Mapping cone
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In mathematics, especially homotopy theory, the mapping cone is a construction Cf of topology.
Given a map 
, the mapping cone Cf is defined to be the quotient topological space of 
with respect to the equivalence relation (x,0)˜(x',0), (x,1)˜f(x) (
). Here I denotes the unit interval [0,1] with its standard topology.
If X is the circle S1, Cf can be considered as the quotient space of the disjoint union of Y with the disk D2 formed by identifying a point x on the boundary of D2 to the point f(x) in Y.
Consider, for example, the case where Y is the circle, and
- f: S1 → Y = S1
the identity function. Then the mapping cone Cf is homeomorphic to two disks joined on their boundary, which is topologically the sphere S2.
The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder joined on one end to a space X1 via the map
- 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, in which one space is a single point.
Given a space X and a loop
- α: S1 → X
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.
[edit] Algebraic analogue
An algebraic manifestation of this construction appears on the chain complex level. In abstract terms this is an important part of the theory of triangulated categories.
