Mapping cylinder

Note that the mapping cone and the mapping cylinder, which will be defined in Section 6.3, are examples of the space attachment constructions. Attaching a cell along its boimdary is another such example, in this case X = S ", A = dW, and the attachment map is an arbitrary continuous map / dB Y. 

Definition 6.8. Let f X — Y be a continuous map between two topological spaces. The mapping cylinder of f is the quotient space... 

Fig. 6.1. A mapping cylinder and a mapping cone of a map taking everything to...

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. 

Proof. Assume first that the inclusion map i A X is a, cofibration. Consider maps / X —> M(f) and H A x I M(f), where the first one is induced by the identity map idx X —> X, and the second one identifies the cylinder Ax I with the corresponding cylinder inside the mapping cylinder M(i). These maps satisfy the conditions of Definition 7.1 hence there must exist a homotopy H X x I M(i) extending the maps H and /. Clearly, this implies that H is the desired retract map H oj = idM(i)-... 

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(/),... 

For the diagram from Example 15.9(2), we see that pb is the canonical projection of the mapping cylinder to an interval, whereas pf collapses the mapping cylinder to its base space. 

We start with dimZ = 0. In this case, the space MhocoUmj" is simply a disjoint union of several mapping cylinders, one for each vertex of A. Since for each v e the map J (v) is a homotopy equivalence, we can apply the mapping cylinder retraction, as in corollary 7.16, to each one of these mapping cylinders, and derive the necessary conclusion. 

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... 

---