Chapter 31 - An update on the elusive fixed-point property

Abstract

This chapter provides an overview of the elusive fixed-point property. R.H. Bing's expository article “The elusive fixed point property,” which appeared in the American Mathematical Monthly in 1969, has been an invaluable guide to the generations of mathematicians. The article consists of 12 questions and a variety of related theorems and examples. A space is said to have the fixed-point property if each map of the space into itself has a fixed point. A continuum is defined as a non-degenerate compact connected metric space. Bing was interested in the fundamental problem of determining which continua have the fixed-point property. Bing's questions, some results, and some related unsolved problems are reviewed in the chapter. The chapter discusses about Bing's first question, “Is there a two-dimensional polyhedron with the fixed-point property which has even Euler characteristic?” This question is still open. It was motivated by W. Lopez's example of an eight-dimensional polyhedron with even Euler characteristic that has the fixed-point property. This chapter discusses many other questions asked by Bing in his article.