In mathematics, specifically algebraic topology, the mapping cylinder^[1] of a continuous function ${\displaystyle f}$ between topological spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is the quotient

${\displaystyle M_{f}=(([0,1]\times X)\amalg Y)\,/\,\sim }$

where the ${\displaystyle \amalg }$ denotes the disjoint union, and ∼ is the equivalence relation generated by

${\displaystyle (0,x)\sim f(x)\quad {\text{for each }}x\in X.}$

That is, the mapping cylinder ${\displaystyle M_{f}}$ is obtained by gluing one end of ${\displaystyle X\times [0,1]}$ to ${\displaystyle Y}$ via the map ${\displaystyle f}$. Notice that the "top" of the cylinder ${\displaystyle \{1\}\times X}$ is homeomorphic to ${\displaystyle X}$, while the "bottom" is the space ${\displaystyle f(X)\subset Y}$. It is common to write ${\displaystyle Mf}$ for ${\displaystyle M_{f}}$, and to use the notation ${\displaystyle \sqcup _{f}}$ or ${\displaystyle \cup _{f}}$ for the mapping cylinder construction. That is, one writes

${\displaystyle Mf=([0,1]\times X)\cup _{f}Y}$

with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone ${\displaystyle Cf}$, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.

Basic properties

The bottom Y is a deformation retract of ${\displaystyle M_{f}}$. The projection ${\displaystyle M_{f}\to Y}$ splits (via ${\displaystyle Y\ni y\mapsto y\in Y\subset M_{f}}$), and the deformation retraction ${\displaystyle R}$ is given by:

${\displaystyle R:M_{f}\times I\rightarrow M_{f}}$

${\displaystyle ([t,x],s)\mapsto [s\cdot t,x],(y,s)\mapsto y}$

(where points in ${\displaystyle Y}$ stay fixed because ${\displaystyle [0,x]=[s\cdot 0,x]}$ for all ${\displaystyle s}$).

The map ${\displaystyle f:X\to Y}$ is a homotopy equivalence if and only if the "top" ${\displaystyle \{1\}\times X}$ is a strong deformation retract of ${\displaystyle M_{f}}$.^[2] An explicit formula for the strong deformation retraction can be worked out.^[3]

Examples

Interpretation

The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense:

Given a map ${\displaystyle f\colon X\to Y}$, the mapping cylinder is a space ${\displaystyle M_{f}}$, together with a cofibration ${\displaystyle {\tilde {f}}\colon X\to M_{f}}$ and a surjective homotopy equivalence ${\displaystyle M_{f}\to Y}$ (indeed, Y is a deformation retract of ${\displaystyle M_{f}}$), such that the composition ${\displaystyle X\to M_{f}\to Y}$ equals f.

Thus the space Y gets replaced with a homotopy equivalent space ${\displaystyle M_{f}}$, and the map f with a lifted map ${\displaystyle {\tilde {f}}}$. Equivalently, the diagram

${\displaystyle f\colon X\to Y}$

gets replaced with a diagram

${\displaystyle {\tilde {f}}\colon X\to M_{f}}$

together with a homotopy equivalence between them.

The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.

Note that pointwise, a cofibration is a closed inclusion.

Applications

Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective.

Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to ${\displaystyle f\colon X\rightarrow Y}$ with the assumption that ${\displaystyle X\subset Y}$ and that ${\displaystyle f}$ is actually the inclusion of a subspace.

Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of ${\displaystyle X}$ to points of ${\displaystyle Y,}$ and hence of embedding ${\displaystyle X}$ within ${\displaystyle Y,}$ despite the fact that the function need not be one-to-one.

See also

• Cofibration
• Mapping cylinder (homological algebra)
• Homotopy colimit
• Mapping path space, which can be viewed as the mapping cocylinder

References

• May, J.P (1999). A Concise Course in Algebraic Topology. The University of Chicago Press. ISBN 978-0-2265-1183-2.