Strong 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. 

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). 

The vulcanization process is necessary to produce most useful rubber articles, like tires and mechanical goods. Unvulcanized rubber is genially not strong, does not retract essentially to its original shape after a large deformation, and it can be very sticky. In short, unvulcanized rubber can have the same consistency as chewing gum. 

The entropy forces are weak. They are strong enough, however, to cause the retraction of the specimen when the applied load is removed. This is because (we emphasize again) the polymer chains are in the liquid state. If the tie points did not exist—if the molecules were not pinned together at particular points—the assembly would flow in a liquid-like manner. It is this combination of long-range, weak entropy forces and liquid molecules which confers on rubbers their twin properties of high deformability and, paradoxically, complete retraction (3.N.4). 

The entropy force is weak. However, it is strong enough to cause a retraction in the liquid state when the applied stress is removed [18]. When the work done by the internal forces is balanced by the work done by the external forces producing the deformation, the total work, W, done for uniaxial drawing is obtained as ... 

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