The Relative Homotopy Equivalence

An important corollary of Proposition 7.11 concerns the homotopy type of CW complexes. 

Corollary 7.12. The homotopy type of a CW complex does not change if the cell attachment maps are replaced by the homotopic maps. 

This is immediate if we take the constructive definition of CW complexes, and use Proposition 7.11 to deform the attaching maps. 

In fact, one can get a very explicit picture. Assume that we are attaching a cell e to a CW complex X over the cell s boundary sphere Let 

Definition 7.13. Let X and Y he topological spaces, and let A be a subspace of both X and of Y. A continuous map f X Y such that f A = id i is called a homotopy equivalence rel A if there exists a continuous map g Y X such that 

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