Homotopy group

In mathematics, specifically in algebraic topology, the homotopy groups are a tool to classify topological spaces. The continuous maps of an n -dimensional sphere in a given space are grouped into equivalence classes called homotopy classes. Two figures are called homotopic if they can be continuously transformed into each other. This homotopy classes form a group, which is called the n-th homotopy the room.

The first homotopy group is also called the fundamental group.

Homotopy equivalent topological spaces have isomorphic homotopy groups. Have two rooms different homotopy groups, so they can not be homotopy equivalent, hence not homeomorphic. For CW complexes is true for a set of Whitehead also a partial reversal.

• 3.1 Example: The Hopf fibration

Definition

In the sphere we choose a point which we call the base point. Be a topological space and a base point. We define as the set of homotopy classes of continuous maps (that it is). More specifically, the equivalence classes are defined by homotopies, which hold the base point. Equivalent could be called the set of mappings defined, ie those continuous maps by the n- dimensional unit cube, which reflect the edge of the cube to the point.

For n ≥ 1 can be provided the set of homotopy classes with a group structure. The construction of the group structure of similar to the case, ie the fundamental group. The idea of ​​the construction of the group operation in the fundamental group is the concatenation of paths passing through, in the more general n-th homotopy group, we proceed similarly, only now we stick together n- cube along one side, ie We define the sum of two images by

In the representation of the sum of two spheres homotopy classes is homotopy that Figure, which is obtained by first contracting the sphere along the equator, and then applies it to the top sphere R, on the lower g. More precisely, is the composition of the ' Äquatorzusammenzurrung ' ( Einpunktvereinigung ) and the figure.

If so is an abelian group. To prove this fact, note that two homotopies from two intertwined dimensions can be "turned". For this is not possible, since the edge of is not path-connected.

Examples

Homotopy groups of spheres

For true, for it follows from the theorem of Hopf that

Is. Jean -Pierre Serre has shown that there must be a finite group.

Eilenberg - MacLane spaces

Topological spaces that cater for all, with Eilenberg - MacLane hot rooms.

Examples of spaces are closed, orientable surfaces with the exception of closed, orientable, prime 3-manifolds with the exception of and all CAT (0 )-spaces, including locally - symmetric spaces of nichtkompaktem type, in particular hyperbolic manifolds.

The long exact sequence of a fibration

Is a Serre fibration with fiber, that is, a continuous map, which owns the Homotopiehochhebungseigenschaft for CW complexes, then there exists a long exact sequence of homotopy groups

The relevant figures are here no group homomorphisms, as it is not gruppenwertig, but they are exactly in the sense that the image of the core (the component of the base point is the excellent element ) is similar.

Example: The Hopf fibration

The base is here and the total area is. is the Hopfabbildung having the fiber. From the long exact sequence

And the fact that for, it follows that for valid. Is particular.

N- equivalences and weak equivalences. The set of Whitehead

A continuous map is called equivalence if the induced map is an isomorphism for and is a surjection. If the mapping is an isomorphism for all, so you call the figure a weak equivalence.

A set of JHC Whitehead says that a weak equivalence between connected CW - complexes is already a homotopy equivalence. Than have the case and dimension smaller so already sufficient that an equivalency is.

Homotopy and homology. The set of Hurewicz

For dotted rooms there are canonical homomorphisms of the homotopy groups in the reduced homology groups

The Hurewicz - homomorphisms are called ( by Witold Hurewicz ). A set of Hurewicz states: If a - connected space, that is true for, then the Hurewicz homomorphism in the case of the Abelisierung and an isomorphism.

Relative homotopy groups

One can also define relative homotopy groups for pairs of spaces, its elements are homotopy classes of maps, illustrations and two such hot here homotopic if there is a homotopy. The absolute homotopy groups are obtained in the special case.

For each pair of rooms, there is a long exact sequence