On Chern classes of almost representations

For a discrete group $\Gamma$, we study vector bundles $E_\rho$ on compact subsets of $B\Gamma$ associated to almost representations $\rho:\Gamma \to U(n)$. We compute the first Chern class of $E_\rho$ in terms of $\rho$. When $\rho$ is both projective and almost multiplicative, we determine its Chern character. These invariants yield obstructions to perturbing almost representations to those arising from projective representations. For residually finite amenable groups, the $K$-theory classes of $E_\rho$ classify almost representations up to stable equivalence. Finally, for $\mathbb{Z}^d$, $\mathbb{Z}\times \mathbb{H}_3$, and $\mathbb{H}_3\times \mathbb{H}_3$, we construct explicit almost representations with prescribed Chern classes.