The CW-Homology Theorem

Abstract

In this chapter we introduce singular homology, and we prove the CWHomology Theorem. The CW-Homology Theorem (Theorem 2.15) states that the singular homology H _* (X,A; Λ) is isomorphic to the homology of the CWchain complex (C _* (X, A; A), a), and it gives a formula for computing the boundary operator ∂ _* in the CW-chain complex in terms of the degrees of the attaching maps. We also prove some basDic theorems from homotopy theory. In particular, we prove that if A C X is closed and the inclusion A ⊆ X is a cofibration, then Hk(X, A; Λ) Hk(X/A, *; Λ) for all k (Corollary 2.31).