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mapping cylinder

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Definition

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

X f Y σ 0 f *(σ 0) X×I (σ 0) *(f) Cyl(f) \array{ X &\stackrel{f}\to& Y \\ {}^{\mathllap{\sigma_0}}\downarrow && \downarrow^{\mathrlap{ f_*(\sigma_0)}} \\ X\times I &\stackrel{(\sigma_0)_* (f)}\to & Cyl(f) }

in Top, where I=[0,1]I = [0,1] (the unit interval) and σ 0:XX×I\sigma_0:X\to X\times I is given by x(x,0)x\mapsto (x,0). By tradition, homotopy theorists sometimes use the inverted (upside-down) mapping cylinder where σ 0\sigma_0 is replaced by σ 1:x(x,1)\sigma_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 II.

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

Properties

As any other pushout, the mapping cylinder has a universal property: for any space ZZ and mapping g 1:X×IZg_1:X\times I\to Z, g 2:YZg_2:Y\to Z such that g 1(x,0)=g 2(f(x))g_1(x,0)=g_2(f(x)) for all xXx\in X, there is a unique k:Cyl(f)Zk:Cyl(f)\to Z, such that the composition X×ICyl(f)kZX\times I\to Cyl(f)\stackrel{k}\to Z equals g 1g_1 and the composition YCyl(f)kZY\to Cyl(f)\stackrel{k}\to Z equals g 2g_2.

Theorem

For f:XYf \colon X\to Y a continuous function, the canonical map jf *(σ 0):YCyl(f)j \coloneqq f_*(\sigma_0) \colon Y\to Cyl(f) is a homotopy equivalence. In fact its homotopy inverse can be chosen a deformation retraction. In particular every continuous function factors as a map into its mapping cylinder followed by a deformation retraction.

Proof

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

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

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

Theorem

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

Theorem

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

Theorem

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

Xσ 1X×I(σ 0) *(f)Cyl(f)X\stackrel{\sigma_1}\to X\times I\stackrel{(\sigma_0)_* (f)}\to Cyl(f)

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

The composition r(σ 0) *(f)σ 1=fr\circ (\sigma_0)_* (f)\circ \sigma_1 = f, hence this is a decomposition of a continuous map into a cofibration followed by a homotopy equivalence.

In homotopy type theory mapping cyclinders can be constructed as higher inductive types. See there.

Revised on June 16, 2016 05:33:07 by Urs Schreiber (131.220.184.222)