Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers

Abstract

Let $\theta$ be an ordinal and let $(N_{\alpha }{)}_{\alpha \le \theta }$ 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 $H_{\alpha }=G{K}_{\alpha }$. The successive and limit differences of the groups $H_{\alpha }$ mark the first stages at which points of $\rho G$ become old quotient points. 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. For a closed normal subgroup $N$ of $G$, let $\hat{q}:\rho G\to \rho (G/N)$ extend the quotient homomorphism and put $K={\overline{N}}^{\rho G}$. Over $gN\in G/N$, the fiber in $\rho G\setminus G$ is $g(K\setminus N)$, homeomorphic to $\rho N\setminus N$, while for $y\in \hat{q}(\rho G)\setminus (G/N)$ the entire $\hat{q}$-fiber lies in $\rho G\setminus G$ and is homeomorphic to $\rho N$. 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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