Decomposing into essentially one-dimensional spaces two-dimensional planar sets that are not homotopy equivalent to anything one-dimensional.
by
Andreas Zastrow
Inst. Math., University of Gdansk , Gdansk, Poland
Coauthors: Umed Karimov (Dushanbe, Tajikistan), Dusan Repovs (Ljubljana, Slovenia), Witold Rosicki (Gdansk, Poland)

This paper has been motivated by an example of one of the authors for a two-dimensional subset of the Euclidean plane that in collaboration with J.Cannon und G.Conner could be proven to be not homotopy equivalent to any one-dimensional space. The original example had an uncountable fundamental group. In the present paper we first give an improved version of such an example with the same dimension-properties that in addition is a cell-like set with only trivial fundamental group. Then we show that any subset X of the Euclidean plane can in such a way be decomposed into two subsets X₁ & X₂ that each of them and their non-empty intersection is homotopy-equivalent to a one-dimensional set. X₁ \cap X₂ contains the entire one-dimensional part of X, and for the two-dimensional part we have that X₁ \cap X₂ \cap Int(X) is open. Finally, by an according example it is also shown, that this decomposition result cannot be improved by finding in any case such X₁ and X₂ that also X₁ \cap X₂ is open.

Date received: August 25, 2002