Wikipedia / Mapping cone (topology)
Wikipedia / Mapping cone (topology)
In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space. Alternatively, it is also called the homotopy cofiber. Its dual, a fibration, is called the mapping fiber. The mapping cone can be understood to be a mapping cylinder 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 is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation 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 with one end (the 0 end) collapsed to a point), and glues the other end onto Y via the map f (the 1 end).
Example of circle
is the circle 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 in Y.
Consider, for example, the case where Y is the standard inclusion 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 via a map
via a map
is a single point.
Dual construction: the mapping fibre
The dual to the mapping cone is the mapping fibre one defines the mapping fiber as
.
Here, I 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 which is dual to the pushout used to construct the mapping cone. In this particular case, the duality is essentially that of currying. 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 representing an element of the fundamental group of X contractible, and therefore the equivalence class 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 is a cofibration, then
,
which follows by applying excision to the mapping cone.
Relation to homotopy (homology) equivalences
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.
be a fixed homology theory induces isomorphisms.
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.