Mapping cone

In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology. Given a map f from the circle S¹ to a topological space Y, C_f 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 C_f 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 cyclinder 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 C_α. The effect of this is to make the loop α contractible in C_α, and therefore the equivalence class of α in the fundamental group of C_α will be simply the identity element.

Algebraic analogue

An algebraic version of the construction can be carried on on the chain complex level. In abstract terms this is an important part of the theory of trangulated categories.