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Milnor K-theory and the graded representation ring | Pierre Guillot
; Jan Minac
; | Date: |
1 Sep 2011 | 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. | Source: | arXiv, 1109.0046 | Services: | Forum | Review | PDF | Favorites |
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