Homotopy

In topology, a homotopy is a continuous deformation between two images of a topological space to another, such as the deformation of a curve into another curve.

  • 5.1 Definition
  • 5.2 Examples
  • 5.3 Unlike the homotopy
  • 5.4 applications

Definition

A homotopy between two continuous maps is a continuous map

With the feature

Where the unit interval. The first parameter corresponds to the original pictures and the second indicates the degree of deformation at. Especially vivid is the definition if you look at the second parameter as "time" imagines (see image ).

One says, is homotopic to and writes. Homotopy is an equivalence relation, the corresponding equivalence classes are called homotopy classes, the amount of these classes is often referred to.

A continuous map is called null-homotopic if it is homotopic to a constant map.

Example

Be the unit circle in the plane and all the plain. The figure is the embedding in, and let the mapping that maps entirely to the origin, ie

Then and homotopic to each other. because

Is continuous and satisfies and.

Relative homotopy

Is a subset of, and agree two continuous maps agree on, so hot and relatively homotopic if there is a homotopy for which each is independent of.

An important special case is the homotopy of paths relative to the endpoints: A path is a continuous map; this is the unit interval. Two paths are called homotopic relative to the endpoints, if they are homotopic relative, ie if the homotopy holds the start and end points. (Otherwise would be way in the same Wegzusammenhangskomponente always homotopic. ) So are two ways in and with and so a homotopy relative endpoints between them a continuous map

A path is called null-homotopic if and only if it is homotopic to the constant path.

The other common case is the homotopy of maps between dotted areas. Are and dotted rooms, two continuous mappings are homotopic as mappings of spaces dotted if they are relatively homotopic.

Example: The fundamental group

The set of homotopy classes of mappings of spaces of punctured according to the fundamental group of the base point.

If, for example, a circle of any selected point, then the way that is described by a single circling of the circle, not homotopic to the path obtained by Resting at the starting point.

Homotopy equivalence

Let and be two topological spaces and continuous maps and. Then the links and each continuous maps are from or on yourself, and you can try to homotopieren this to the identity on X and Y, respectively.

If there are such and that homotopic to and is homotopic, it is called and homotopy equivalent or of the same homotopy type. The illustrations and then called homotopy equivalences.

Homotopy equivalent spaces have in common, most topological properties. If and are homotopy equivalent, shall apply

  • If path-connected, as well.
  • If and path-connected, then the fundamental groups and the higher homotopy groups are isomorphic.
  • The homology and cohomology groups - of and are equal.
  • X and Y are Deformationsretrakte a topological space Z.

Isotopy

Definition

If given two homotopic pictures and belong to a particular Regularitätsklasse or other additional properties have, one may wonder whether the two can be connected by a path with one another within that class. This leads to the concept of isotopy. An isotopy is a homotopy

Above, where all intermediate images ( for fixed t ) should also have the required additional properties.

Examples

Two homeomorphisms are isotopic so if there is a homotopy so that all homeomorphisms are. Two diffeomorphisms are isotopic, if all diffeomorphisms itself. ( They are then referred to as diffeotop. ) Two embeddings are isotopic, if all embeddings are.

Unlike the homotopy

To require that two figures are isotopic, may actually be a stronger requirement than requiring that they are homotopic. For example, the homeomorphism the unit disc is defined by f ( x, y) = ( -x,- y), the same as a 180 - degree rotation around the zero point, so the identity mapping and F isotope, because they can be connected to each other by turns. In contrast, the image on the interval [ -1,1] in, defined by f ( x ) = -x is not isotopic to the identity. The reason is that any homotopy of the two images must swap the two end points of each other at any given time; at this time they are mapped to the same point and the corresponding figure is not a homeomorphism. However, f is homotopic the identity, for example by homotopy H: [-1,1 ] x [0,1] → [-1,1] is given by H (x, t) = x - 2tx.

Applications

In Geometric topology isotopies are used to create equivalence relations.

For example, in knot theory - when there are two nodes and treat them as the same? The intuitive idea of ​​deforming a knot in the other, leads to a path of homeomorphisms is asked: An isotopy that begins with the identification of three-dimensional space and ends with a homeomorphism h such that h node in the node transferred. Such an isotopy of the ambient space is also called Umgebungsisotopie.

Another important application is the definition of the mapping class group Mod ( M) of a manifold M. One considers diffeomorphisms of M " up to isotopy ", that is, Mod ( M) is the (discrete ) group of diffeomorphisms of M, the group modulo of diffeomorphisms that are isotopic to the identity.

Homotopy can be used in numerical mathematics for a robust initialisation for the solution of differential- algebraic equations (see homotopy ).

Chain homotopy

Two Kettenhomomorphismen

Between chain complexes and hot kettenhomotop if there is a homomorphism

With

There.

If homotopic maps between topological spaces, then the induced images of the singular chain complexes

Kettenhomotop.

Dotted homotopy

Two dotted pictures

Homotopic to say if it is a continuous map

There. The set of homotopy classes is denoted by dotted pictures.

240223
de