Result 316, Topology

Curtis’s conjecture

Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.

Proof

The bigger picture

Why it matters

Maps between spheres encode subtle topological information, but homology sees only part of it. This result claims to identify exactly what survives a standard comparison, restricting the visible information to two distinguished families of classes.

What changes?

Stable homotopy classes record maps between spheres after stabilization, meaning both sphere dimensions can increase together. The Hurewicz map extracts homology data, here with coefficients modulo two. The manuscript reports that in every positive degree its image is spanned by the images of the Hopf-invariant-one classes eta, nu and sigma and the Kervaire-invariant-one classes that exist. Spanned means every image is a sum of these; no additional Kervaire classes are asserted to exist.

What does that help mathematicians do?

The claimed classification gives a concrete obstruction: a homology class outside the specified span cannot arise under this Hurewicz map. In a positive degree where all the listed images vanish, every stable class must have zero Hurewicz image. Researchers could therefore rule out proposed detections without first classifying all stable sphere maps. The abstract also states that Eccles's conjecture follows for every sphere of dimension greater than zero.

Are there practical applications?

The immediate value is foundational: it would establish the exact limits of this homology-based way of detecting stable sphere maps. It separates information that this comparison can reveal from information requiring other tools. The supplied material describes a consequence within topology, not a practical algorithm or an application outside mathematics.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Stable Hurewicz Image of the Sphere at Two

September 25, 2026 37 pages

We prove Curtis's conjecture: in every positive degree, the mod-two stable Hurewicz image of the sphere is spanned by the images of η, ν, σ and the Kervaire-invariant-one classes that exist. Consequently, Eccles's conjecture holds for every sphere Sn with n > 0.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026,
  author = {{OpenAI}},
  title = {{The Stable Hurewicz Image of the Sphere at Two}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026/paper.pdf}{OAI:The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.