Homotopy Theory of CW-complexes

Abstract

In Chapter II we proved that if X is an n-cellular extension of the pathwise connected space A, then the pair (X, A) is (n — l)-connected, so that π _i (X, A) = 0 for all i < n. The next step is the determination of π _n (X, A). By the results of §2 of Chapter II, H _n (X, A) is a free abelian group with one basis element for each n-cell of (X, A). We have seen that the Hurewicz map ρ: π _n (X, A) → H _n (X, A) is an epimorphism whose kernel is generated by all elements of the form α — τ’ _ξ (α) with α ∈ π _n (X, A), ξ ∈ π ₁ (A). If II = π ₁ (A) operates trivially on π _n (X, A), then ρ is an isomorphism. But this condition is not easy to verify a priori.