Mapping cone (topology)

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 ${\displaystyle \forall x,x'\in X,(x,0)\sim (x',0)\,}$, ${\displaystyle (x,1)\sim f(x)}$. 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) 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 ${\displaystyle C_{f}=Y/f(X)}$; 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, ${\displaystyle (x_{0},t)\sim (x_{0},t')\,.}$ Thus one end and the "seam" are all identified with ${\displaystyle y_{0}.}$

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]

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

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.

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]

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]

• Cofibration
• Mapping cone (homological algebra)