A finite-dimensional version of Fredholm representations

We consider pairs of mappings from a discrete group Γ to the unitary group. The deficiencies of these mappings 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 ⁰ group of the classifying space of the group. We also show that a Fredholm representation of Γ determines balanced pairs.