On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X such that b 2 ( X ) ⩽ 1 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 H 1 ( X ; Q ) is non-zero.