Chapter 13 - Stable Homotopy and Iterated Loop Spaces

Abstract

The chapter discusses stable homotopy and iterated loop spaces. The homology theory has been an effective tool in the study of homotopy invariants for topological spaces. An important reason for this is the fact that it is often easy to compute homology groups. For instance, if one is given a finite simplicial complex, computing its homology becomes a straightforward problem in the linear algebra of finitely generated free modules over integers. Homotopy groups are difficult to compute. For instance, there are no finite cobra venom factor (CVF) complexes except the classifying spaces of certain infinite groups, for example, bouquet of circles or compact closed surfaces, whose homotopy groups are known completely. The difficulty in carrying out this calculation can be traced in part to the nonexistence of an excision theorem for homotopy groups,in the consequent nonexistence of long exact Mayer-Vietoris sequences, and in long exact sequences of cofibrations.