Mapping cone (topology)

In the mathematical branch of topology mapping cone is a construction that associates a third kind of space a continuous function between two topological spaces. It is closely related to the concept of the cone over a topological space; as well as that of the imaging cone is mainly considered in algebraic topology. More generally, it is in the homological algebra of the mapping cone chain maps between chain complexes.

Definition

Be two topological spaces and a continuous function between them is further above the cone.

The Figure cone you get now ( as indicated in the drawing) by gluing and virtue.

Specifically, this means:

You identified in the disjoint union each with each, as is implicitly an equivalence relation.

Figure cone of the parameter space is then provided with the ratio with respect to the canonical topology projection.

Reduced Figure cone

In the category of the dotted topological areas - are punctured and is therefore - considering the most reduced image cone. This arises from the fact that one in Figure cone also the interval - or more precisely its image under the projection - identified.

Analog may be in the above construction of the imaging cone also assumed equal to the reduced cone.

Properties

• The space is naturally a subspace of, since each of its points is preserved under the projection.
• Is injective and relatively open, ie a homeomorphism onto its image, so also are included in and thus.
• Looking at the identity, then the homeomorphism applies.

All the above relations are also valid for the reduced cone illustration, in the case of punctured spaces and basispunkterhaltendem and, where appropriate, must it be passed to the reduced cone.

• If the anklebende figure in a CW - complex to the skeleton, so the figure cone is homeomorphic to the skeleton.

This is one of the main applications of the imaging cone in algebraic topology. Especially for the reduced cone illustration also applies:

• Are punctured and constant, applies the reduced device to attach to and the wedge product call.
• For a well- dotted area of ​​Figure reduced cone is homotopy equivalent to the normal cone mapping.

• A picture of exactly then induces an isomorphism of homology theory though.

Role in homotopy theory

If two continuous maps homotopic, so are their picture cones and homotopy equivalent.

If a closed subspace and the inclusion is a Kofaserung so is homotopy equivalent to the quotient space. It can also show that the inclusion is always a closed Kofaserung. Thus we obtain that the mapping cone is homotopy equivalent to, in which case the call of device to attach. Continuing in the same way, it follows that the mapping cone of the inclusion of by the device to attach of results etc.

If one has more steady in a topological space, then the composition is exactly then is homotopic to a constant map, where is continuable to an image. In the event that the result is still somewhat clearer: a picture is exactly then is homotopic to a constant map if it can be continued to an image. To construct the mapping, we just use the homotopy that is constant on.

If you look dotted spaces and base point preserving mappings, this means that the following sequence is exact:

This exact sequence is also called doll series.

Figure cone of a chain map

Be two chain complexes with differentials ie,

And correspondingly for

For a chain map to define the mapping cone oderr as the chain complex:

With differential

Herein, the chain complex with and. Explicitly, the differential is calculated as follows:

If a continuous map between topological spaces and the induced chain map between the singular chain complexes, then