On the degree of cocycles with values in the groupSU(2)

Abstract

In this paper are presented some properties of smooth cocycles over irrational rotations on the circle with values in the groupSU(2). It is proved that the degree of anyC ²-cocycle (the notion of degree was introduced in [2]) belongs to 2πℕ (ℕ={0, 1, 2,...}). It is also shown that if the rotation satisfies a Diophantine condition, then everyC ^∞-cocycle with nonzero degree isC ^∞-cohomologous to a cocycle of the form

$$\mathbb{T} \ni x \mapsto \left[ \begin{gathered} e^{2\pi i(rx + w)} 0 \hfill \\ 0 e^{2\pi i(rx + w)} \hfill \\ \end{gathered} \right] \in SU(2)$$

, where 2πr is the degree of the cocycle andw is a real number. The above statement is false in the case of cocycles with zero degree. The proofs are based on ideas presented by R. Krikorian in [6].