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.  2004 Elsevier B.V. All rights reserved.MSC: primary 54F15, 55M10; secondary 54D05

We consider the classical problem of a position of n-dimensional manifold M n in R n+2 . We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n + 2)-cycle for a fundamental quandle of a knotting M n → R n+2 . In particular, we show that for any fixed quandle, quandle coloring, and shadow quandle coloring, of a diagram of M n embedded in R n+2 we have (n + 1)-and (n + 2)-(co)cycle invariants (i.e. invariant under Roseman moves).

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