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.
Definition
Given a map, the mapping cone is defined to be the quotient topological space of the mapping cylinder with respect to the equivalence relation, on X. Here denotes the unit interval with its standard topology. Note that some authors use the opposite convention, switching 0 and 1. Visually, one takes the cone on X, and glues the other end onto Y via the map f. Coarsely, one is taking the quotient space by the image 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 , 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.
Double mapping cylinder
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, in which one of is a single point.
Dual construction: the mapping fibre
The dual to the mapping cone is the mapping fibre. Given the pointed map one defines the mapping fiber as Here, I is the unit interval and is a continuous path in the space . The mapping fiber is sometimes denoted as ; however this conflicts with the same notation for the mapping cylinder. It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback which is dual to the pushout used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone has the curried form where is simply an alternate notation for the space of all continuous maps from the unit interval to. The two variants are related by an adjoint functor. Observe that the currying preserves the reduced nature of the maps: in the one case, to the tip of the cone, and in the other case, paths to the basepoint.
Given a space X and a loop representing an element of the fundamental group of X, we can form the mapping cone. The effect of this is to make the loop contractible in, and therefore the equivalence class of in the fundamental group of will be simply the identity element. Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.
Homology of a pair
The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient. Namely, if E is a homology theory, and is a cofibration, then which follows by applying excision to the mapping cone.
Relation to homotopy (homology) equivalences
A map between simply-connected CW complexes is a homotopy equivalenceif and only if its mapping cone is contractible. More generally, a map is called n-connected if its mapping cone is n-connected, plus a little more. Let be a fixed homology theory. The map induces isomorphisms on, if and only if the map induces an isomorphism on, i.e.,. Mapping cones are famously used to construct the long coexact Puppe sequences, from which long exact sequences of homotopy and relative homotopy groups can be obtained.
