Sarah Whitehouse (University of Sheffield)
S.Whitehouse@sheffield.ac.uk
The complex K-theory of a space or spectrum may be usefully endowed with operations. The most well-known of these are the Adams operations, arising out of the geometry of vector bundles. For many applications one wants to consider stable operations; the Adams operation Ψn is only stable if n is a unit in the coefficients one is working over. From work of Adams and Clarke, we know that the ring of operations is uncountable. Therefore there are plenty of operations which are not finite linear combinations of Adams operations and the challenge is to explicitly describe the elements and the algebraic structures involved. We solve this problem completely in the p-local setting. Operations are given in a simple, natural way, in terms of a single Adams operation, by allowing certain infinite sums. The results allow us to study the rather complicated ring structures and to show that these rings of operations are not Noetherian.
This is an account of joint work with Francis Clarke and Martin Crossley.