0.6. Absolute neighborhood retracts 13 (3) Y is k-connected and locally k-connected. It follows easily from Theorem 0.6.1 that every finite-dimensional poly- hedron or CW complex is an ANR. The Hawaiian earring is not an ANR neither is the sine (1/x) curve (Figure 2.13). The next theorem identifies an especially useful property of ANRs. Theorem 0.6.3. Let Y be a compact ANR and a positive number. There exists δ 0 such that for any two maps f0,f1 : S Y of an arbitrary space S to Y with d(f0,f1) δ, there is a homotopy H : S × I Y such that H0 = f0, H1 = f1, and diam H({s} × I) for all s S. Sketch of Proof. Embed Y in the Hilbert cube I∞ = Π∞ n=1 [0, 1/n], where it has a neighborhood U that retracts to the embedded copy Y of Y . Choose δ small enough that the straight line homotopy from f0 to f1 stays in U and still has diameter less than after being retracted into Y . Theorem 0.6.4 (Estimated Homotopy Extension Theorem). Let Y be an ANR, X a normal space, f : X Y a map, b : Y (0, ∞] another map, A a closed subset of X, U a neighborhood of A in X, and µ : A × I Y a homotopy such that µ0 = f|A and diam µ({a} × I) b(µ(a, t)) for all a A and t I. Then there exists a homotopy H : X × I Y such that H0 = f, H|A × I = µ, H({x} × I) = f(x) for all x X U, and diam H({x} × I) b(H(x, t)) for all x X and t I. Proof. Define a map F on Z = (X × {0}) (A × I) X × I as f on X ×{0} and µ on A×I. Since Y is an ANR, F has an extension F : O Y over some neighborhood O of Z in X × I. Find an open subset V of X, A V U, such that V × I O and diam F({v} × I) b(F(v, t)) for all v V and t I. Apply Urysohn’s Lemma to obtain a map η : X [0, 1] for which η(X V ) = 0 and η(A) = 1. Finally, define H : X × I Y as H(x, t) = F(x, η(x)t). Corollary 0.6.5. Suppose R : Y A is a retraction of an ANR Y to a compact subset A, and suppose η : A × I Y is a homotopy between η0 = inclA and an embedding λ = η1. Then Y retracts to λ(A) moreover, if λ moves points less than c 0 and diam Rη({a} × I) b for all a A, then there is a retraction R : Y λ(A) such that d(R (y),R(y)) b + 2c for all y Y . Proof. The homotopy η = λRη(λ−1 × Id) : λ(A) × I λ(A) satisfies η1 = λR|λ(A) and η0 = Id obviously η1 extends to λR : Y λ(A). Exercises 0.6.1. Every n-cell is an AR.
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