Algebraic Topology: Lecture 4

We showed that for a relative CW-complex (X, A), the union of the subspaces X x {0} and A x [0,1] of X x [0,1] is a strong deformation retract. In particular, the union of X x {0} and A x [0,1] is a retract of X x [0,1] and this, in turn, is equivalent to the statement that the inclusion of A in X has the Homotopy Extension Property. We used this to prove the following Compression Lemma: Let (X, A) be a relative CW-complex, and let (Y, B) be a pair of a space Y and a subspace B. Assume that the inclusion of B in Y is a weak equivalence. Then every map f : (X, A) --> (Y, B) is homotopic relative to A to a map from X to B. If, in particular, (X, A) is a relative CW-complex and if the inclusion of A in X is a weak equivalence, then A is a strong deformation retract of X. The compression lemma implies the following theorem of J.H.C. Whitehead: Let f : X --> Y be a weak equivalence between two CW-complexes. Then f is a homotopy equivalence. Finally, we briefly discussed the homotopy category of pointed spaces. For reading see Hatcher, Chap. 0 and 4, or Spanier, Chap. 7, Sect. 6.