In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted C_f. Alternatively, it is also called the homotopy cofiber and also notated Cf. Its dual, a fibration, is called the mapping fiber. The mapping cone can be understood to be a mapping cylinder Mf with the initial end of the cylinder collapsed to a point. Mapping cones are frequently applied in the homotopy theory of pointed spaces.

Definition

Given a map, the mapping cone C_f is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation ,. Here I 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 X \times I with one end (the 0 end) collapsed to a point), and glues the other end onto Y via the map f (the 1 end). 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 (so ), one also identifies all of. Formally,. Thus one end and the "seam" are all identified with y_0.

Example of circle

If X is the circle S^1, the mapping cone C_f can be considered as the quotient space of the disjoint union of Y with the disk D^2 formed by identifying each point x on the boundary of D^2 to the point f(x) in Y. Consider, for example, the case where Y is the disk D^2, and is the standard inclusion of the circle S^1 as the boundary of D^2. Then the mapping cone C_f is homeomorphic to two disks joined on their boundary, which is topologically the sphere S^2.

Double mapping cylinder

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder X \times I joined on one end to a space Y_1 via a map and joined on the other end to a space Y_2 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 Y_1, Y_2 is a single point.

Dual construction: the mapping fibre

The dual to the mapping cone is the mapping fibre F_f. Given the pointed map one defines the mapping fiber as Here, I is the unit interval and \omega is a continuous path in the space (the exponential object) Y^I. The mapping fiber is sometimes denoted as Mf; 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 X\times_f Y which is dual to the pushout X\sqcup_f Y 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 I\to Y is simply an alternate notation for the space Y^I of all continuous maps from the unit interval to Y. 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.

Applications

CW-complexes

Attaching a cell.

Effect on fundamental group

Given a space X and a loop representing an element of the fundamental group of X, we can form the mapping cone C_\alpha. The effect of this is to make the loop \alpha contractible in C_\alpha, and therefore the equivalence class of \alpha in the fundamental group of C_\alpha 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 equivalence if and only if its mapping cone is contractible. More generally, a map is called n-connected (as a map) if its mapping cone is n-connected (as a space), plus a little more. Let be a fixed homology theory. The map induces isomorphisms on H_, if and only if the map induces an isomorphism on H_, 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.