16 0. Prequel For years the classic monograph by W. Hurewicz and H. Wallman (1948) stood as the standard dimension theory reference. The more recent book by J. van Mill (1989) is an excellent alternative. 0.8. The Hurewicz Isomorphism Theorem and its localization The Hurewicz Theorem relates homotopy and homology groups. The fol- lowing statement appears on page 394 of Spanier (1966) and on page 366 of Hatcher (2002). Theorem 0.8.1 (Hurewicz Isomorphism). Let X be a (k − 1)-connected space, k ≥ 2, with x0 ∈ X. Then there is a natural isomorphism πk(X, x0) → Hk(X). Corollary 0.8.2. If X is 1-connected and Hi(X) ∼ 0 for 1 ≤ i ≤ k, then πi(X, x0) ∼ 0 for i ≤ k. There is also a useful local version of the theorem that does not appear in any of the standard references on algebraic topology. Theorem 0.8.3 (Local Hurewicz). Suppose V ⊂ U0 ⊂ · · · ⊂ Uk, k ≥ 2, are open sets such that Hk(V ) → Hk(U0) is trivial and πq(Uq) → πq(Uq+1) is trivial for 0 ≤ q ≤ k − 1. Then πk(V ) → πk(Uk) is trivial. Proof. Consider any map α : Sk → V . As [α] = 0 in Hk(U0), there exist a subdivision L of Sk and a singular (k + 1)-chain c = Σjnjσj carried by U0 such that Σiα#(τi) = ∂c, where {τi} denotes a collection of 1–1 simplicial maps ∆k → L, one for each k-simplex of L, determined by some ordering of the vertices. Let K denote a geometric realization of the finite, singular complex determined by the {σj} here K contains L as a subcomplex, and α : L → V ⊂ U0 has a natural extension β : |K| → U0. Let K be the union of K and the cone on its (k − 1)-skeleton. Since πq(Uq) → πq(Uq+1) is trivial, we can extend β over successive skeleta to a map β : |K | → Uk. Now [Sk] is zero in Hk(K) and hence in Hk(K ). One can easily check that K is (k − 1)-connected. By the Hurewicz Isomorphism Theorem, [Sk] = 0 in πk(K ). Application of β confirms that [α] = 0 in πk(Uk). Theorem 0.8.3 is also known as the Eventual Hurewicz Theorem—see (Ferry, 1979, Proposition 3.1) and (Quinn, 1979, Theorem 5.2). Several applications require a relative version of the Hurewicz Theorem. A complete statement of the relative Hurewicz Theorem must take account of the action of π1 on the higher homotopy groups. In order to avoid that technicality we state the relative theorem only in the simply connected case.

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