On G-structures, the likeness relation, monotone and refinable maps.
by
Pawel Krupski
University of Wroclaw, PL
Coauthors: D. Cichoń and K. Omiljanowski

Locally connected continua which admit monotone maps onto graphs are characterized. The notion of a G-structure is introduced for any graph G as a generalization of a linear or circular chain cover (of arbitrary finite length) of a continuum by its subcontinua. As an application, the following results will be presented:

(1) a locally connected continuum X has a G-structure iff G is X-like;

(2) any nondegenerate locally connected continuum has an arc-structure or a circle-structure;

(3) some new invariants of the likeness relation are found;

(4) a map f from a G-like continuum onto a graph G is refinable iff f is monotone;

(5) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no non-boundary indecomposable subcontinuum admits a monotone map onto G;

(6) if the bonding maps of the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. This fact is used to show that refinable maps do not preserve completely regular nor totally regular continua.

Date received: April 27, 2007