Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers
Year: 2026
Type: preprint
Abstract: Let \( \theta \) be an ordinal and let \( (N_\alpha)_{\alpha\leq\theta} \) be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group \( G \), with \( N_0=\{e\} \). Write \( \rho G \) for the Raikov completion of \( G \), and put \( K_\alpha=\overline{N_\alpha}^{\rho G} \) and \( H_\alpha=GK_\alpha \). The successive differences and the corresponding limit differences of the groups \( H_\alpha \) mark the first stages at which points of the Raikov completion become old quotient points. The set of stages with nonempty strata can be prescribed arbitrarily even under pseudocompactness. For every nonzero ordinal \( \theta \) and every set \( S\subseteq(0,\theta] \), there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in \( S \). The terminal quotient can simultaneously be chosen isomorphic to \( \mathbb Z/2\mathbb Z \), and for every \( \delta<\theta \) the successive factor \( N_{\delta+1}/N_\delta \) is compact exactly when \( \delta+1\notin S \). Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. The realization theorem uses an exact description of the fibers of quotient extensions. For a closed normal subgroup \( N \) of \( G \), let \( \widehat q:\rho G\to\rho(G/N) \) extend the quotient homomorphism and write \( K=\overline N^{\rho G} \). The fiber in \( \rho G\setminus G \) over an embedded quotient point \( gN \) is \( g(K\setminus N) \), homeomorphic to the Raikov remainder \( \rho N\setminus N \), while for a point\( y\in\widehat q(\rho G)\setminus(G/N) \) the whole \( \widehat q \)-fiber lies in \( \rho G\setminus G \) and is a copy of \( \rho N \). For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. For compact kernels, pseudocompactness of the source and quotient remainders is equivalent. For noncompact precompact kernels, the source remainder maps onto the entire quotient completion, and countable Abelian examples exhibit both pseudocompact and non-pseudocompact quotient remainders.
Source: Preprints.org
Author: Xing-Yu Hu
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204753417