Mapping cone (topology)

An illustration of a mapping cone; that is, a cone is glued to a space along some function ${\displaystyle f\colon X\to Y}$.

In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted ${\displaystyle C_{f}}$. Alternatively, it is also called the homotopy cofiber and also notated ${\displaystyle Cf}$. Its dual, a fibration, is called the mapping fiber. The mapping cone can be understood to be a mapping cylinder ${\displaystyle Mf}$ with the initial end of the cylinder collapsed to a point. Mapping cones are frequently applied in the homotopy theory of pointed spaces.

Definition

Given a map ${\displaystyle f\colon X\to Y}$, the mapping cone ${\displaystyle C_{f}}$ is defined to be the quotient space of the mapping cylinder ${\displaystyle (X\times I)\sqcup _{f}Y}$ with respect to the equivalence relation ${\displaystyle \forall x,x'\in X,(x,0)\sim \left(x',0\right)\,}$, ${\displaystyle (x,1)\sim f(x)}$. Here ${\displaystyle I}$ denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1.

Visually, one takes the cone on X (the cylinder ${\displaystyle X\times I}$ with one end (the 0 end) collapsed to a point), and glues the other end onto Y via the map f (the 1 end).

Coarsely, one is taking the quotient space by the image of X, so ${\displaystyle C_{f}=Y/f(X)}$; 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 ${\displaystyle f\colon (X,x_{0})\to (Y,y_{0})}$ (so ${\displaystyle f\colon x_{0}\mapsto y_{0}}$), one also identifies all of ${\displaystyle x_{0}\times I}$. Formally, ${\displaystyle (x_{0},t)\sim \left(x_{0},t'\right)}$. Thus one end and the "seam" are all identified with ${\displaystyle y_{0}.}$

Example of circle

If ${\displaystyle X}$ is the circle ${\displaystyle S^{1}}$, the mapping cone ${\displaystyle C_{f}}$ can be considered as the quotient space of the disjoint union of Y with the disk ${\displaystyle D^{2}}$ formed by identifying each point x on the boundary of ${\displaystyle D^{2}}$ to the point ${\displaystyle f(x)}$ in Y.

Consider, for example, the case where Y is the disk ${\displaystyle D^{2}}$, and ${\displaystyle f\colon S^{1}\to Y=D^{2}}$ is the standard inclusion of the circle ${\displaystyle S^{1}}$ as the boundary of ${\displaystyle D^{2}}$. Then the mapping cone ${\displaystyle C_{f}}$ is homeomorphic to two disks joined on their boundary, which is topologically the sphere ${\displaystyle S^{2}}$.

Double mapping cylinder

The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder ${\displaystyle X\times I}$ joined on one end to a space ${\displaystyle Y_{1}}$ via a map

${\displaystyle f_{1}:X\to Y_{1}}$

and joined on the other end to a space ${\displaystyle Y_{2}}$ via a map

${\displaystyle f_{2}:X\to Y_{2}}$

The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one of ${\displaystyle Y_{1},Y_{2}}$ is a single point.

Dual construction: the mapping fibre

Main article: Homotopy fiber

The dual to the mapping cone is the mapping fibre ${\displaystyle F_{f}}$. Given the pointed map ${\displaystyle f\colon (X,x_{0})\to (Y,y_{0}),}$ one defines the mapping fiber as ^[1]

${\displaystyle F_{f}=\left\{(x,\omega )\in X\times Y^{I}:\omega (0)=y_{0}{\mbox{ and }}\omega (1)=f(x)\right\}}$.

Here, I is the unit interval and ${\displaystyle \omega }$ is a continuous path in the space (the exponential object) ${\displaystyle Y^{I}}$. The mapping fiber is sometimes denoted as ${\displaystyle Mf}$; 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 ${\displaystyle X\times _{f}Y}$ which is dual to the pushout ${\displaystyle X\sqcup _{f}Y}$ used to construct the mapping cone. ^[2] In this particular case, the duality is essentially that of currying, in that the mapping cone ${\displaystyle (X\times I)\sqcup _{f}Y}$ has the curried form ${\displaystyle X\times _{f}(I\to Y)}$ where ${\displaystyle I\to Y}$ is simply an alternate notation for the space ${\displaystyle Y^{I}}$ of all continuous maps from the unit interval to ${\displaystyle Y}$. 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.

Applications

CW-complexes

Attaching a cell.

Effect on fundamental group

Given a space X and a loop ${\displaystyle \alpha \colon S^{1}\to X}$ representing an element of the fundamental group of X, we can form the mapping cone ${\displaystyle C_{\alpha }}$. The effect of this is to make the loop ${\displaystyle \alpha }$ contractible in ${\displaystyle C_{\alpha }}$, and therefore the equivalence class of ${\displaystyle \alpha }$ in the fundamental group of ${\displaystyle C_{\alpha }}$ 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. Namely, if E is a homology theory, and ${\displaystyle i\colon A\to X}$ is a cofibration, then

${\displaystyle E_{*}(X,A)=E_{*}(X/A,*)={\tilde {E}}_{*}(X/A)}$,

which follows by applying excision to the mapping cone. ^[2]

Relation to homotopy (homology) equivalences

A map ${\displaystyle f\colon X\rightarrow Y}$ 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. ^{[3] [ page needed ]}

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

Mapping cones are famously used to construct the long coexact Puppe sequences, from which long exact sequences of homotopy and relative homotopy groups can be obtained. ^[1]

See also

• Cofibration
• Mapping cone (homological algebra)

References

1. Rotman, Joseph J. (1988). An Introduction to Algebraic Topology. Springer-Verlag. ISBN   0-387-96678-1. See Chapter 11 for proof.
2. May, J. Peter (1999). A Concise Course in Algebraic Topology (PDF). Chicago Lectures in Mathematics. ISBN   0-226-51183-9. See Chapter 6.
3. • Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN   9780521795401.