Dubrovnik VI – Geometric Topology September 30 – October 7, 2007
	ABSTRACTS	pdf-version of this abstract

Danuta Kołodziejczyk
Warsaw University of Technology, Warsaw, Poland

On some questions concerning decompositions of shapes into Cartesian factors

Following K. Borsuk, a shape Sh(X) is said to be prime, if it is not trivial and can not be decomposed into the product of two nontrivial shapes.

In 1968, at the Topological Conference in Herceg-Novi, K. Borsuk asked: Does there exist for every Sh(X) ≠ 1 a prime factor?

The above problem was also published in a few of Borsuk's papers from the seventies and eighties, for example in [Fund. Math. 67 (1970), 221-240].

We answer this question, showing that there exists a continuum X such that Sh(X) ≠ 1 has no prime factor.

We also consider some other problems on decompositions of shapes into factors from Borsuk's papers and his monograph Theory of Shape, and from the collection [J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Warsaw, 1981].

For example, we prove that for each integer n ≥ 3, there exists a continuum X such that Sh(X) = Shⁿ(X), but Sh(X) ≠ Sh^n-1(X). This answers in the negative the following question: Suppose that Sh(X) = Shⁿ(X) for some n ≥ 3. Is it true that Sh(X) = Sh²(X) ?

Similar problems in the homotopy category of CW-complexes, related to my previous work on homotopy dominations of polyhedra, are also discussed.