Article

# On the Cohomology of Galois Groups Determined by Witt Rings

Department of Mathematics, University of Wisconsin, Madison, Wisconsin, 53706, f1E-mail: {adem,dikran}@math.wisc.eduf1; Department of Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7f2E-mail: minac@uwo.caf2

Advances in Mathematics 08/1998; DOI: 10.1006/aima.1999.1847 - [Show abstract] [Hide abstract]

**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.08/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**This is a report of a talk given at the Oberwolfach workshop on "cohomology of finite groups: Interactions and applications" which was held during July 25th - July 31st, 2010. It is an announcement of some of the results (with motivation) and their applications from the paper "Quotients of absolute Galois groups which determine the entire Galois cohomology"; see arXiv:0905.1364.01/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the relations in Galois groups of maximal p-extensions of fields, the structure of their natural filtrations, and their relationship with the Bloch-Kato conjecture proved by Rost and Voevodsky with Weibel's patch. Our main focus is on the third degree, but we provide examples for all degrees.12/2010;

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