On the cohomology of Galois groups determined by Witt rings

Alejandro Adem, Dikran B. Karagueuzian and Jan Minac

Abstract
Let $F$ denote a field of characteristic different from two. In this paper we
describe the mod $2$ cohomology of the Galois group $\mathcal G_F$ (called the
$W$-group of $F$) which is known to essentially characterize the Witt ring
$WF$ of anisotropic quadratic modules over $F$. We show that 
$H^*(\mathcal G_F,\mathbb F_2)$ contains the mod $2$ Galois cohomology of $F$ 
and that its structure will reflect important properties of the field. We 
construct a space $X_F$ endowed with an action of an elementary abelian group 
$E$ such that the computation of $\mathcal G_F$ reduces to calculating the
equivariant cohomology $H^*_E(X_F,\mathbb F_2)$. For the case of a field which
is not formally real this amounts to computing the cohomology of an explicit
Euclidean space form, an object which is interesting in its own right. We 
provide a number of examples and a substantial combinatorial computation for
the cohomology of the universal $W$-groups.