BASES FOR CO-OPERATIONS IN COMPLEX K-THEORY

rst part of the talk will cover background material on spectra, cohomology theories, operations and co-operations, with particular reference to the example of complexK-theory. Then the results of Adams, Harris and Switzer [4] describing the structure of K (K) will be given. It was shown by Adams and Clarke that additively K0(K) is a free abelian group [3]. We introduce a family of bases for K0(K), using Gaussian polynomials. A basis was already given by Johnson [8], but those given here admit a convenient multiplication formula and are easily adapted to give bases for the conjugation invariant subring. We also obtain a more explicit form of a theorem of Johnson which characterises operations in K-theory in terms of their action on the coecient groups. Our constructions are performed, in the rst instance, localised at a prime. We do odd primes rst and then explain the modications needed for the prime 2. At the end we explain how the results may be lifted to the global case.