Arc-reduced forms for Peano continua
In [Topology Appl. 159, No. 15, 3253--3262 (2012; Zbl 1255.55002)], the authors introduced the deforestation of a Peano contionuum as a certain strong deformation retraction. The deforestation yields the unique minimal deformation retract if the space is one-dimensional and non-contractible. In this paper the authors simplify the homotopy representative of a Peano continuum even more by introducing the arc reduced forms. The one dimensional set \(I(X)\) of a space \(X\) is the set of points with one-dimensional neighborhoods. The set of bad points \(B(X)\) of a space \(X\) is the set of points which possess no simply connected neighborhood. The space \(X\) is arc-reduced if \(I(X)-B(X)\) is either a disjoint union of open arcs or a finite boquet of circles. The main theorem: Every Peano continuum is homotopy equivalent to an arc-reduced continuum. As an application the authors discuss the case of one-dimensional Peano continua.
scientific article; zbMATH DE number 6092998