Topological Necessary Conditions for Control Dynamics

Efthimios Kappos

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper states Theorem 7 (equality of images im(φ_*) = im(G_{-∇h*})) but gives no proof and even misstates the domain of φ_* (using H_*(DB\S) instead of H_*(DB\S\Σ)). The model’s proof captures the intended strategy but relies on a false step: it treats s_U as if it allowed every class in H_*(DB\S\Σ) to be represented on s_U(B\S), which fails because removing Σ changes the fiberwise homotopy type. Thus both are incomplete as written.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents a coherent program for deriving topological necessary conditions for control design via Gauss maps and index theory, and it collects several useful statements (including Lemma 1 and homotopy comparisons of Gauss maps). However, core results such as Theorem 7 are stated without proofs, and there are precision issues (notably the domain of φ and its induced maps). These gaps prevent verification of correctness. With detailed proofs, careful handling of domains, and explicit homotopy constructions (or spectral-sequence arguments) where needed, the contribution could be solid for specialists.