Mathematics – K-Theory and Homology
Scientific paper
2011-08-31
Mathematics
K-Theory and Homology
Scientific paper
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.
Guillot Pierre
Minac Jan
No associations
LandOfFree
Milnor K-theory and the graded representation ring does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Milnor K-theory and the graded representation ring, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Milnor K-theory and the graded representation ring will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-224409