×

On a problem of S. Ulam concerning Cartesian squares of 2-dimensional polyhedra. (English) Zbl 0621.57002

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.

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

Citations:

Zbl 0485.01013
PDFBibTeX XMLCite
Full Text: DOI EuDML