---
title: "A counterexample to Wall's finite D(2) problem"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 321; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "topology"
  - "unformalized"
seed_rank: 2176
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "A Counterexample to Wall's D(2) Problem"
    url: "https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# A counterexample to Wall's finite D(2) problem

## One-sentence takeaway

OpenAI's result family 321 (Topology) claims: Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model.

## Why it matters here

Topology results are background for knot, surface and configuration-space reasoning in geometry code. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.
- *A Counterexample to Wall's D(2) Problem*: We give a negative answer to Wall's finite $D(2)$ problem. We construct a finite connected three-dimensional CW complex satisfying the $D(2)$ finiteness condition but not homotopy equivalent to any finite CW complex of dimension at most two. The example has infinite fundamental group.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [A Counterexample to Wall's D(2) Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math