10 0. Prequel A map ψ : (X, x0) → (Y, y0) leads to an induced homomorphism ψ# : πn(X, x0) → πn(Y, y0) defined by ψ#([f]) = [ψ ◦ f]. The nth homotopy group has the expected functoriality properties with respect to such base- point-preserving maps of spaces. Unlike the fundamental group, however, πn(X, x0) is always abelian when n 1. Application of lifting theorems such as (Munkres, 2000, Lemma 79.1) to a covering map Θ : X → X quickly yields that Θ# : πn(X , x) → πn(X, Θ(x)) is an isomorphism for n 1. Consequently, πn(X, x0) = 0 for all n 1 if the universal covering space of X is contractible. Note that X is path connected if and only if π0(X, x0) = 0 and X is simply connected if and only if both π0(X, x0) and π1(X, x0) are trivial. There is a corresponding kind of connectivity in every dimension and it is detected by the higher homotopy groups. Definition. A space X is said to be k-connected, k ≥ 0, if each map ∂In+1 → X can be extended to a map In+1 → X for n = 0, 1,...,k. Theorem 0.5.1. Let X be a pathwise connected space with basepoint x0. For each k ≥ 0 the following are equivalent. (1) X is k-connected. (2) πn(X, x0) = 0 for n ≤ k. (3) If K is a k-dimensional polyhedron and L is a closed subpolyhedron of K, then any map (K, L) → (X, x0) is homotopic, rel L, to the constant map K → x0. Proof. Exercise 0.5.2. There is also a local version of connectivity, the usefulness of which will be illustrated in the next two sections. Definition. A space X is said to be locally connected in dimension k (ab- breviated k-LC) if for every x ∈ X and for every neighborhood U of x in X there exists a neighborhood V of x such that each map Sk → V extends to a map Bk+1 → U. The space X is said to be locally k-connected (abbreviated LCk) if X is n-LC for 0 ≤ n ≤ k. We now define a relative homotopy group πn(X, A, x0), which is associ- ated with a space X, a nonempty subset A of X, and a basepoint x0 ∈ A. In order to define the relative group, it is convenient to identify In−1 with the face In−1 ×{0} ⊂ ∂In and to use Jn−1 to denote the union of the remaining faces of In (so ∂In = In−1 ∪Jn−1 and In−1 ∩Jn−1 = ∂In−1). An element of πn(X, A, x0) is represented by a map of triples (In,In−1,Jn−1) → (X, A, x0). Two such maps are equivalent if they are homotopic via a homotopy of triples.

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