Projective Fra\"ıssé limits of trees

Abstract

The following paper has been withdrawn from consideration for publication because there are mistakes. In particular, Theorem 3.9 does not hold. Examples were found of finite trees with monotone epimorphisms which do not amalgamate. Further, finite rooted trees with monotone epimorphisms do not amalgamate. A revision, with additional co-authors A. Kwiatkowska and S. Yang, is posted on arXiv ({\it Projective Fra\"ıssé limits of trees with confluent epimorphisms} 2312.16915). In that article, it is shown that the family of finite trees having ramification vertices of order at most 3 with monotone epimorphisms does form a projective Fra\"ıssé family and the topological realization of its Fra\"ıssé limit is the Wa\. zewski dendrite $D_3$. Further, two families of finite rooted trees with restrictions on what confluent epimorphisms are allowed are also shown to form projective Fra\"ıssé families. The topological realization of the Fra\"ıssé limit of one of these families is shown to be the Mohler-Nikiel universal dendroid. We continue study of projective Fra\"ıssé limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, and light mappings from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the projective Fra\"ıssé limits we obtain the dendrite $D_3$ as well as quite new, interesting continua for which we do not yet have topological characterizations.