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The Upper Triangular Technology

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Part of the Progress in Mathematics book series (PM, volume 273)

Abstract

The object of this chapter is to establish the basic result which relates the upper triangular group to operations in connective K-theory. This result will identify a certain group of operations with the infinite upper triangular group with entries in the 2-adic integers. This identification will be canonical up to inner automorphisms. The 2-adic integers enter here because we are going to work in the stable homotopy category of 2-localised spectra in the sense of [43] (see also Chapter 1 §1.3.4(iv)). Frank Adams was rather fond of dicta and one of his favorites was something to the effect that it is preferable to have a modest example of something mathematical providing it was canonical. That sort of sentiment is the motivation behind my upper triangular technology — that it is better to have only a subset of the operations in connective K-theory providing it is canonical. However, I shall go further and ask for “rigidity”. In our case that means turning the 2-adic integral group-ring into a ring of K-theory operations in order that manipulating them becomes as rigid as group-ring algebra.

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© Birkhäuser Verlag AG 2009

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