On Chern classes of almost representations

For a discrete group $\mathrm{\Gamma }$, we study vector bundles $E_{\rho }$ on compact subsets of $B\mathrm{\Gamma }$ associated to almost representations $\rho :\mathrm{\Gamma }\to U(n)$. 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.