Selected intractable problems in continuum theory, paths and inverse limits of simplicial trees
by
Piotr Minc
Auburn University

The collection of paths (either in or near a given continuum) plays an important role in several stubborn open problems in continuum theory. Problems of this kind include the fixed point problem for non-separating plane continua, the problem of small retractions of dendroids onto trees, and the problems concerning embedding of continua into the plane. Understanding combinatorics of the collection of paths would certainly be very useful. To study such combinatorics, it is necessary to consider objects that are finitely defined and locally not so complicated as arbitrary continua can be. The inverse limits of simplicial trees (or graphs) with simplicial bonding maps are very convenient in this context, especially because many of the problems may be reduced to this simpler and combinatorially defined class.

Date received: February 26, 2008