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

Oliver Röndigs and Paul Arne Østvær

Geometry & Topology 20 (2016) 1157–1212
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.

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Keywords
motivic cohomology, quadratic forms, slices of hermitian $K$–theory and Witt theory
Mathematical Subject Classification 2010
Primary: 11E04, 14F42, 55P42
Secondary: 19D50, 19G38, 55T05
References
Publication
Received: 29 January 2015
Revised: 2 May 2015
Accepted: 23 June 2015
Published: 28 April 2016
Proposed: Jesper Grodal
Seconded: Paul Goerss, Ronald Stern
Authors
Oliver Röndigs
Institut für Mathematik
Universität Osnabrück
D-49069 Osnabrück
Germany
Paul Arne Østvær
Department of Mathematics
University of Oslo
PO Box 1053
0316 Oslo
Norway