Mapping cone (topology)
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.
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 [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.
Visually, one takes the cone on X (the cylinder 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 Cf "=" Y/f(X); 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
Example of circle
Consider, for example, the case where Y is the disc D2, and
- f: S1 → Y = D2
is the standard inclusion of the circle S1 as the boundary of D2. Then the mapping cone Cf is homeomorphic to two disks joined on their boundary, which is topologically the sphere S2.
Double mapping cylinder
- f1: S1 → X1
and joined on the other end to a space X2 via the map
- f2: S1 → X2.
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.
Dual construction: the mapping fibre
Here, I is the unit interval and is a continuous path in the space (the exponential object) . 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.
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 Cα. The effect of this is to make the loop α contractible in Cα, and therefore the equivalence class of α in the fundamental group of Cα 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:
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.
Let be a fixed homology theory. The map induces isomorphisms on , if and only if the map induces an isomorphism on , i.e. .