  Given a finite, connected 2-complex $X$ such that $b_2(X)\le1$ we establish
two existence results for representations of the fundamental group of $X$ into
compact connected Lie groups $G$, with prescribed values on certain loops. If
$b_2(X)=1$ we assume $G=SO(3)$ and that the cup product on the first rational
cohomology group of $X$ is non-zero.
