Slices of Hermitian \(K\)-theory and Milnor’s conjecture on quadratic forms. (English) Zbl 1416.19001

Röndigs, Oliver; Østvær, Paul

Geom. Topol. 20, No. 2, 1157-1212 (2016).

Summary: 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.

MSC:

19G38	Hermitian \(K\)-theory, relations with \(K\)-theory of rings
11E70	\(K\)-theory of quadratic and Hermitian forms
11E04	Quadratic forms over general fields
14F42	Motivic cohomology; motivic homotopy theory
55P42	Stable homotopy theory, spectra
55T05	General theory of spectral sequences in algebraic topology

Keywords:

motivic cohomology; quadratic forms; slices of Hermitian \(K\)-theory and Witt theory