Exact Sequences

Abstract

In the first section we use the ideas of Chapter 3 to derive several basic exact sequences. The main sequences that we consider are two long exact sequences of homotopy sets. One is associated to a fiber sequence F → E → B. The terms are the homotopy sets [X,Y], where the X’s are the iterated suspensions of some fixed space and the Y’s are the successive spaces of the fiber sequence. The other sequence is associated to a cofiber sequence A → X→ Q. The terms are the homotopy sets [X,Y], where the Y ’s are the iterated loop spaces of some fixed space and the X’s are the successive spaces of the cofiber sequence. As special cases we obtain the exact homotopy sequence of a fibration and the exact cohomology sequence of a cofibration. We next discuss the action of an H-space on a space and the coaction of a co-H-space on a space. The former is illustrated in a fiber sequence by the action of the loops of the base ΩB on the fiber F, and the latter in a cofiber sequence by the coaction of the suspension ∑ A on the cofiber Q. These then yield operations of one homotopy set on another and give sharper exactness results at the end terms of the two exact sequences of homotopy sets mentioned above. In the final section we return to homotopy groups. We give equivalent characterizations of homotopy groups, define the homotopy groups of a pair, establish the exact homotopy sequence of a pair, and discuss the relative Hurewicz homomorphism. We conclude by considering certain maps, called excision maps, which are associated to maps X → Y → Z whose composition is trivial. These maps are heavily used in Chapter 6.