Darboux injections from closed manifolds: Banakh–Banakh Problems 1.7 and 1.8
Banakh and Banakh (2020) proved that connectedness-preserving (Darboux) injections are continuous in several compact settings — into 1-manifolds from compact sources, from closed surfaces into surfaces, and from closed 3-manifolds with finite into 3-manifolds — and asked whether every Darboux bijection of (Problem 1.7) and of (Problem 1.8) is a homeomorphism. Answer: yes. For every , every Darboux injection from a connected closed -manifold into an -manifold is a homeomorphism onto a connected component of the target; no homology hypothesis and no surjectivity are needed
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Continuum theory / Darboux maps
- Posed by
- Iryna Banakh and Taras Banakh, Topology and its Applications 275 (2020), Problems 1.7–1.8
- Year posed
- 2020
- Years open
- 6y
- Solved
- 2026-06
- Model
- GPT-5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Peter L.
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Both problems are answered affirmatively, in every dimension at once: every connectedness-preserving injection from a connected closed -manifold into an -manifold is a homeomorphism onto a component, so in particular every Darboux self-bijection of and of every closed manifold is a homeomorphism. This removes the finite- hypothesis of the 2020 theorem for 3-manifolds and extends it above dimension 3. Compactness of the source is essential: the companion preprint (Zenodo 10.5281/zenodo.22346412) shows the corresponding statement fails for , . The note does not address noncompact sources, manifolds with boundary, or targets of different dimension.
What the AI did
The note's acknowledgment: it was produced with substantial assistance from large language models, principally GPT-5.5 Pro, in a research program directed by the author, who selected, checked and assembled the arguments. The proof uses Banakh–Banakh's framework of -varieties and componnectedness, Alexander–Lefschetz duality with coefficients, and an induction on minimal carriers of nonzero Čech cohomology classes; a separate proof that metrizable -manifolds are -varieties, and a one-dimensional base case, are supplied. The same theorem was later re-derived by the same method, independently and without access to the note, by GPT-6 (Codex) in a subsequent phase of the program; that re-derivation is in the program's records.
Verification
Unreviewed. The 17-page note (Zenodo 10.5281/zenodo.22347647, dated June 2026, posted 5 September) was read here in full; the theorem, the method (Alexander–Lefschetz duality with F2 coefficients, induction on minimal carriers of Čech cohomology classes, in Banakh–Banakh's framework of n-varieties) and the disclosure match the submission, and Problems 1.7 and 1.8 were confirmed verbatim in arXiv 1809.00401. Nobody outside the author's program has read the argument; the re-derivation by a second model inside the same program is internal corroboration. Candidate as submitted.
Sources
Submitted by WittyHeron892 on