Some Families of Operations in Morava K-Theory

Abstract

This paper gives an account of operations of Steenrod type in Morava K-theory K(n)*( ). Our research was originally stimulated by the paper of U. Wurgler [61; unfortunately, equation (2.11) does not appear to follow and this seems to prevent his Oi from being stable operations. We therefore attack the problem of giving an algebraically defined set of using the relationship between K(n)-theory and Lubin-Tate formal group laws. We hope to return to the tom Dieck approach considered by Wurgler in future joint work with him; we will also consider Bockstein type operations in a joint sequel to the present work. We would like to thank Urs Wurgler for correspondence on this material, and suggesting Theorem 18. We would also like to thank the SERC and Universitat Bern for support whilst this research was in progress. NOTE: For all background information we refer to Ravenel's book [5]. We consider the problem of constructing operations in K(n)-theory analogous to the classical Steenrod reduced powers, by using a multiplicative operation in an enlarged version of K(n)-theory, and then picking off certain terms in this. We relate the operations thus produced to elements in the dual cooperation ring and show that we have enough indecomposables to generate the operation algebra K(n)*(K(n)). We also find that our reduced powers satisfy a cyclic Cartan formula (see Theorem 14). Finally we use an infinite dimensional generalisation of Hensel's Lemma to lift operations to E(n)-theory.