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∞

= Πn=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.