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0.5. Higher homotopy groups 11 The group operation is defined exactly as in the absolute case. The only catch is that the coordinate tn plays a special role in the definition of the relative homotopy group, and the definition of the sum of two equiv- alence classes uses the coordinate t1 in an independent way. As a re- sult, the definition of the group operation only makes sense if n ≥ 2 π1(X, A, x0) is simply considered to be a set of equivalence classes with no natural group structure (just like π0(X, x0)). As in the absolute case, a continuous map ψ : (X, A, x0) → (Y, B, y0) induces a homomorphism ψ∗ : πn(X, A, x0) → πn(Y, B, y0) defined by ψ∗([f]) = [ψ ◦ f]. We may identify the absolute group πn(X, x0) with the relative group πn(X, {x0},x0), so there is a natural (inclusion-induced) homomorphism j∗ : πn(X, x0) → πn(X, A, x0). There is also an inclusion-induced homo- morphism i∗ : πn(A, x0) → πn(X, x0). An element of πn+1(X, A, x0) is represented by a map f : In+1 → X such that f(In) ⊂ A and f(Jn) = {x0}. In particular, f(∂In) = {x0}, so f|In may be viewed as representing an el- ement of πn(A, x0). This correspondence induces a natural homomorphism ∂ : πn+1(X, A, x0) → πn(A, x0). The homomorphism ∂ is called the bound- ary operator because f|Jn is constant and so f|In is essentially f|∂In+1. The three homomorphisms just described combine to form an extremely valuable long exact sequence, the homotopy sequence of the pair (X, A) with base point x0: · · · − πn+1(X, A, x0) ∂ − πn(A, x0) i∗ − πn(X, x0) j∗ − πn(X, A, x0) − · · · A proof of the exactness of the homotopy sequence of a pair can be found on pages 344 and 345 of (Hatcher, 2002). Definition. A topological pair (X, A) is said to be k-connected, k ≥ 0, if each map (In × {0},∂In × {0}) → (X, A) can be extended to a map (In+1,Jn) → (X, A) for n = 0, 1,...,k. The final theorem in the section will be applied in the proof of the basic engulfing theorem of Chapter 3. Theorem 0.5.2. Let X be a pathwise connected space with subset A and basepoint x0 ∈ A. For each k ≥ 0 the following are equivalent. (1) The pair (X, A) is k-connected. (2) πn(X, A, 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, A) is homotopic, rel L, to a map K → A. Proof. Exercise 0.5.4.