Deformation Retracts and Collapses

The cells of XjY are precisely those cells of X that are not in Y, together with one 0-cell, which denotes the result of shrinking y to a point. The new attaching maps are best described as compositions of the old ones with the map that shrinks all cells of y to a point. 

We are now ready to see, using Facts 1 and 3, that there is a natural induced CW structure on M(/). In this structure we have the cells of X, the cells of y, and, additionally, the cells of X multiplied by an interval glued in the appropriate way. This is because M(/) = X x I) U j Y, where / maps the bottom copy of X inside X x / to the CW complex Y along the cellular map /. 

Observe that it is important in Example 6.9 to choose the CW structure on the interval [0,1] so that the map / is cellular. For instance, if we take the endpoints to be the vertices and the open interval (0,1) to be the 1-cell, then the induced cell structure on M(/) will not be well-defined, since we are not allowed to glue the boundaries of cells onto the interiors of cells of the same dimension or higher. 

When both X and Y are CW complexes and the map / is cellular, there is a natural induced CW structure on Cone(/) as well. This is because Cone(/) = M(/)/X, where we quotient the top copy of X inside of X x / to a point, and use Fact 2. 

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

---