deformation retract



Homotopy theory

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Paths and cylinders

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Basic facts




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.


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

i:AX i : A \to X

(the deformation retract itself) is another morphism

r:XA r : X \to A

such that

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


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

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

X d 0 Id X IX σ X X d 1 Id X 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:AXi : A \to X is a morphism r:XAr : X \to A such that ri=id Ar \circ i = id_A and such that there exists a morphism η:IXX\eta : I \otimes X \to X fitting into a diagram

X r A d 0 i IX η X d 1 Id X X. \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 η\eta is a strong deformation retract if moreover

η(Ii)=σ X(Ii) \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.


In topological spaces

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

With respect to the corresponding notion of left homotopy, if XX is a topological space and AXA\subset X a subspace, then AA is a strong deformation retract of XX if there exists a continuous map H:X×IXH \colon X\times I\to X such that H(a,t)=aH(a,t) = a for all aAa\in A, tI=[0,1]t\in I=[0,1], H(x,0)=xH(x,0) = x for all xXx\in X and H(x,1)AH(x,1)\in A for all xXx\in X.

Equivalently, there are continuous maps i:AXi \colon A\to X and r:XAr \colon X\to A such that ri=id Ar \circ i = id_A and irid X(relA)i\circ r\sim id_X (rel A), where (relA)\sim (rel A) denotes homotopy with fixed AA. More generally, for any continuous map j:ZYj \colon Z\to Y we say that it is deformation retractable if there is r:YZr \colon Y\to Z such that jrid Yj\circ r\sim id_Y and rj=id Zr\circ j = id_Z.

A pair (X,A)(X,A) is an NDR-pair if there is a pair of continuous maps, u:XI,H:X×IXu \colon X\to I,\; H \colon X\times I\to X such that H(a,t)=aH(a,t)=a for all aAa\in A and all tt, H(x,0)=xH(x,0)=x for all xXx\in X, u 1(0)=Au^{-1}(0)=A and H(x,1)AH(x,1)\in A for all xx such that u(x)<1u(x)\lt 1. If (X,A)(X,A) is an NDR-pair, then the inclusion has a left homotopy inverse iff AA is also a retract of XX (in Top, in the standard category-theoretic sense).

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

In chain complexes


(Eilenberg-Zilber/Alexander-Whitney deformation retraction)


and denote

Then there is a deformation retraction


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

Last revised on September 18, 2021 at 09:57:05. See the history of this page for a list of all contributions to it.