FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1998, VOLUME 4, NUMBER 1, PAGES 81-100
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 $B=S^n$ 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 $T(S^n)$ .
All articles are published in Russian.
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Last modified: April 8, 1998