FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1998, VOLUME 4, NUMBER 1, PAGES 81-100

Algebraic structure of function rings of some universal spaces

A. V. Zarelua

Abstract

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Using an algebraic characterisation of zero-dimensional mappings the author constructed universal compacts $Z(B,H)$ for the spaces possessing zero-dimensional mappings into the given compact $B$, where $H$ is a collection of functions on $B$ which separates points and closed subsets. By the characterisation theorem due to M. Bestvina for $B=Sn$ and an appropriate $H$ it is proved that the compact $Z(B,H)$ coincides with the Menger's universal compact $\mu n$. As an application one gets a description of the ring $C$_R( μ ⁿ) as the closure of the polynomial ring $C$_R(Sⁿ)[u₁,u₂,...,u_k,...] on elements $u$_k such that $u$_k²=h_k⁺ for some $h$_k⁺ ∈ C_R(Sⁿ). Another application is an representation of $\mu n$ as the inverse limit of real algebraic manifolds. The complexification of this construction leads to some compact $E2n$ which is the inverse limit of compactifications of complex algebraic manifolds without singularities and contains $\mu n$ as the fixed set of the involution generated by the complex conjugation. On $E2n$ an action of the countable product of order 2 cyclic groups is defined; the orbit-space of this action is a compactification of the tangent bundle $T(Sn)$.

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Last modified: April 8, 1998