The fixed-point property for simply connected plane continua

Abstract

We answer a question of R. Mańka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let M M be a plane continuum with the property that every simple closed curve in M M bounds a disk in M M . Then every map of M M that sends each arc component into itself has a fixed point. Hence every deformation of M M has a fixed point. These results are corollaries to the following general theorem. If M M is a plane continuum, D \mathcal {D} is a decomposition of M M , and each element of D \mathcal {D} is simply connected, then every map of M M that sends each element of D \mathcal {D} into itself has a fixed point.