q . One can thus define the following application

$\backslash \; mapsto\; **\; ^\; \{-\; 1\}$

who carries out obviously an isomorphism of the fundamental group $\backslash \; pi\_1\; (X,\; Q)$ towards the fundamental group $\backslash \; pi\_1\; (X,\; p)$ whose reciprocal isomorphism is the application:

$\backslash \; mapsto\; ^\; \{-\; 1\}\; **$

Thus, one can speak about the fundamental group (with a nonsingle isomorphism near) of topological space X , which one notes $\backslash \; pi\_1\; (X)$.

Compatibility with the continuous applications respecting the points bases

That is to say F an equivalence of homotopy respecting the points bases; then $\backslash \; pi\_1\; (F)$ is an isomorphism of groups.

Corollary : Two homeomorphic spaces have isomorphous fundamental groups.

fundamental group of a product

The fundamental group of the product of two pointed topological spaces is the product of the fundamental groups.

Bond with the first group of homology

It is shown that the first group of homology (of a related space by arcs) is abélianisé of the fundamental group.

It is a particular case of the Théorème of Hurewicz.

Methods of calculating and applications

Theorem of van Kampen

To calculate the fundamental group of a topological space which is not Simplement related is a difficult exercise, because it should be proven that certain laces are not homotopic. The theorem of van Kampen , also called theorem of Seifert-Van Kampen , makes it possible to solve this problem when topological space can be broken up into simpler spaces whose fundamental groups are already known. This theorem makes it possible to calculate the fundamental group of a very broad range of spaces.

See also: Theorem of van Kampen

Theorem of the cone and fundamental group of projective spaces

If X is a topological space, one defines the cone X as space quotient $\backslash \; frac\; \{I\; \backslash \; times\; X\}\; \{0\; \backslash \; times\; X\}$ where $I$ indicates the segment. If X is a circle, one obtains part of a circular cone. The fundamental group of the cone of a related space by arc is commonplace, in other words, if X is related by arcs, C (X) is Simplement related. There is a canonical inclusion $X=1\; \backslash \; times\; X\; \backslash \; subset\; C\; (X)$.

If F is a continuous application between two topological spaces $f:\; X\; \backslash \; to\; Y$, one defines the cone of the application F : $C\; (F)$ as space obtained by resticking $X\; \backslash \; sub\; C\; (X)$ and $f\; (X)\; \backslash \; sub\; Y$ along X.

Example: If F is the application of degree 2 in the circle $S^1\; \backslash \; to\; S^1:\; Z\; \backslash \; mapsto\; z^2$, one obtains $C\; (F)\; =P\_2\; (\backslash \; mathbb\; R)$. The cone of F is the projective Plan real.

The theorem of the cone affirms that the fundamental group of C (F) is isomorphous with the quotient of $\backslash \; pi\_1\; (Y)$ by standardized sub-group of $\backslash \; pi\_1\; (Y)$ image of F.

Application: the projective spaces (real) have isomorphous fundamental groups with $\backslash \; mathbb\; Z/2\; \backslash \; mathbb\; Z$.

Fundamental group of the graphs, surfaces and the polyhedrons

• the fundamental group of the Graphe S is a free Groupe.
• the fundamental group of the polyhedrons admits a presentation by generators and relations. A relation being provided by each face of the polyhedron.
• the fundamental group of a Surface compacts directional admits a presentation with 2g generating $a\_1,\; \backslash \; b\_1,\; \backslash \; a\_2,\; \backslash \; b\_2,\; \backslash \; ldots\; \backslash \; a\_g,\; \backslash \; b\_g$ and only one relation ($a\_1b\_1a\_1^\; \{-\; 1\}\; b\_1^\; \{-\; 1\}\; \backslash \; a\_2b\_2a\_2^\; \{-\; 1\}\; b\_2^\; \{-\; 1\}\; \backslash \; ldots\; \backslash \; a\_gb\_ga\_g^\; \{-\; 1\}\; b\_g^\; \{-\; 1\}\; =1$, one can also choose the presentation $a\_1\; \backslash \; ldots\; \backslash \; a\_g\; \backslash \; b\_1\; \backslash \; ldots\; \backslash \; b\_g\; \backslash \; a\_1^\; \{-\; 1\}\; \backslash \; ldots\; \backslash \; a\_g^\; \{-\; 1\}\; \backslash \; b\_1^\; \{-\; 1\}\; \backslash \; ldots\; \backslash \; b\_g^\; \{-\; 1\}\; =1$). The entirety G is only determined by surface and is called kind surface.

