Mapping cone (topology)

In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted

Alternatively, it is also called the homotopy cofiber and also notated

Its dual, a fibration, is called the mapping fiber.

with the initial end of the cylinder collapsed to a point.

Mapping cones are frequently applied in the homotopy theory of pointed spaces.

is defined to be the quotient space of the mapping cylinder

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

The above is the definition for a map of unpointed spaces; for a map of pointed spaces

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

Consider, for example, the case where Y is the disk

is homeomorphic to two disks joined on their boundary, which is topologically the sphere

joined on one end to a space

via a map and joined on the other end to a space

one defines the mapping fiber as[1] Here, I is the unit interval and

is a continuous path in the space (the exponential object)

; 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

[2] In this particular case, the duality is essentially that of currying, in that the mapping cone

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.

representing an element of the fundamental group of X, we can form the mapping cone

The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient.

is a cofibration, then which follows by applying excision to the mapping cone.

between simply-connected CW complexes is a homotopy equivalence if and only if its mapping cone is contractible.

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.