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

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

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

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.

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:

(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 ${\displaystyle M_{f}}$. An explicit formula for the strong deformation retraction can be worked out.

For a fiber bundle ${\displaystyle \pi :P\to X}$ with fiber ${\displaystyle F}$, the mapping cylinder