Mapping cone

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In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology. It is also called the homotopy cofiber, and also notated $C f$ .

1 Definition
- 1.1 Example of circle
- 1.2 Double mapping cylinder
2 Applications
3 See also

[edit] Definition

Given a map $f\colon X \to Y$ , the mapping cone $C f$ is defined to be the quotient topological space of $(X \times I) \sqcup Y$ with respect to the equivalence relation $(x, 0) \sim (x',0)\,$ , $(x,1) \sim f(x)\,$ on X. Here $I$ denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.

[edit] Example of circle

If $X$ is the circle S¹, 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 disc D², and

f: S¹ → Y = D²

is the standard inclusion of the circle S¹ as the boundary of D². Then the mapping cone C_f is homeomorphic to two disks joined on their boundary, which is topologically the sphere S².

[edit] Double mapping cylinder

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 (also known as the homotopy pushout), in which one space is a single point.

[edit] Applications

[edit] CW-complexes

Attaching a cell

[edit] Effect on fundamental group

Given a space X and a loop

$\alpha\colon S^1 \to X$

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.

Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.

[edit] Homology of a pair

The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:

If E is a homology theory, and $i\colon A \to X$ is an inclusion, then $E_*(X,A) = E_*(X/A,*) = \tilde E_*(X/A)$ , which follows by applying excision to the mapping cone.