Milnor K-theory and the graded representation ring

TL;DR

This paper constructs a new ring homomorphism linking Milnor K-theory of a field to the graded representation ring of its Galois group, revealing deep connections with algebraic topology and classical invariants.

Contribution

It introduces a novel homomorphism from mod 2 Milnor K-theory to the graded representation ring, and develops an analog of the Chern character using real vector bundles and Steenrod operations.

Findings

The homomorphism to the W-group is an isomorphism in all studied cases.

A new Chern character analog relates real K-theory to mod 2 cohomology.

The machinery has potential applications in algebraic topology.

Abstract

Let F be a field, let G be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k_*(F) to the graded ring gr R(G, k) associated to Grothendieck's \gamma-filtration. We study this map in particular cases, as well as a related map involving the W-group of F rather than G. The latter is an isomorphism in all cases considered. Naturally this echoes the Milnor conjecture (now a theorem), which states that k_*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring gr W(F). The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the real K-theory ring K(X) of the topological space X, again with respect to the \gamma -filtration, and whose target is a certain subquotient of the mod 2 cohomology of X. In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.