# The Upper Triangular Technology

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## 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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