Hereditarily Weakly Confluent Mappings onto S

Results are obtained about the existence and behavior of hereditarily weakly confluent maps of continua onto the unit circle S1. A simple and useful necessary and sufficient condition is given for a map of a continuum, X, onto S1 to be hereditarily weakly confluent (HWC). It is shown that when X is arcwise connected, an HWC map of X onto S1 is monotone with arcwise connected fibers. A number of theorems about HWC irreducible maps of X onto S1 are proved; for example, such maps are monotone with nowhere dense fibers, and a complete determination of the structure of X is obtained when X admits an HWC irreducible map onto S1 and X is arcwise connected. Among other results, the arcwise connected semilocally-connected continua that admit an HWC map onto S1 are completely determined, and it is shown how the map must be defined.