Topology Atlas Document # zaaa-38.htm | Production Editor: R. Flagg
Topology Atlas Invited Contributions, vol. 1, issue 3 (1996), pages 39-40.

© 1996, Topology Atlas

The Sierpinski universal plane curve provides an answer to a question of Ofelia Teresa Alas

by

Wayne Lewis

(Texas Texh University, Lubbock, TX 79409-1042, U.S.A.)

In Question no. 2 on openly homogeneous spaces, Topology Atlas Invited Contributions, vol. 1, issue 2 (1996), p. 14, Ofelia Teresa Alas (alas@ime.usp.br) asked the following question:

Is there an example of an openly homogeneous space X such that no power X to kappa (=infinite cardinal) is homogeneous?

I believe it can be shown that the Sierpinski universal plane curve S is such a continuum.

S is openly homogeneous

Janusz Prajs of Opole, Poland has announced that the Sierpinski curve is homogeneous with respect to open monotone maps (I do not have a preprint or reference to this proof, but it is announced in Openly homogeneous continua in 2-manifolds: A generalization of a theorem of Bing, in Continua with the Houston Problem Book (Cook et. al. ed.) Marcel Dekker 305-314 (1995) ISBN 0-8247-9650-0). Carl Seaquist of our department has independently obtained this result and is in the process of writing it up.

The result on open homogeneity of the Sierpinski curve does not yet appear in print, but I know that in Seaquist's case he hopes to soon have a version ready for submission for publication. His proof centers on constructing continuous decompositions of the Sierpinski curve with decomposition space also the Sierpinski curve.

No power of S is homogeneous

Phelps (now Judy Kennedy), in Homeomorphisms of Products of Universal Curves, Houston J. Math. 6 (1980) 127-134, has shown that every homeomorphism of an arbitrary product of Menger universal curves is a product of homeomorphisms of the individual factor spaces composed with a homeomorphism which permutes coordinates. She does point out that the same proof applies to products of Sierpinski universal plane curves and hence no product of such is homogeneous.

While Phelps (Kennedy) considers arbitrary products of Sierpinski or Menger universal curves her result was preceded by K. Kuperberg, W. Kuperberg, and W. R. R. Transue (On the two homogeneity of Cartesian products, Fund. Math. 110 (1980) 131-134) for finite products of Menger curves and by Robert Cauty (Sur les homeomorphismes de certains produits de courbes, Bull. Acad. Polon. des Sci. 27 (1979) 413-415) for finite products of locally connected one-dimensional continua every open subset of which contains a simple closed curve.

Possible modifications of the original question

The disc is openly homogeneous and its countably infinite product with itself is homogeneous. Perhaps this suggests that some assumption of locally (homotopically) trivial structure is necessary. In the case of the Sierpinski universal plane curve, the nonhomogeneity of its products seems related to the fact that its space of homeomorphisms is totally disconnected.

I do believe that there is still much of interest to be discovered regarding the various generalizations of homogeneity.

Received by the editors: December 26, 1995. Revised: December 27, 1995.