Chapter 15 - Unstable Operations in Generalized Cohomology

Abstract

This chapter deals with the discussion of unstable operations in generalized cohomology. There are three kinds of cohomology operation: stable operations, which form the endomorphism ring E*(E,o) of E ; unstable operations, defined on En(X) for spaces X and fixed n, which form E*{En); and additive unstable operations r on En(-) (that satisfy r{x+y) = r(x) + r(y)), which form the subset PE*(En). In the classical case E = H(Fp), for which E*(E, o) is the Steenrod algebra, it is true that: (a) every additive operation comes from a stable operation; (b) the additive operations generate multiplicatively all the unstable operations. Difficulties stem from the fact that for MU and BP, both (a) and (b) arc false. The chapter proposes to describe completely the algebraic structure that has to be present on an E*-module or E*-algebra to make it an unstable object, with particular attention to the case E = BP. These definition lead to structure theorems.