Alex Chigogidze

Cohomological dimension of Tychonov spaces

Abstract The concept of the cohomological dimension dim G X is defined for any Tychonov (i.e., completely regular and Hausdorff) space X and any countable abelian group G . We prove that dim G X = dim G νX , where νX denotes the Hewitt realcompactification of X . We also give a spectral characterization of the dimension dim G . These two results allow us to reduce several problems concerning the dimension dim G of general spaces to the corresponding problems for Polish (i.e., completely metrizable and separable) spaces. For instance, we show that the inequality dim Z ( A ∪ B ) ⩽ dim Z A + dim Z B + 1, validity of which was known for metrizable spaces (Rubin [1991]), remains true in general. We also establish the existence of universal spaces of a given cohomological dimension and of a given weight.

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex P the following conditions are equivalent: (1)

The Extension Dimension and C-Spaces

P \in AE(X)

Nöbeling spaces and pseudo-interiors of Menger compacta

iff

Universal metric spaces and extension dimension

P \in AE(\beta X)

Compact groups and fixed point sets

for any space X; (2) There exists a P-invertible map of a metrizable compactum X with

On some dimensional properties of 4-manifolds

P \in AE(X)

Topological characterization of torus groups

onto the Hilbert cube.

On absolute extensors modulo a complex

Some generalizations of the classical Hurewicz formula are obtained for extension dimension and C-spaces. A characterization of the class of metrizable spaces which are absolute neighborhood extensors for all metrizable C-spaces is also given.

SECTIONS OF SERRE FIBRATIONS WITH 2-MANIFOLD FIBERS

Abstract For each k ⩾ 1, we introduce the categorical and the geometric pseudo-interiors of the k -dimensional universal Menger compacta and prove that they are homeomorphic to the universal Nobeling space N k 2 k + 1 .