Mapping cone (topology)

In mathematics, especially homotopy theory, the mapping cone is a construction formula_1 of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated formula_2. Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder formula_3, 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 formula_4, the mapping cone formula_1 is defined to be the quotient space of the mapping cylinder formula_6 with respect to the equivalence relation formula_7, formula_8. Here formula_9 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 formula_10 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 image of "X", so formula_11; 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 formula_12 (so formula_13), one also identifies all of formula_14; formally, formula_15 Thus one end and the "seam" are all identified with formula_16
Example of circle.
If formula_17 is the circle formula_18, the mapping cone formula_1 can be considered as the quotient space of the disjoint union of "Y" with the disk formula_20 formed by identifying each point "x" on the boundary of formula_20 to the point formula_22 in "Y".
Consider, for example, the case where "Y" is the disk formula_20, and formula_24 is the standard inclusion of the circle formula_18 as the boundary of formula_20. Then the mapping cone formula_1 is homeomorphic to two disks joined on their boundary, which is topologically the sphere formula_28.
Double mapping cylinder.
The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder formula_10 joined on one end to a space formula_30 via a map
and joined on the other end to a space formula_32 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 formula_34 is a single point.
Dual construction: the mapping fibre.
The dual to the mapping cone is the mapping fibre formula_35. Given the pointed map formula_36 one defines the mapping fiber as
Here, "I" is the unit interval and formula_38 is a continuous path in the space (the exponential object) formula_39. The mapping fiber is sometimes denoted as formula_3; 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 formula_41 which is dual to the pushout formula_42 used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone formula_43 has the curried form formula_44 where formula_45 is simply an alternate notation for the space formula_39 of all continuous maps from the unit interval to formula_47. 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 formula_48 representing an element of the fundamental group of "X", we can form the mapping cone formula_49. The effect of this is to make the loop formula_50 contractible in formula_49, and therefore the equivalence class of formula_50 in the fundamental group of formula_49 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 formula_54 is a cofibration, then
which follows by applying excision to the mapping cone.
Relation to homotopy (homology) equivalences.
A map formula_56 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 formula_57 be a fixed homology theory. The map formula_56 induces isomorphisms on formula_59, if and only if the map formula_60 induces an isomorphism on formula_59, i.e., formula_62.
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.