Mapping cone (topology)

The mapping cone of $f: X \to Y$ is obtained by gluing the cone over X to Y.

In mathematics, especially homotopy theory, the mapping cone is a construction $C_f$ of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated $Cf$.

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.

Visually, one takes the cone on X (the cylinder $X \times I$ 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 $f\colon (X,x_0) \to (Y,y_0),$ (so $f\colon x_0 \mapsto y_0$), one also identifies all of ${x_0}\times I$; formally, $(x_0,t) \sim (x_0,t')\,.$ Thus one end and the "seam" are all identified with $y_0.$

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.

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.

Applications

CW-complexes

Attaching a cell

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.

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 a cofibration, then $E_*(X,A) = E_*(X/A,*) = \tilde E_*(X/A)$, which follows by applying excision to the mapping cone.^[1]

Relation to homotopy (homology) equivalences

A map $f\colon X\rightarrow Y$ 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. See A. Hatcher Algebraic Topology.

Let $\mathbb{}H_*$ be a fixed homology theory. The map $f:X\rightarrow Y$ induces isomorphisms on $\mathbb{}H_*$, if and only if the map $\{\cdot\}\hookrightarrow C_f$ induces an isomorphism on $\mathbb{}H_*$, i.e. $\mathbb{}H_*(C_f,pt)=0$.

See also

• Mapping cone (homological algebra)

References

1. Peter May "A Concise Course in Algebraic Topology", section 14.2