Fundamental group

The fundamental group is used in algebraic topology for the study of geometric objects or topological spaces. Each topological space can be assigned to a fundamental group. However, it is itself an object from the algebra and may also use these methods are investigated. If two topological spaces different fundamental groups, one concludes that the two spaces topologically different, that does not mean homeomorphic are. Henri Poincaré introduced in 1895 as first the concept of a fundamental group.

• 4.1 Overlays
• 4.2 set of Seifert -van Kampen -
• 4.3 Implications of certain fundamental groups
• 4.4 context of homology

Intuitive Explanation on the example of the torus

First, the idea of the fundamental group will be explained with an example: As a topological space of the (two-dimensional ) torus is considered, and it marked a base point.

From this point, there are grinding, that is a closed curve, the start point of the base, extend on the torus surface and ends again at the base point. Some of the loops can be on the torus to a point move in together, others do not. To give an example, imagine that the loops are made of rubber and can be stretched freely, compressed and displaced, but always so that the beginning and end of stay firmly in the base point and the loops must always remain on the torus (ie only on the surface and not pass through the " dough " of the donut ). Such deformation is called homotopy; we also say, a loop is homotopiert. Two loops that can be converted into each other by a homotopy, called homotopic.

All loops which are homotopic to each other, one summarizes a homotopy class. The different homotopy classes form the elements of the fundamental group.

The two loops and in the right figure includes, for example to different homotopy classes: you can not deform into each other and therefore describe different elements of the fundamental group. Other items you get by one of the two grinding passes several times through before closing the loop: A loop that goes around twice around the hole can not be in a deform the leads around it three times, etc.

More generally, can be two loops combine to a third by first one, then the other goes through, so the end of the first with the beginning of the second link (because the point of attachment now is an interior point of the loop, it no longer needs to remain on the base point, but may be pushed away from him). This link is from the set of homotopy classes a group called the fundamental group. The neutral element is the class of loops that can be contracted at the base point. The inverse element to a class of loops is obtained by going through it backwards.

Mathematical definition

Be a topological space and a base point. A loop is a continuous map, which connects with itself, ie.

A homotopy between two loops and is a continuous family of loops that combines both loops, ie: is a continuous map with the properties

• And.

The first parameter describes the progress of the loop deformation (corresponding and equivalent ). The second parameter corresponds to the initial loop parameters.

Two loops are called homotopic if there is a homotopy between them. The homotopy between loops defines an equivalence relation and the equivalence classes are called homotopy classes. The set of homotopy classes is the fundamental group of the base point.

The group structure is obtained by the link above, so by concatenating the loops:

With

Since one can construct from homotopies between various representatives also a homotopy between the nested loops, the resulting homotopy class is independent of the choice of the representatives.

The neutral element of the fundamental group is the homotopy class of the constant loop and the inverse element of the homotopy is the homotopy class of the loop that cycles through the loop backwards.

Independence from the base point

Since all the loops on the base point to start, the fundamental group measures only the properties of Wegzusammenhangskomponente. Therefore, it is sensible to assume that is path connected. Then, however, is also the choice of the base point for the fundamental group is not essential. Rather, there is a group isomorphism if and by a curve

Are connected. This group isomorphism defined by

While the loops and different, they are still homotopic and the group isomorphism can be defined by both loops.

So is path-connected one speaks generally of leaves and the base point away. However, is not path-connected, then the fundamental group may well depend on the chosen base point. According to the above argument, the fundamental group is then only dependent on the Wegzusammenhangskomponente of.

Examples

• On a sphere from dimension 2 can each loop to contract to a point. Therefore, the fundamental group of the sphere is trivial, ( for ).
• The Torusrand described above has the fundamental group: the two loops and are generators of the fundamental group. It is in this case is abelian: The loop can be contracted to a point ( cut the Torusrand along and on, the result is a quadrilateral whose boundary curve is accurate and can be tightened inside the rectangle ). Therefore applies, ie.
• For one -dimensional Torusrand.
• The two-dimensional plane with a hole, the fundamental group, as well as the 1- sphere (a simple circle). The homotopy class of a loop is determined by how many times the loop around the hole runs (eg clockwise).
• Has the two-dimensional plane with two holes as the fundamental group of a free group in two generators, namely the two loops that run around once around one of the holes. This group is not abelian.
• Fundamental groups need not be torsion: so are the fundamental groups of the real projective plane or the group of rotations in space, isomorphic to the cyclic group of order 2
• One can show that there are to each group a so-called classifying space whose fundamental group is isomorphic to.
• The fundamental group of a Knotenkomplements is called a node group. It can be calculated with the Wirtinger algorithm.

Properties and Applications

Overlays

The fundamental group plays an important role in the classification of interference. For rooms that have a universal covering, the fundamental group is isomorphic to the deck transformation group of the universal covering. This isomorphism is one of the most important tools for the calculation of the fundamental group.

Set of Seifert -van Kampen -

An important tool for the calculation of the fundamental group is the set of Seifert -van Kampen - which allows to decompose the space into overlapping regions and calculate the fundamental group of from the (simpler ) fundamental groups of spheres and overlapping.

Consequences of certain fundamental groups

Knowledge of the fundamental group often allows conclusions on the topological space. If, for example, a manifold is a finite fundamental group, so they can wear any metric that not everywhere positive curvature. The only closed surface with trivial fundamental group is the sphere. The now proven Poincaré conjecture says that an analogous statement is also true for three-dimensional manifolds.

Connection with homology

In the general case the fundamental group need not be abelian (as in Torusrand ). But they can abelian " clean" by one out shares the commutator subgroup. The group, which is then obtained is isomorphic to path-connected spaces to the first homology group.

Generalizations

The fundamental group is the first homotopy group, hence comes the name. Since the definition used one-dimensional loops, the fundamental group can only recognize the one-dimensional topological structure. A hole in a two-dimensional surface can be determined by grinding a hole in three-dimensional space, however (about ) is not: let go by it.

The generalization to the -th homotopy groups therefore used instead of grinding spheres of dimension.

If so says the set of Hurewicz ( by Witold Hurewicz ) that the first non-trivial homotopy group coincides with the first non-trivial homology group.

With one does not denote a group, but only the set of path components of. As can be understood as a homotopy path in the loop space is

The relationship between and manufactured.