12 0. Prequel Exercises 0.5.1. Sk is (k − 1)-connected. 0.5.2. Prove Theorem 0.5.1. 0.5.3. If X is k-connected and A is (k − 1)-connected, then (X, A) is a k-connected pair. More generally, if the inclusion-induced homo- morphism πi(A) → πi(X) is an epimorphism for i = k and an isomorphism for i k, then (X, A) is a k-connected pair. 0.5.4. Prove Theorem 0.5.2. 0.5.5. πn(X, x0) is abelian provided n ≥ 2. 0.5.6. πn(X, A, x0) is abelian provided n ≥ 3. Show by example that π2(X, A, x0) need not be abelian. 0.6. Absolute neighborhood retracts Although many spaces considered in this text are polyhedra, there are some situations in which more general kinds of spaces are the appropriate ones to use. The class of ANRs is particularly relevant. Definition. A metric space Y is an absolute retract (abbreviated AR) if for every metric space X and for every embedding e : Y → X such that e(Y ) is a closed subset of X, there is a retraction r : X → e(Y ). We say that Y is an absolute neighborhood retract (abbreviated ANR) if for every metric space X and for every embedding e : Y → X such that e(Y ) is a closed subset of X, there exist a neighborhood U of e(Y ) in X and a retraction r : U → e(Y ). Finite-dimensional ANRs can be characterized in terms of local con- nectivity and finite-dimensional ARs can be characterized in terms of both connectivity and local connectivity. Detailed proofs of the next two theorems may be found in (Hu, 1965, Chapter V). Theorem 0.6.1. Let Y be a separable metric space of dimension k ∞. Then the following are equivalent. (1) Y is an ANR. (2) Y is locally contractible. (3) Y is locally k-connected. Theorem 0.6.2. Let Y be a separable metric space of dimension k ∞. Then the following are equivalent. (1) Y is an AR. (2) Y is contractible and locally contractible.

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