Periodicity in groups

Periodicity of Hermitian K-groups

Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with …nite coe¢ cients, similar results are consequences of the Milnor and Bloch-Kato conjectures, a¢ rmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodici...

Clifford Periodicity from Finite Groups

We deduce the periodicity 8 for the type of Pin and Spin representations of the orthogonal groups O(n) from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include the case of arbitrary signature O(p, q). The changes in the type of representation can be seen as a rotation in the complex plane. The essential result is that adding a (+) dimen...

Periodicity and Jumps in Cohomology of R-torsion-free Groups

A discrete group G has periodic cohomology over R if there is an element α ∈ Ext RG (R,R) of positive degree and an integer n ≥ 0 such that the cup product map by α induces an isomorphism of Ext RG (R,−) and Ext i+|α| RG (R,−) for i ≥ n. Adem and Smith [1] showed if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a spher...

Hermitian Periodicity and Cohomology of Infinite Orthogonal Groups

As an application of our papers [2], [4], in favourable cases we prove the periodicity of hermitian K-groups with a shorter period than previously obtained. We also compute the homology and cohomology with …eld coe¢ cients of in…nite orthogonal and symplectic groups of speci…c rings of integers in a number …eld.

Periodicity, Quasiperiodicity, and Bieberbach's Theorem on Crystallographic Groups