  Let G be the fundamental group of the complement of a K(G,1) hyperplane
arrangement (such as Artin's pure braid group) or more generally a
homologically toroidal group (as defined in the paper). The subgroup of
elements in the complex K-theory of BG which arises from complex unitary
representations of G is shown to be trivial. In the case of real K-theory, this
subgroup is an elementary abelian 2-group, which is characterized completely in
terms of the first two Stiefel-Whitney classes of the representation.
Furthermore, an orthogonal representation of G gives rise to a trivial bundle
if and only if the representation factors through the spinor groups.
  In addition, quadratic relations in the cohomology algebra of the pure braid
groups which correspond precisely to the Jacobi identity for certain choices of
Poisson algebras are shown to give the existence of certain homomorphisms from
the pure braid group to generalized Heisenberg groups. These cohomology
relations correspond to non-trivial Spin representations of the pure braid
groups which give rise to trivial bundles.
