Contens

Definition

A continuous map $i:A↪X$ is a Hurewicz cofibration if it satisfies the homotopy extension property.

This means that

• for any topological space $Y$,

• all continuous maps $f:A\to Y$, $\stackrel{˜}{f}:X\to Y$ such that $\stackrel{˜}{f}\circ i=f$

• and a homotopy $F:A×I\to Y$ such that $F\left(-,0\right)=f$

there is a homotopy $\stackrel{˜}{F}:X×I\to Y$ such that

• $\stackrel{˜}{F}\circ \left(i×{\mathrm{id}}_{I}\right)=F$

• and $\stackrel{˜}{F}\left(-,0\right)=\stackrel{˜}{f}$.

$\begin{array}{ccc}A& \stackrel{f}{\to }& Y\\ ↓& {↗}_{\stackrel{˜}{f}}\\ X\end{array}$\array{ A &\stackrel{f}{\to}& Y \\ \downarrow & \nearrow_{\mathrlap{\tilde f}} \\ X }

We say that a Hurewicz cofibration $i:A\to X$ is a closed cofibration if $i\left(A\right)$ is closed in $X$.

If $A\subset X$ is closed and the inclusion is a cofibration, then the pair $\left(X,A\right)$ is an NDR-pair.

There is also a version for pointed spaces.

Properties

Closedness

Every Hurewicz cofibration $i$ is an injective map and if the image $i\left(A\right)$ is closed then it is a homeomorphism onto its image. In the category of weakly Hausdorff compactly generated spaces, $i\left(A\right)$ is always closed (the same in the category of all Hausdorff spaces), but in the category of all topological spaces there are pathological counterexamples. The simplest example (see the classical monograph Dieck, Kamps, Puppe, Homotopietheorie, LNM 157) is the following: let $A=\left\{a\right\}$ and $X=\left\{a,b\right\}$ be the one and two element sets, both with antidiscrete topology (only $X$ and $\varnothing$ are open in $X$), and $i:A↪X$ is the inclusion $a↦a$. Then $i$ is a non-closed cofibration (useful exercise!).

Model structure

The collections

• closed Hurewicz cofibrations

make one of the standard Quillen model category structures on the category Top of all topological spaces; see Strøm's model category.

Interaction with pullbacks

Theorem

Let

$\begin{array}{ccc}{X}_{0}& ↪& X\\ {}^{{p}_{0}}↓& & {↓}^{p}\\ {B}_{0}& ↪& B\\ ↑& & ↑\\ {E}_{0}& ↪& E\end{array}$\array{ X_0 &\hookrightarrow & X \\ {}^{\mathllap{p_0}}\downarrow && \downarrow^{\mathrlap{p}} \\ B_0 &\hookrightarrow& B \\ \uparrow && \uparrow \\ E_0 &\hookrightarrow& E }

be a commuting diagram of topological spaces such that

• the horizontal morphisms are closed cofibrations;

• the morphisms ${p}_{0}$ and $p$ are Hurewicz fibrations.

Then the induced morphism on pullbacks is also a closed cofibration

${X}_{0}{×}_{{B}_{0}}{E}_{0}↪X{×}_{B}E\phantom{\rule{thinmathspace}{0ex}}.$X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E \,.

This is stated and proven in (Kieboom).

Corollary

The product of two closed cofibrations is a closed cofibration.

References

• Dieter Puppe, Bemerkungen über die Erweiterung von Homotopien, Arch. Math. (Basel) 18 1967 81–88; MR0206954 (34 #6770) doi

• Arne Strøm, Note on cofibrations, Math. Scand. 19 1966 11–14 file MR0211403 (35 #2284); Note on cofibrations II, Math. Scand. 22 1968 130–142 (1969) file MR0243525 (39 #4846)

The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in

• R. W. Kieboom, A pullback theorem for cofibrations (1987) , (web)

Revised on November 9, 2011 13:19:01 by Urs Schreiber (131.174.41.104)