Homogeneous Continua
by
Charles L. Hagopian
California State University, Sacramento

A continuum is a nondegenerate compact connected metric space. A continuum M is homogeneous if for each two points p and q of M there is a homeomorphism of M onto M that takes p to q. The three known topologically distinct examples of homogeneous plane continua are a circle, a pseudo-arc, and a circle of pseudo-arcs. Does there exist a fourth homogeneous plane continuum? The answer to this open question is “no” if every homogenous plane continuum that does not separate the plane is chainable. If there exists a homogeneous nonseparating plane that is not chainable, we know it must have span zero, be hereditarily indecomposable, and fail to be almost chainable in the sense of C. E. Burgess. In 1960, R. H. Bing proved every homogeneous plane continuum that contains an arc is a simple closed curve. In 1975, F. B. Jones gave a short proof of Bing’s theorem. Jones’s elegant argument involved a transformation group theorem of E. G. Effros. Effros’s theorem was used again in 1990, when P. R. Prajs generalized Bing’s theorem by proving every homogeneous continuum in euclidean n-space that contains an (n-1)-cell is an (n-1)-manifold. It is not known if there exists a homogenous indecomposable continuum of dimension greater than 1.

Date received: April 7, 2000