Degree of an application of the circle in itself $S^1\; \backslash \; to\; S^1$

See also: Degree of an application

Other theorems

See also: Theorem of Borsuk-Ulam, Theorem of Alembert-Gauss, Theorem of the hairy ball, Théorème of the point fixes of Brouwer

Fundamental group and theory of the coatings

There is equivalence between the sub-groups with conjugation close to the fundamental group and the coatings except for isomorphism. In this equivalence, the sub-groups normal correspond to the coatings galoisiens.

In theory of the coatings, one shows that if space admits a related coating Simplement in particular (if space is Semi-locally simply related i.e. if space is not too " sauvage" , for example if it is locally contractile) that the fundamental group is isomorphous with the group of the automorphisms of a universal coating.

Generalizations

Fundamental Groupoïde (or groupoïde of Poincaré)

A category is called a groupoïde if the objects and the arrows form a unit (it is a " small catégorie") and if all the arrows are invertible (are isomorphisms). The groupoïdes form a category whose morphisms are the functors between groupoïdes. The groups are groupoïdes (with only one object).

Either G a groupoïde, one defines the relation of equivalence $x\; \backslash \; equiv\; \{\}\; \backslash ,\; y$ if G (X, there) is nonempty, it defines a groupoïde quotient noted $\backslash \; pi\_0\; (G)$. $\backslash \; pi\_0$ defines a functor ( component related ) of the category of the groupoïdes towards the category of the units.

To each topological space one will associate in a fonctorielle way a groupoïde $\backslash \; pi\; X$.

Either X a topological space, one takes for whole of objects $\backslash \; pi\; X$ the unit subjacent with X. the arrows of source X and of goal are there the classes of homotopy of the ways (= continuous arcs) of X towards Y. the relation of homotopy is compatible with the composition of the ways and thus defines a groupoïde $\backslash \; pi\; X$ called the fundamental groupoïde of X. The theorem of Van Kampen is expressed also simply by using the fundamental groupoïdes.

That is to say G a groupoïde, and $x$ an object of G (one says also a point of G). The law of composition between the arrows of $G\; (X,\; X)$ restricted with it under-groupoïde is a law of group. One notes $\backslash \; pi\_1\; (G,\; X)$ this group. Note: $\backslash \; pi\_1$ does not define not a functor of the category of the groupoïdes towards the category of the groups.

The fundamental group is defined by $\backslash \; pi\_1\; (X,\; x\_0)\; =\; \backslash \; pi\_1\; (\backslash \; pi\; X,\; x\_0)$

Higher groups of homotopy

The fundamental group is in fact the first Groupe of homotopy, from where index 1 in the notation $\backslash \; pi\_1\; (X)$.

Fundamental group and algebraic geometry

The fundamental group can also be defined in an abstract way like the Groupe of the Automorphismes of the Foncteur fiber, which, with a basic coating   $(X,\; p)$, associates fiber of the coating with the point  $p$.

This alternative definition opens the way with generalization in algebraic Géométrie, where the definition given previously in terms of basic laces p does not apply naturally. In this generalization, coatings being replaced by the slack morphisms: the fundamental group of space pointed $(X,\; p)$ is the group of the automorphisms of the functor fiber which, with a morphism spreads out $E\; \backslash \; to\; X$, associates the fiber $E\; (p)$ with the point $p$.

This generalization is due to Grothendieck and Claude Chevalley.

This theory makes it possible to explain the bond between the theory of Welsh of the coatings of the surfaces of Riemann (group of automorphisms…) and the Theory of Welshman of the bodies of functions.

Bibliography (in French)

• Dolbeault : Analyze complexes
• Zisman: elementary algebraic Topology
• Jean Dieudonné: Elements of Analysis , volume 3
• Andre Gramain: Topology of Surfaces

External bond (English)

• Algebraic Topology, by Allen Hatcher

External bond

• Animations which introduces with the fundamental group, by Nicolas Delanoue

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