Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers
Abstract
Let θ be an ordinal and let (Nα)α≤θ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group G, with N0={e}. Write ρG for the Raikov completion of G, and put Kα=Nα̅ρG and Hα=GKα. The successive and limit differences of the groups Hα mark the first stages at which points of ρG become old quotient points. For every nonzero ordinal θ and every set S⊆(0,θ], 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 ℤ/2ℤ, and for every δ<θ the successive factor Nδ+1/Nδ is compact exactly when δ+1∉ S. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup N of G, let q̂:ρG→ρ(G/N) extend the quotient homomorphism and put K=N̅ρG. Over gN∈ G/N, the fiber in ρG∖ G is g(K∖ N), homeomorphic to ρN∖ N, while for y∈q̂(ρG)∖(G/N) the entire q̂-fiber lies in ρG∖ G and is homeomorphic to ρN. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.
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BibTeX
@misc{hu2026arbitrary,
title = {{Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02212},
eprint = {2610.02212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02212},
}
RIS
TY - UNPB
TI - Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers
AU - Xing-Yu Hu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.02212
UR - https://arxiv.org/abs/2610.02212
AB - Let θ be an ordinal and let (N_α)_α≤θ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group G, with N_0={e}. Write ρG for the Raikov completion of G, and put K_α=N_α̅^ρG and H_α=GK_α. The successive and limit differences of the groups H_α mark the first stages at which points of ρG become old quotient points. For every nonzero ordinal θ and every set S⊆(0,θ], 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 ℤ/2ℤ, and for every δ<θ the successive factor N_δ+1/N_δ is compact exactly when δ+1∉ S. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup N of G, let q̂:ρG→ρ(G/N) extend the quotient homomorphism and put K=N̅^ρG. Over gN∈ G/N, the fiber in ρG∖ G is g(K∖ N), homeomorphic to ρN∖ N, while for y∈q̂(ρG)∖(G/N) the entire q̂-fiber lies in ρG∖ G and is homeomorphic to ρN. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.
ER -