Le theoreme de periodicite en K-theorie hermitienne

TL;DR

This paper extends Bott periodicity in topological K-theory to a discrete setting involving all classical groups, generalizing previous results and demonstrating that higher Witt groups over finite fields of characteristic 2 are isomorphic to Z/2.

Contribution

It generalizes the periodicity theorem in hermitian K-theory to include all classical groups and characteristic 2 cases, using new ideas like enlarged orthogonal groups and cup-products.

Findings

Higher Witt groups of finite fields of characteristic 2 are isomorphic to Z/2.

Generalization of previous results to all classical groups.

Extension of periodicity theorem in hermitian K-theory.

Abstract

Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The present paper generalizes previous results of the author [K1] and [K2], where 2 was assumed to be invertible in the rings involved. For the proof, two important ideas have to be mentioned : the first one is due to Ranicki [R] who introduced a kind of "enlarged" orthogonal group ; the second one is a genuine cup-product between quadratic forms due to Clauwens [C]. As an example of results obtained, we prove that the higher Witt groups of a finite field of characteristic 2 are all isomorphic to Z/2. They generalize in some sense the Dickson and Arf invariants.