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.