Algebraic topology - Errata by Allen Hatcher

By Allen Hatcher

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If the graph homomorphism is left-covering, then our estimates are both simpler to state and easier to prove. But we remind the reader that the case of the a homomorphism δn : GN (π) → GN −1 (π) cannot be assumed to have this property. 1. Let G, H be graphs and θ : H → G be a graph homomorphism. (1) If the associated map on the shift spaces is an s-resolving factor map, then there is a constant Kθ ≥ 0 such that, if e, f are in ΣH and are stably equivalent and k0 is an integer such that θ(e)k = θ(f )k , for all k ≥ k0 , then ek = f k , for all k ≥ k0 + Kθ .

Suppose that each point of (Y, ψ) is non-wandering. Then π is s-bijective. The proof will be done in a series of Lemmas, beginning with the following quite easy one. 9. Let π : Y → X be a continuous map and let x0 be in X with π −1 {x0 } = {y1 , y2 , . . , yN } finite. For any > 0, there exists δ > 0 such that π −1 (X(x0 , δ)) ⊂ ∪N n=1 Y (yn , ). Proof. If there is no such δ, we may construct a sequence xk , k ≥ 1 in X converging to x0 and a sequence y k , k ≥ 1 with π(y k ) = xk and y k not in ∪N n=1 Y (yn , ).

ZM ) in GK L,M such that e[−K0 ,K−K0 ] (yl , zm ) = pl,m , for all l, m. 6 with l1 = l, l2 = α(l) and any m1 = m2 = m. It follows that ek (yl , zm ) = ek (yα(l) , zm ) with k ≤ 0. Applying k = −K yields the result. The proof of the last statement is analogous to the fifth and we omit it. CHAPTER 3 Dimension groups In this chapter, we present background material on Krieger’s theory of dimension group invariants for shifts of finite type. The first section presents some very simple observations on free abelian groups.

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