Rosicki, Witold On a problem of S. Ulam concerning Cartesian squares of 2-dimensional polyhedra. (English) Zbl 0621.57002 Fundam. Math. 127, 101-125 (1987). In 1933 the following problem was posed by S. Ulam: ”Assume that A and B are topological spaces and \(A^ 2=A\times A\) and \(B^ 2=B\times B\) are homeomorphic. Is it true that A and B are homeomorphic ?” [See ”The Scottish Book”, ed. R. D. Mauldin (1981; Zbl 0485.01013)] In general, this question has a negative answer. It is proved in this paper, however, that the problem has a positive answer if A and B are compact, connected 2-polyhedra. The positive answer for 3-manifolds implies the positive solution of the Poincaré conjecture. Recently, the author has additionally proved that if the Cartesian powers \(A^ k\) and \(B^ k\) are homeomorphic then A and B are homeomorphic for some classes of 2-polyhedra. Cited in 3 ReviewsCited in 2 Documents MSC: 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57N10 Topology of general \(3\)-manifolds (MSC2010) 54B10 Product spaces in general topology 57Q05 General topology of complexes Keywords:Cartesian squares of 2-complexes; Cartesian squares of 3-manifolds; Poincaré conjecture; Cartesian powers Citations:Zbl 0485.01013 PDFBibTeX XMLCite \textit{W. Rosicki}, Fundam. Math. 127, 101--125 (1987; Zbl 0621.57002) Full Text: DOI EuDML