A finitedimensional version of Fredholm representations
  We consider pairs of maps from a discrete group to the unitary group. The deficiencies of these maps from being homomorphisms may be great, but if they are close to each other then we call such pairs balanced. We show that balanced pairs determine elements in the K-theory group of the classifying space of the discrete group. We also show that a Fredholm representation determines balanced pairs. 
V. Manuilov
math.KT math.OA