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

Abstract

Given a finite, connected 2-complex $X$ such that $b_2(X)\le 1$ 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 the first rational cohomology group of $X$ is non-zero.

arXiv comment22 pages, to appear in `Topology and its Applications'. v2: The title was changed, reflecting the fact that Cor. 1.1 was already known. The old Theorem 1.5 was omitted, as it is easily proved using a result in the new appendix. v3: Only minor changes. v4: The proof of Prop. 2.1 was omitted, because the result was already known. Minor changes following referee's suggestions