A weakly chainable uniquely arcwise connected continuum without the fixed point property (Q2928549)

A continuum \(X\) is called uniquely arcwise connected if any two points in it are the endpoints of a unique arc. For such continua it is known that every map \(f: X \to X\) has a fixed point as long as \(f\) is a homeomorphism (Hagopian) or \(X\) is planar (Mohler). However, this is not the case in full generality, the first construction was due to Young.NEWLINENEWLINEThis paper gives an example of a uniquely arcwise connected continuum \(X\) and a fixed point free map \(f: X \to X\) such that \(X\) is also weakly chainable, that is, the continuous image of a chainable (inverse limit of arcs) continuum. This answers negatively a question posed in [\textit{D. P. Bellamy}, Lect. Notes Pure Appl. Math. 170, 27--35 (1995; Zbl 0833.54023)]. Previous work on the relation of weak chainability and the fixed point property (f.p.p.) has been carried out by Minc.