Norms and the Cohomology of Wreath Products

Abstract

In this chapter we introduce the norm map and develop some of its properties. A major application of the norm will appear in the next chapter when we define the Steenrod operations. The norm was first invented by Leonard Evens [69] and it is often called the Evens norm map. The norm is a function H^s (H, k) → H^ns (G, k) where H is a subgroup of G of index |G: H| = n. The map is not usually additive, but it has some multiplicative properties that make it very useful. The norm first appeared in Evens’ proof of the finite generation of the cohomology ring of a group as an algebra over the base ring k of coefficients. A required assumption is that k is noetherian. Evens also proved that if M is a finitely generated kG-module, then H* (G, M) is a finitely generated module over H*(G, k). An independent proof of the finite generation was given by Venkov [144].