### Periodicity of hermitian K-groups, by A. Jon Berrick, Max Karoubi, and Paul Arne Østvær

Bott periodicity for the unitary, orthogonal and symplectic groups is
fundamental to topological K-theory. Analogous to unitary topological K-theory,
for algebraic K-groups with finite coefficients similar periodicity results are
consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky,
Rost and others. More generally, we prove that periodicity of the algebraic
K-groups for any ring implies periodicity for its hermitian K-groups, analogous
to orthogonal and symplectic topological K-theory.
The proofs use in an essential way higher KSC theories extending those of
Anderson and Green. They also provide an upper bound for the higher hermitian
K-groups in terms of the higher algebraic K-groups.
We also relate periodicity to etale hermitian K-groups by proving a hermitian
version of Thomason's etale descent theorem. The results are illustrated in
detail for local fields, rings of integers in number fields, smooth complex
algebraic varieties, rings of continuous functions on compact spaces, and group
rings.

A. Jon Berrick <berrick@math.nus.edu.sg>

Max Karoubi <max.karoubi@gmail.com>

Paul Arne Østvær <paularne@math.uio.no>