Hereditarily weakly confluent mappings onto \(S^1\) (Q2725456)

A continuum means a nonempty, compact, connected, metric space, and \(S^1\) stands for the unit circle. A map \(f:X\to Y\) between continua \(X\) and \(Y\) is weakly confluent if each subcontinuum of \(Y\) is the image under \(f\) of a subcontinuum of \(X\). It is said to be hereditarily weakly confluent (HWC) if for each subcontinuum \(K\) of \(X\) the restriction \(f|K\) is weakly confluent. Results are obtained about the existence and behavior of HWC maps of continua onto \(S^1\). Samples of them are the following: (1) For each continuum \(X\) a map \(f:X \to S^1\) is HWC if and only if for all subcontinua \(K\) of \(X\) such that \(f(K) =S^1\) the restriction \(f|K\) is essential. (2) If \(X\) is arcwise connected, then an HWC map of \(X\) onto \(S^1\) is monotone with arcwise connected fibers. (3) Each HWC irreducible map of a continuum onto \(S^1\) is monotone with nowhere dense fibers. (4) If a continuum is arcwise connected, then it admits an HWC irreducible map onto \(S^1\) if and only if it is a compactification of the real line with a continuum as the remainder. Among other results, the arcwise connected semi-locally connected continua that admit an HWC map onto \(S^1\) are completely determined.NEWLINENEWLINENEWLINEChapters are: 1. Introduction; 2. A characterization theorem for HWC maps onto \(S^1\); 3. A theorem about irreducible HWC maps onto \(S^1\); 4. Monotoneity of HWC maps of arcwise connected continua onto \(S^1\); 5. Subcontinua on which HWC maps of arcwise connected continua onto \(S^1\) are irreducible; 6. Nonexistence of HWC maps of Cartesian products onto \(S^1\); 7. The cyclic element retraction; 8. HWC maps of arcwise connected semi-locally connected continua onto \(S^1\); 9. A problem.