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

Algebraic structure of function rings of some universal spaces

A. V. Zarelua


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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 μ n. As an application one gets a description of the ring CR( μ n) as the closure of the polynomial ring CR(Sn)[u1,u2,...,uk,...] on elements uk such that uk2=hk+ for some hk+ ∈ CR(Sn). Another application is an representation of μ 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 μ 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