Deformation retraction

Correspondingly, A is called a retract, a deformation retract, or a strong deformation retract of X. 

A useful example of a strong deformation retraction is provided by the following. 

Proposition 6.14. A sequence of collapses yields a strong deformation retraction, in particular, a homotopy equivalence. 

Another source of strong deformation retracts is provided by mapping cylinders. 

Although strong deformation retraction seems like a much stronger operation than homotopy equivalence, it turns out that two topological spaces are homotopy equivalent if and only if there exists a third space that can be strong deformation retracted both onto X and onto Y. One possible choice for this third space is simply the mapping cylinder of the homotopy equivalence map see Corollary 7.16. 

In either case. Van Kampen s theorem can be applied. The space A is contractible, the intersection A fl B is homeomorphic to a circle, and the space B deformation retracts to a circle. To see the last of these facts, simply retract the punctured unit disk to its boundary, and note that the antipodal self-identification of the boundary again produces a circle. We have nfiA) = 0 and 7Ti(B) = Tri(AnB) = Z. The fundamental group homomorphisms induced by the inclusion maps i AC B A and An B B are the following ... 

Definition 7.4. A pair of topological spaces (X,A), A C X, is called an NDR-pair (which is an o66remot ora/or neighborhood deformation retract j if there exist continuous maps u X I (think of it as a separation map) and h X x I X (think of it as a homotopy) such that... 

On the other hand, symmetrically, a strong deformation retraction of X x 7 onto Xx l uXx7 induces a strong deformation retraction of Z onto... 

Proof. Applying Theorem 7.14 to the map i A X gives that i is actually a homotopy equivalence relative to A. This means that there exists a map r X A that is a homotopy inverse of i relative to A. Untangling definitions, this translates to r A = id, and ior idx rel A, which means precisely that A is a strong deformation retract of X. ... 

A continuous map between topological spaces f X — Y is a homotopy equivalence if and only if X is a strong deformation retract of the mapping cylinder M(/),... 

Intuitively, a cellular collapse is a strong deformation retract that pushes the interior of a maximal cell in, using one of its free boundary cells as the starting point, much like compressing a body made of clay. The cellular collapses can be defined for arbitrary CW complexes. 

The reduction X ne Y implies the existence of a collapsing sequence X, Y, which, in turn implies that, viewed as a topological space, Y is a strong deformation retract of X. 

Ascending and descending closure operators induce strong deformation retractions of Z (P) onto A p P)). Here we give a short and self-contained inductive proof of the following stronger fact. 

Under the conditions of Theorem 13.22(b), the topological space A(Q) is a strong deformation retract of the topological space A P). 

It is obvious that MhocoUm. r deformation retracts onto hocolimP2i and it remains to be checked that it also deformation retracts onto hocolim T>i. We shall construct this retraction by induction on the dimension of the underlying trisp A. 

Assume now that dim. 4 = n > 0. We already know by the induction hypothesis that the space MJ jQj. UhocolimI>i deformation retracts onto hocolimPi, where denotes the part of the mapping cylinder lying... 

It is not difficult to see that the pair (X, A) is NDR. Therefore it is enough to show that the inclusion map A X is a homotopy equivalence. As we have said, A deformation retracts onto hocolimX>i, which in turn, by the construction of homotopy colimit, deformation retracts onto On the other hand, again by the construction of homotopy colimit, the whole space X deformation retracts onto the mapping cylinder of the map X n) An —> Bn-Since X(n) is a homotopy equivalence, we conclude that A is... 

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