On a question of R.H. Bing concerning the fixed point property for two-dimensional polyhedra

In 1969 R.H. Bing asked the following question: Is there a compact two-dimensional polyhedron with the fixed point property which has even Euler characteristic? In this paper we prove that there are no spaces with these properties and abelian fundamental group. We also show that the fundamental group of an example cannot have trivial Schur multiplier.