Contents

# Contents

## Idea

In algebraic topology and homotopy theory, Hurewicz cofibrations are a kind of cofibration of topological spaces, hence a kind of continuous function satisfying certain extension properties.

Specifically, a continuous function is a Hurewicz cofibration (Strøm 1966) if it satisfies the homotopy extension property for all target spaces and with respect to the standard notion of left homotopy of topological spaces given by the standard topological interval object/cylinder object.

A pointed topological space $(X,x)$ for which the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a Hurewicz cofibration is called a well-pointed topological space. A simplicial topological space all whose degeneracy maps are Hurewicz cofibrations is called a good simplicial topological space.

In point-set topology Hurewicz cofibrations are often just called cofibrations,for short. If their image is a closed subspace they are called closed cofibrations.

Beware that there are other relevant classes of cofibrations between topological spaces, notably the Serre cofibrations, and (in homotopy theory and model category-theory, at least) also often just called “cofibrations”. But “closed cofibration” always refers to closed Hurewicz cofibrations.

In fact, the notion of homotopy extension property makes sense in any category with a chosen cylinder object; and in this generality one also speaks of h-cofibrations (following Mandell, May, Schwede & Shipley 2001, p. 16).

## Definition

###### Definition

(Hurewicz cofibration)
A map (continuous function) $i \colon A \xrightarrow{\;} X$ betwee topological spaces is a Hurewicz cofibration if it satisfies the homotopy extension property for all spaces, hence if

• for any map $f \colon X \longrightarrow Y$ and any left homotopy $\eta \,\colon\, A \times [0,1] \longrightarrow Y$ such that $\eta(-,0) = f \circ i$, there exists a left homotopy $\widehat{\eta} \,\colon\, X \times [0,1] \longrightarrow Y$ such that $\widehat{\eta} \circ (i \times id) = \eta$;

equivalently:

• given a commuting diagram in TopSp of solid arrows as shown below, there exists a dashed arrow $\widehat \eta$ making all sub-diagrams commute:

###### Remark

(re-formulation in terms of right homotopies)
In a convenient cartesian closed category $TopSp$ of topological spaces (such as that of compactly generated topological spaces) the product $\dashv$ internal hom-adjunction

$TopSp \underoverset {\underset{ (-)^{[0,1]} }{\longrightarrow}} {\overset{ (-) \times [0,1] }{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} TopSp$

allows to pass to adjunct morphisms in Def.

$\array{ TopSp \big( A \times [0,1] ,\, Y \big) &\simeq& TopSp \big( A ,\, Y^{[0,1]} \big) \\ \eta &\leftrightarrow& \eta' }$

and equivalently express the above diagram of left homotopies as the following (somewhat more transparent) diagram of right homotopies, where $ev_0 \,\colon\, Y^{[0,1]}\to Y$ denotes evaluation at 0 $\gamma \mapsto \gamma(0)$:

In this equivalent formulation, the homotopy extension property is simply the right lifting property against the evaluation map $ev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y$ out of the path space of $Y$.

###### Definition

(closed cofibrations)
A Hurewicz cofibration $i \colon A\to X$ (Def. ) is called a closed cofibration if the image $i(A)$ is a closed subspace in $X$. In this case the pair $(A,X)$ is also called an NDR-pair.

## Properties

### (Non-)Closed images

###### Proposition

Every Hurewicz cofibration $i \colon A \to X$ is injective and a homeomorphism onto its image. If $X$ is Hausdorff, then $i$ is closed.

(tDKP 1970 (1.17)).

###### Proposition

In the category of weakly Hausdorff compactly generated spaces, the image $i(A)$ of a Hurewicz cofibration is always closed. The same holds in the category of all Hausdorff spaces.

But in the plain category Top of all topological spaces there are pathological counterexamples:

###### Example

Let $A =\{a\}$ and $X=\{a,b\}$ be the one and two element sets, both with the codiscrete topology (only $X$ and $\varnothing$ are open subsets of $X$), and $i:A\hookrightarrow X$ is the inclusion $a\mapsto a$. Then $i$ is a non-closed cofibration.

