Towards bivariant K-theory: how to classify extensions

Abstract

Many important maps between K-theory groups are constructed as index maps of certain extensions. We have seen one instance of this in our proof of Bott periodicity, where we have constructed the periodicity isomorphism as such an index map. As the notation suggests, more examples arise in index theory. Often it is important to compose index maps with homomorphisms or with other index maps. For such purposes, it is useful to have a (graded) category in which ordinary bounded algebra homomorphisms and extensions give morphisms (of degrees 0 and 1, respectively). In this chapter, we construct such a category, which is denoted ΣHo, and show that it is triangulated. This additional structure allows us to treat long exact sequences efficiently. Moreover, many important constructions in topology and homological algebra may be rephrased in the language of triangulated categories and then carry over to ΣHo and related categories.