Steenrod Operations in Spectral Cohomology Theories

摘要

Certain properties of Eilenberg-Maclane spectra that allow definition of Steenrod operations are abstracted, and the notion of a Gamma (a)-spectrum is arrived at. This is a ring spectrum E, whose characteristics are given. An expression defining operations is derived which gives cohomology operations if E is an Omega-spectrum. The process for achieving stability and definability for non Omega-spectra is shown. Internal operations using slant product with sequences of defined elements are given and the case for considering finite coefficients presented. Examples of Gamma (a)-spectra are formulated. The dependence of stable internal slant product operations on the E-cohomological structure of the group is discussed. As a comment on the relationship between the slant product and the Kuenneth formula approach to internal operations, a natural homomorphism from the Ext to the Tor Kuenneth formula in ordinary cohomology is offered.