(Strøm 1966, p. 5)

### Characterization for closed subspace inclusions

When the image of a Hurewicz cofibration is a closed subspace – which is automatically the case in weakly Hausdorff topological spaces, by Prop. HurewiczCofibrationsInCGWHSpacesAreClosed – then there are a number of equivalent reformulations of the defining homotopy extension property (Def. ).

#### Via retracts of the pushout-product with $0 \hookrightarrow [0,1]$

###### Proposition

A closed topological subspace-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if its pushout product with the endpoint inclusion $0 \,\colon\, \ast \xhookrightarrow{\;} [0,1]$ into the topological interval

(1)$X \!\times\! \{0\} \,\cup\, A \!\times\! [0,1] \; \xrightarrow{\;\;\;\;\;\;} \; X \times [0,1]$

admits a retraction $r$:

(Strøm 1968 Thm. 2; review in May 1999, Sec. 6.4, p. 45 (53 of 251); AGP 2002, Thm. 4.1.7; Gutiérrez, Prop. 8.3)

###### Example

Let $A \xhookrightarrow{i} X$ be a closed topological subspace-inclusion of compactly generated topological spaces which is a Hurewicz cofibration. Then for $Y$ any compactly generated topological space, the k-ified product space-construction $Y \times A \xhookrightarrow{ id_Y \times i } Y \times X$ is itself a Hurewicz cofibration.

(e.g. AGP 2002, Ex. 4.2.16)
###### Proof

Since compactly generated spaces with the k-ified product space construction form a cartesian closed category, the operation $Y \times (-)$ is a left adjoint and hence preserves the pushout in (1). It follows that the retract which exhibits, according to Prop. , the cofibration property of $i$, may be extended as a constant function in $Y$ to yield a retract that exhibits the cofibration property of $Y \times i$.

#### Via neighbourhood deformations

###### Proposition

A closed topological subspace-inclusion $A \xhookrightarrow{\;i\;} X$ is a Hurewicz cofibration precisely if the following condition holds:

There exists:

(1) a neighbourhood $U$ of $A$ in $X$

$A \xhookrightarrow{\;} U \xhookrightarrow{\;} X$

(2) a left homotopy $\eta$ shrinking this neighbourhood into $A$ relative to $A$, in that it makes the following diagram commute:

(3) another continuous function $\phi \colon X \xrightarrow{\;} [0,1]$ which makes the following diagram commute:

and in fact such that only $A$ is the preimage of zero: $A \,=\, \phi^{-1}(\{0\})$.

(This is due to Strøm 1966, Thm. 2; recalled, e.g., in Bredon 1993, Thm. 1.5 on p. 431)

### Assorted facts

###### Proposition

If $X$ is a normal topological space then any inclusion $A \xhookrightarrow{\;} X$ is an h-cofibration iff its factorization $A \hookrightarrow{\;} U_A$ through any open neighbourhood $U_A \subset X$ of $A$ is an h-cofibration.

(AGP 2002, Prop. 4.1.9)

###### Proposition

The composition of two h-cofibrations is itself an h-cofibration.

(e.g. AGP 2002, Ex. 4.2.17)

### Interaction with (fiber) producs

###### Proposition

If

$\array{ A_1 & & A_2 \\ \big\downarrow {}^{\mathrlap{i_1}} & , & \big\downarrow {}^{\mathrlap{i_2}} \\ X_1 & & X_2 }$

is a pair of Hurewicz cofibrations, with $i_1(A_1) \underset{clsd}{\subset} X_1$ closed, then the inclusion

$\array{ A_1 \times X_2 \;\cup\; X_1 \times A_2 \\ \big\downarrow \\ X_1 \times X_2 }$

into the product space of the codomains is itself a Hurewicz cofibration.

(Strøm 1968, Thm. 6)

###### Proposition

(fiber product of Hurewicz fibrations preserves Hurewicz cofibrations)
Let

$\array{ X_0 &\hookrightarrow & X \\ {}^{\mathllap{p_0}}\downarrow && \downarrow^{\mathrlap{p}} \\ B_0 &\hookrightarrow& B \\ \uparrow && \uparrow \\ E_0 &\hookrightarrow& E }$

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 fiber products

$X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E$

is also a closed cofibration.

(Kieboom 1987, Thm. 1).

