A fixed point problem
by
Kenneth R. Kellum
San Jose State U.

Consider a simple triod in the plane together with a spiral closing down on it in the simplest way. Call the resullting continuum M. It is not known if the cone over M has the fixed point property.

A function k:X --> Y is inverse graph almost continuous if whenever k:Y --> X is continuous with x =/= k(f(x)) for all x there exists a continuous function g:X --> Y with x =/= k(g(x)) for all x. A continuum X is a pseudo AR if, whenever X is embedded in an AR Z, there exists an inverse graph almost continuous function r:Z --> Z such that r(Z) = Z and r is the identity on X.

A pseudo AR must have the fixed point property. I have proved that M is a pseudo AR, but I do not know if its cone is. Using this notion, I have been able to establish some properties that a fixed point free map from cone(M) to itself must have, if such a map indeed exists.

Date received: February 27, 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-27.