Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms

Abstract

We advance the understanding of $K$–theory of quadratic forms by computing the slices of the motivic spectra representing hermitian $K$–groups and Witt groups. By an explicit computation of the slice spectral sequence for higher Witt theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian $K$–groups in terms of motivic cohomology.