Contents

# Contents

## Idea

A deformation retract is a retract which is also a section up to homotopy. Equivalently, it is a homotopy equivalence one of whose two homotopies is in fact an identity.

## Definition

Let $\mathcal{C}$ be a category equipped with a notion of homotopy between its morphisms. Then a deformation retraction of a morphism

$i : A \to X$

(the deformation retract itself) is another morphism

$r : X \to A$

such that

$\array{ && X \\ & {}^{\mathllap{i}}\nearrow &\Downarrow^=& \searrow^{\mathrlap{r}} \\ A &&\stackrel{=}{\to}&& A }$

and

$\array{ && A \\ & {}^{\mathllap{r}}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{i}} \\ X &&\stackrel{=}{\to}&& X } \,.$

In particular, if “homotopy” in $\mathcal{C}$ means left homotopy with respect to an cylinder object $I \otimes X$

$\array{ X \\ \downarrow^{\mathrlap{d_0}} & \searrow^{\mathrlap{Id_X}} \\ I \otimes X &\stackrel{\sigma_X}{\to}& X \\ \uparrow^{\mathrlap{d_1}} & \nearrow_{\mathrlap{Id_X}} \\ X } \,,$

then a deformation retract of $i : A \to X$ is a morphism $r : X \to A$ such that $r \circ i = id_A$ and such that there exists a morphism $\eta : I \otimes X \to X$ fitting into a diagram

$\array{ X &\stackrel{r}{\to}& A \\ \downarrow^{\mathrlap{d_0}} && \downarrow^{\mathrlap{i}} \\ I \otimes X &\stackrel{\eta}{\to}& X \\ \uparrow^{\mathrlap{d_1}} & \nearrow_{\mathrlap{Id_X}} \\ X } \,.$

Hence a deformation retract is a (left) homotopy equivalence where one of the two homotopies occuring is in fact an identity.

If the cylinder object assignment here is functorial, we say that $\eta$ is a strong deformation retract if moreover

$\eta \circ (I \otimes i) = \sigma_X \circ (I \otimes i)$

(hence if the homotopy restricted to the inclusion is “constant” as seen by the chosen cylinder object).

In parts of the literature, deformation retracts are required to be strong by default.

## Examples

### In topological spaces

In the category Top of topological spaces the standard cylinder object is given by cartesian product with the interval $I \coloneqq [0,1]$.

With respect to the corresponding notion of left homotopy, if $X$ is a topological space and $A\subset X$ a subspace, then $A$ is a strong deformation retract of $X$ if there exists a continuous map $H \colon X\times I\to X$ such that $H(a,t) = a$ for all $a\in A$, $t\in I=[0,1]$, $H(x,0) = x$ for all $x\in X$ and $H(x,1)\in A$ for all $x\in X$.

Equivalently, there are continuous maps $i \colon A\to X$ and $r \colon X\to A$ such that $r \circ i = id_A$ and $i\circ r\sim id_X (rel A)$, where $\sim (rel A)$ denotes homotopy with fixed $A$. More generally, for any continuous map $j \colon Z\to Y$ we say that it is deformation retractable if there is $r \colon Y\to Z$ such that $j\circ r\sim id_Y$ and $r\circ j = id_Z$.

A pair $(X,A)$ is an NDR-pair if there is a pair of continuous maps, $u \colon X\to I,\; H \colon X\times I\to X$ such that $H(a,t)=a$ for all $a\in A$ and all $t$, $H(x,0)=x$ for all $x\in X$, $u^{-1}(0)=A$ and $H(x,1)\in A$ for all $x$ such that $u(x)\lt 1$. If $(X,A)$ is an NDR-pair, then the inclusion has a left homotopy inverse iff $A$ is also a retract of $X$ (in Top, in the standard category-theoretic sense).

The pair $(X,A)$ is a DR-pair if it is a deformation retract and there is a function $u \colon X\to I$ such that $A=u^{-1}(0)$ (i.e. it gives simultaneously a deformation retract and a NDR-pair). If $(X,A)$ is an NDR-pair then the inclusion $A\hookrightarrow X$ is a homotopy equivalence iff $A$ is a deformation retract of $X$. Any map $f:X\to Y$ is a homotopy equivalence iff $X$ is the deformation retract of the mapping cylinder of $f$. If $(X,A)$ is an NDR-pair and $A$ is contractible, then the quotient map $X\to X/A$ is a homotopy equivalence.

### In chain complexes

###### Proposition

Let

and denote

Then there is a deformation retraction

where

For unnormalized chain complexes, where we have a homotopy equivalence, this is the original Eilenberg-Zilber theorem (Eilenberg & Zilber 1953, Eilenberg & MacLane 1954, Thm. 2.1). The above deformation retraction for normalized chain complexes is Eilenberg & MacLane 1954, Thm. 2.1a. Both are reviewed in May 1967, Cor. 29.10. Explicit description of the homotopy operator is given in Gonzalez-Diaz & Real 1999.

Textbook accounts