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

TL;DR

This paper addresses Bing's 1969 question by proving that no compact two-dimensional polyhedron with the fixed point property and even Euler characteristic can have an abelian fundamental group, and discusses constraints on the fundamental group's Schur multiplier.

Contribution

It proves the non-existence of such polyhedra with abelian fundamental groups and explores limitations on their fundamental groups' properties.

Findings

No such polyhedra exist with abelian fundamental groups.

Fundamental groups of these polyhedra cannot have trivial Schur multiplier.

Provides constraints on the algebraic properties of fundamental groups for these spaces.

Abstract

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.