On two-dimensional planar compacta not homotopically equivalent to any one-dimensional compactum

摘要

The paper provides examples of planar "homotopically two-dimensional" compacta, (i.e., of compact subsets of the plane that are not homotopy equivalent to any one-dimensional set) that have different additional properties than the first such constructed examples (amongst them cell-like, trivial π_1, and "everywhere" homotopically two-dimensional). It also points out that open subsets of the plane are never homotopically two-dimensional and that some homotopically two-dimensional sets cannot be in such a way decomposed into homotopically at most one-dimensional sets that the Mayer-Vietoris Theorem could be straightforwardly applied.