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Fundamentalnaya i Prikladnaya Matematika, 1998, Volume 4, Issue 1, Pages 81–100
(Mi fpm302)
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This article is cited in 1 scientific paper (total in 1 paper)
Research Papers Dedicated to the 100th Anniversary of P. S. Alexandroff's Birth
Algebraic structure of function rings of some universal spaces
A. V. Zarelua M. V. Lomonosov Moscow State University
Abstract:
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 $B=S^n$ 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 $C_{\mathbb R}(\mu^n)$ as the closure of the polynomial ring $C_{\mathbb R}(S^n)[u_1,u_2,\dots,u_k,\dots]$ on elements $u_k$ such that $u_k^2=h_k^+$ for some $h_k^+\in C_{\mathbb R}(S^n)$. 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 $E^{2n}$ 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 $E^{2n}$ 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 $T(S^n)$.
Received: 01.10.1997
Citation:
A. V. Zarelua, “Algebraic structure of function rings of some universal spaces”, Fundam. Prikl. Mat., 4:1 (1998), 81–100
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