By Andrew H. Wallace
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Example text
We use the usual argument to show that the mod k Hurewicz isomorphism theorem for spaces implies a similar mod k isomorphism theorem for pairs of spaces. 1 Basic definition In order to relate integral homology and integral cohomology, it is convenient to introduce the following two distinct notions of duality. 1. (A) If F is a finitel generated torsion free abelian group, let F ∗ = Hom(F, Z). (B) If T is a finit abelian group, let T ∗ = Hom(T, Q/Z). Thus, Z∗ ∼ = Z generated by the identity map 1Z : Z → Z and (Z/kZ)∗ ∼ = Z/kZ generated by the map which sends 1 to 1/k.
This completes the inductive step in the proof. Hence the mod k Hurewicz theorem is true for all the Postnikov stages Em , . Since X = limm →∞ Em , is an inverse limit which is finit in each degree, it follows that the mod k Hurewicz theorem is true for all X. Exercise (1) Suppose k and are positive integers. Suppose either that X is simply connected or that X is a connected H-space. Show that ϕ : πj (X; Z/kZ) → Hj (X; Z/kZ) is an isomorphism for all 1 ≤ j < and an epimorphism for j = if and only if the same is true for ϕ : πj (X; Z/kr Z) → Hj (X; Z/kr Z) where r is a fi ed positive integer.
Let X be a nilpotent space with abelian fundamental group and let n ≥ 1. Suppose πi (X; Z/kZ) = 0 for all 1 ≤ i ≤ n − 1. Then the mod k Hurewicz homomorphism ϕ : πi (X; Z/kZ) → Hi (X; Z/kZ) is: (a) an isomorphism for all 1 ≤ i ≤ n. (b) an epimorphism for i = n + 1 if n ≥ 2. (c) an isomorphism for i = n + 1 and an epimorphism for i = n + 2 if n ≥ 3 and k is odd. Proof: The strategy of this proof is as follows: (1) First, for all n ≥ 1, show that it is true for Eilenberg–MacLane spaces. (2) Second, for all n ≥ 1, show that it is true for a general space by considering its Postnikov system.