In the mathematical subfield of topology , the mapping cone is a construction that assigns a third such space to a continuous function between two topological spaces . It is closely related to the concept of the cone over a topological space; just like this, the mapping cone is mainly considered in algebraic topology . More generally, there is the mapping cone of chain mappings between chain complexes in homological algebra .
definition
Let two topological spaces and be a continuous function between them, let the cone over .
The mapping cone is now obtained (as indicated in the drawing) by bonding of and in virtue .
More precisely this means:
One identified in the disjunctive association each with each , the result is an implicit equivalence relation .
The mapping cone is then the factor space provided with the quotient topology with respect to the canonical projection .
Reduced imaging cone
In the category of dotted topological spaces - that is, dotted and applies - one usually considers the reduced imaging cone . This is created by also identifying the interval in the imaging cone - more precisely, its image under the projection .
Analogously, in the above construction of the imaging cone , the reduced cone can also be assumed.
properties
- The space is naturally a subspace of , since each of its points is preserved under the projection .
- Is injective and relatively open , so a homeomorphism to its image, so too , and thus to contain.
- If one looks at identity , homeomorphism applies .
All of the above relationships also apply the reduced mapping cone in the case of punctured spaces and and basispunkterhaltendem , may be required to ensure the reduced cone are gone.
- If the image adhering to the skeleton is in a CW complex , the imaging cone is homeomorphic to the skeleton .
This is one of the main uses of the mapping cone in algebraic topology. The following also applies specifically to the reduced imaging cone:
- If dotted and constant , the following applies , where the reduced suspension of and denotes the wedge product .
- For a well-dotted space , the reduced imaging cone is homotopy-equivalent to the normal imaging cone.
- A mapping induces an isomorphism for a homology theory if and only if .
Role in homotopy theory
If two continuous maps are homotopic , then their mapping cones and are homotopy equivalent.
If a closed subspace and the inclusion is a cofiber , then homotopy is equivalent to the quotient space . It can also be shown that inclusion is always a closed cofibre. It is thus obtained that the imaging cone is equivalent to homotopy , where the suspension of denotes here. If one continues in the same way, it follows that the mapping cone of the inclusion of to results in the suspension of , etc.
If one also has a continuous in a topological space , then the composition is homotopic to a constant mapping if and only if it can be continued to a mapping . In the event that the result is a little clearer: a mapping is homotopic to a constant mapping if and only if it can be continued into a mapping . To construct the map , one simply uses the homotopy , which is constant on .
If one considers dotted spaces and base point preserving maps, it means that the following sequence is exact :
This exact sequence is also called a doll sequence .
Image cone of a chain image
Let two chain complexes with differentials d. H.,
and accordingly for
For a chain mapping one defines the mapping cone or as the chain complex:
with differential
-
.
Here the chain complex is designated with and . The differential is explicitly calculated as follows:
If there is a continuous mapping between topological spaces and the induced chain mapping between the singular chain complexes , then is
-
.
See also
literature
-
Glen E. Bredon : Topology and Geometry. Revised 3rd printing. Springer, New York a. a. 1997, ISBN 0-387-97926-3 ( Graduate Texts in Mathematics 139).
- Robert M. Switzer: Algebraic Topology - Homology and Homotopy. Reprint of the 1975 edition. Springer, Berlin a. a. 2002, ISBN 3-540-42750-3 ( Classics in Mathematics ).