Tree-like Continua have the Fixed-Point Property for Deformations
by
Charles L. Hagopian
California State University, Sacramento

Suppose f is a map of a tree-like continuum M that sends each arc-component of M into itself.

Then f must leave some point of M fixed. Hence every map of a tree-like continuum that is homotopic to the identity has a fixed point.

This result answers a question of D.P. Bellamy. My argument is somewhat similar to L.E.J. Brouwer's proof that a 2-sphere cannot admit a continuous nonvanishing tangent vector field. Many related problems remain unsolved. At the center of this area is the classical problem of extending the Brouwer fixed-point theorem to all nonseparating plane continua.

Date received: February 18, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-08.