nLab
mapping cylinder

Context

Homotopy theory

In Top
- Definition
- Properties
Related entries

In Top

Definition

Given a continuous map $f : X \to Y$ of topological spaces, one can define its mapping cylinder as a pushout

\begin{matrix} X & \overset{f}{\to} & Y \\ ^{σ_{0}} ↓ & ↓^{f_{*} (σ_{0})} \\ X \times I & \overset{(σ_{0})_{*} (f)}{\to} & Cyl (f) \end{matrix}

in Top, where $I = [0, 1]$ (the unit interval) and $σ_{0} : X \to X \times I$ is given by $x \mapsto (x, 0)$ . By tradition, homotopy theorists sometimes use the inverted (upside-down) mapping cylinder where $σ_{0}$ is replaced by $σ_{1} : x \mapsto (x, 1)$ . Of course the two mapping cylinders are homeomorphic so it is matter of convention which one to use, of course, compatibly with other constructions depending on the orientation of $I$ .

Set-theoretically, the mapping cylinder is usually represented as the quotient space $(X \times I ∐ Y) / \sim$ where $\sim$ is the smallest equivalence relation identifying $(x, 0) \sim f (x)$ for all $x \in X$ .

Properties

As any other pushout, the mapping cylinder has a universal property: for any space $Z$ and mapping $g_{1} : X \times I \to Z$ , $g_{2} : Y \to Z$ such that $g_{1} (x, 0) = g_{2} (f (x))$ for all $x \in X$ , there is a unique $k : Cyl (f) \to Z$ , such that the composition $X \times I \to Cyl (f) \overset{k}{\to} Z$ equals $g_{1}$ and the composition $Y \to Cyl (f) \overset{k}{\to} Z$ equals $g_{2}$ .

Theorem

Let $f : X \to Y$ be any continuous map. The canonical map $j : = f_{*} (σ_{0}) : Y \to Cyl (f)$ is a homotopy equivalence. In fact its homotopy inverse can be chosen a deformation retraction.

Proof

We exhibit $j$ as a homotopy equivalence by constructing its homotopy inverse $\tilde{f}$ given by $\tilde{f} : [x, t] \mapsto f (x)$ , where $[x, t]$ is a class of $(x, t) \in X \times I$ and $\tilde{f} ([y]) = [y]$ for $y \in Y$ . Clearly this map is well-defined and $\tilde{f} \circ j = {id}_{Y}$ . On the other hand, $(j \circ \tilde{f}) [x, t] = [f (x)]$ . Homotopy $H : Cyl (f) \times I \to Y$ is given by

H ([x, t], τ) = [x, t (1 - τ)], H ([y], τ) = [y] .

It is easy to see that $H (-, 0) = {id}_{Cyl (f)}$ , $H (-, 1) = [-, 0] = [f (-)]$ hence $j \circ \tilde{f} \sim {id}_{Cyl (f)}$ .

Theorem

A continuous map $i : A \to X$ is a Hurewicz cofibration iff there is a retraction $r : X \times I \to Cyl (f)$ for the canonical map $X \times I \to Cyl (f)$ .

Theorem

A continuous map $f : X \to Y$ is a homotopy equivalence iff $X = X \times {0}$ is a deformation retract of the cylinder $Cyl (f)$ .

Theorem

For any $f : X \to Y$ , the composition

X \overset{σ_{1}}{\to} X \times I \overset{(σ_{0})_{*} (f)}{\to} Cyl (f)

is a Hurewicz cofibration. Furthermore, the map $r : Cyl (f) \to Y$ determined by $r ([x, t]) = f (x)$ (for all $x \in X$ and $t \in I$ ) and $r ([y]) = y$ (for $y \in Y$ ) is well defined and a homotopy equivalence.

The composition $r \circ (σ_{0})_{*} (f) \circ σ_{1} = f$ , hence this is a decomposition of a continuous map into a cofibration followed by a homotopy equivalence.

nLab
mapping cylinder

Context

Homotopy theory

Background

Variations

Definitions

Paths and cylinders

Homotopy groups

Theorems

Contents

In Top

Definition

Properties

Theorem

Proof

Theorem

Theorem

Theorem

Related entries

nLab mapping cylinder

Context

Homotopy theory

Background

Variations

Definitions

Paths and cylinders

Homotopy groups

Theorems

Contents

In Top

Definition

Properties

Theorem

Proof

Theorem

Theorem

Theorem

Related entries

nLab
mapping cylinder