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. 
Jonathan Ariel Barmak, Iv\'an Sadofschi Costa
math.AT