Abstract
We study the Boolean topological group associated with the last-step characteristic ρ3 of walks on the countable ordinal ω1. The construction is based on a C-sequence on ω1 and the characteristic ρ3(α, β) = 1 exactly when the last step of the minimal walk from β to α has maximal ρ1-weight. We review the construction of the corresponding Boolean group Gρ3(ω1) and its canonical neighborhood system, and we explain the relation between this neighborhood system and Tukey reducibility. The principal purpose of the paper is to clarify the precise cofinal-type question for Gρ3(ω1). We prove directly the basic topological properties required for the Tukey analysis, including the group axioms, the neighborhood-base properties, non-metrizability, and the Fréchet property. We then formulate the Tukey problem in terms of finite subsets of ω1 and identify the combinatorial conditions on the family F1(α) = {ξ < α : ρ3(ξ, α) = 1} which would yield the maximal Tukey type [ω1]<ω. This formulation separates the topological part of the argument from the set-theoretic combinatorics of walks on ordinals. The paper also compares the ρ3 construction with the ρ1 and ρ2 constructions and indicates several directions for further investigation, including higher ordinal analogues and Tukey spectra of Boolean groups arising from generalized walk characteristics.