Mapping cone (topology)

In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces. Given a map , the mapping cone is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation , . Here denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1. Visually, one takes the cone on X (the cylinder with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end). Coarsely, one is taking the quotient space by the of X, so ; this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected map. The above is the definition for a map of unpointed spaces; for a map of pointed spaces (so ), one also identifies all of ; formally, Thus one end and the "seam" are all identified with If is the circle , the mapping cone can be considered as the quotient space of the disjoint union of Y with the disk formed by identifying each point x on the boundary of to the point in Y. Consider, for example, the case where Y is the disk , and is the standard inclusion of the circle as the boundary of . Then the mapping cone is homeomorphic to two disks joined on their boundary, which is topologically the sphere . The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder joined on one end to a space via a map and joined on the other end to a space via a map The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one of is a single point.

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Mapping cylinder

In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction.

Cofibration

In mathematics, in particular homotopy theory, a continuous mapping between topological spaces is a cofibration if it has the homotopy extension property with respect to all topological spaces . That is, is a cofibration if for each topological space , and for any continuous maps and with , for any homotopy from to , there is a continuous map and a homotopy from to such that for all and . (Here, denotes the unit interval .

Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups.

Explores homotopy pushouts, standard models, equivalence properties, and the importance of strict commutativity in pushout constructions.

Explores tower sequences, homomorphisms, and their applications in topology, including the computation of homology and the construction of telescopes.

Explores mapping cylinders and cones, key in exact sequences and topology.