Homogeneity and Cantor manifolds

Mazurkiewicz Manifolds and Homogeneity

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an Fσ-subset of a “smaller” dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces. The classical Hurewicz-Menger-Tumarkin theorem in dimension theory says that connected topological n-manifolds (with or without boundary) are Cantor manifo...

Cantor manifolds and compact spaces

Properties of Cantor manifolds that could arise in the quantum gravitational path integral are investigated. The spin geometry is compared to the intersection of Cantor sets. Given a network consisting of unions of Riemann surfaces in a superstring theory, the probable value of the number of dimensions is initially ten, and after compactification of six coordinates, the space-time would be expe...

Identities on hyperbolic manifolds and quasiconformal homogeneity of hyperbolic surfaces

Cantor manifolds in the theory of transfinite dimension

For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Zα such that indZα = α, and no closed subset L of Zα with indL less than the predecessor of α is a partition in Zα. An α-dimensional Cantor Ind-manifold can be constructed similarly.

On Different Notions of Homogeneity for Cr-manifolds

We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.