Polynomial operations on stable cohomotopy

Abstract

Following ideas of G. Segal [24] and [25] we give a conceptual framework for the study of a wide class of unstable operations from the 0-th stable cohomotopy πs0 to other homotopy functors, which we call polynomial operations. Using a generalization of the group completion theorem we prove that it is equivalent to study polynomial operations from πs0 as to study characteristic classes of finite coverings. Finally using the Segal conjecture we give a description of the ring of polynomial operations on πs0 in terms of polynomial operations on Burnside rings.