###### Corollary

(Cartesian product preserves Hurewicz cofibrations)
If

$\array{ A_1 & & A_2 \\ \big\downarrow {}^{\mathrlap{i_1}} & , & \big\downarrow {}^{\mathrlap{i_2}} \\ X_1 & & X_2 }$

is a pair of Hurewicz cofibrations, then their image under the product functor is, too:

$\array{ A_1 \times A_2 \\ \big\downarrow \\ X_1 \times X_2 }$

###### Proof

This is the special case of Prop. for $B$ and $B_0$ being the point space

### Strøm’s model structure

The collections

• closed Hurewicz cofibrations (def. , def. ),

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

The identity functor $id \colon Top \to Top$ is left Quillen from the classical model structure on topological spaces (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.

$Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \,.$

This means in particular that any retract of a relative cell complex inclusion is a closed Hurewicz cofibration.

## Examples

### Relative cell complex inclusions

###### Example

Any relative CW complex-inclusion is an h-cofibration.

Proofs may be found spelled out in: Bredon 1993, Cor. I.4 on p. 431, Félix, Halperin & Thomas 2000, Prop. 1.9, Mathew 2010b

More generally, every retract of a relative cell complex inclusion is a closed Hurewicz cofibration.

This is part of the statement of the Quillen adjunction between then classical model structure on topological spaces and the Strøm model structure (see below).

### Points in locally Euclidean Hausdorff spaces

###### Proposition

Any point-inclusion into a (finite-dimensional) locally Euclidean Hausdorff space (e.g. a topological manifold) is an h-cofibration.

This is a simple special case of the general Prop. below, but we give an explicit proof:
###### Proof

By definition of locally Euclidean spaces, any point has a neighbourhood $U$ which is chart, being a Euclidean space that may be identified with a vector space $\mathbb{R}^n$ with the given point being the origin $0 \,\in\, \mathbb{R}^n$:

$\{0\} \;\in\; \mathbb{R}^n \;\simeq\; U \;\subset\; X \,.$

Now let:

• $\eta \colon U \times [0,1]\to X$ the homotopy given by $(\vec x, t) \mapsto (1-t)\cdot \vec x$;

• $K \;\coloneqq\; B_{\leq 1}(0)$ the closed ball of unit radius around the origin in $\mathbb{R}^n$ (hence compact by Heine-Borel);

• $\phi \colon X \to[0,1]$ be given by:

1. $x \mapsto min\big( \| x\|, 1 \big)$ on $U$ (the distance from the origin cut off at 1),

2. $x \mapsto 1$ on the complement $X \setminus K$.

It is manifest that this is a well-defined function and that the restrictions $\phi|_{U}$ and $\phi|_{X\setminus K}$ are continuous functions. Moreover, since compact subspaces of Hausdorff spaces are closed, $X \setminus K$ is open and $\big\{ U ,\, X \setminus K \big\}$ is an open cover of $X$. Therefore (by the sheaf-property of continuous functions), $\phi$ is continuous on all of $X$.

It is immediate to see that this data satisfies the conditions discussed in Prop. . Since Hausdorff spaces are $T_1$, so that all of their points are closed, that proposition applies and implies the claim.

### Closed submanifolds of Banach manifolds

###### Proposition

Let $X$ be an absolute neighbourhood retract (ANR) and $A \xhookrightarrow{i} X$ a closed subspace-inclusion. Then $i$ is a Hurewicz cofibration iff $A$ is itself an ANR.

(Aguilar, Gitler & Prieto 2002, Thm. 4.2.15)

###### Proposition

Let $X$ be a paracompact Banach manifold. Then the inclusion $A \hookrightarrow X$ of any closed sub-Banach manifold is a Hurewicz cofibration.

###### Proof

Being a closed subspace of a paracompact space, $A$ is itself paracompact (by this Prop.). But paracompact Banach manifolds are absolute neighbourhood retracts (this Prop.) Therefore the statement follows with Prop. .

### Further examples

###### Example

(point inclusion into PU(ℋ))
The projective unitary group PU(ℋ) on an infinite-dimensional separable Hilbert space is

Named after:

Original articles:

Textbook accounts:

Lecture notes:

Exposition:

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

The terminology “h-cofibration” is due to: