Curtis’s conjecture

A paper claims to prove Curtis's 1975 conjecture: only η, ν, σ and Kervaire classes show up in mod-2 sphere homology.

1 paper September 25, 2026 Explained by AI

Stable homotopy groups record the ways one sphere can wrap around another, after enough room is added. The Hurewicz map turns each such wrapping into a homology class, a count of holes taken mod 2. Curtis guessed in 1975 that almost every wrapping becomes zero this way. The paper claims to prove that only the classes η, ν, σ and the Kervaire-invariant-one classes survive. It follows that the surviving image sits in at most nine degrees, all at or below 126.

First step

1975

Edward Curtis, study of the Dyer–Lashof and lambda algebras

Significance

Major

It claims to settle a fifty-year-old conjecture about which stable homotopy classes of spheres homology can see.

Trust

Claimed, not checked

Field

Topology

1 paper

The explanation

Start with the eighth stable stem. It is the group of ways an (n + 8)-dimensional sphere can wrap around an n-dimensional sphere, when n is large. This group has four elements. Now ask a coarser question about each element. Turn the wrapping into a map into a large space called Q₀S⁰. Then read off the homology class it hits, counting holes with coefficients taken mod 2. This rule is the mod-2 Hurewicz map. In degree 8 the paper says every element maps to zero. Homology cannot tell any of the four wrappings apart from doing nothing.

Some wrappings do survive. The class η lives in degree 1, ν in degree 3, and σ in degree 7. These are the Hopf-invariant-one classes. A second family is the Kervaire-invariant-one classes θⱼ. A class has Kervaire invariant one when a matching framed manifold carries a certain invariant equal to one. These classes can only sit in degrees 2^(j+1) − 2. For j from 1 to 6 those degrees are 2, 6, 14, 30, 62 and 126.

Curtis proposed that these are the only survivors, in his study of the Dyer–Lashof and lambda algebrasSource 1. Later, May's account recorded a gap that Wellington found in the proposed proofSource 2. So the question stayed open. The main theorem says the image in positive degrees is spanned by the images of η, ν, σ and the θⱼ that existSource 3. It claims that Curtis's conjecture holds in all positive degreesSource 4.

The degrees become finite because of an earlier result. Hill, Hopkins and Ravenel proved that θⱼ can exist only for j at most 6Source 5. So the image vanishes above degree 126, the paper claimsSource 6. That leaves 9 possible degrees between 1 and 126, namely 1, 2, 3, 6, 7, 14, 30, 62 and 126. The other 117 degrees in that range always map to zero. This is only an upper bound. The paper does not claim a nonzero image in every listed degreeSource 7. Separately, Lin, Wang and Xu claim in a preprint that θ₆ exists in degree 126Source 8.

Why was this hard? A homology class hit by a sphere must pass simple tests. It must be primitive, which means it splits in the simplest possible way under the diagonal map. It must also be killed by the positive Steenrod operations, a standard set of tools that act on homology. But these tests alone do not show that a sphere really hits the classSource 9. The homology algebra is large, and many classes pass the tests. The obstacle is to rule them out one by one, using only that an actual sphere map exists.

The paper follows a class through repeated suspensions, each one shifting the picture up a dimension. It looks at the last step where the class is still nonzero. There it shows the class is a square. At the bottom level, it rules out fourth and higher powers. Eccles and Zare had already reached that conclusion by other meansSource 10. The paper then uses the cone of the sphere map and an algebraic argument about Dickson algebras. In the last case, Kuhn's theorem and Browder's work identify the class as a Kervaire classSource 11. As a consequence, the paper claims Eccles's conjecture for every sphere Sⁿ with n > 0Source 12.

How sure is this? The paper gives a full proof, but no Lean check covers it. No specialist review has been published. The argument is long and builds on deep earlier results. Treat the result as a claim until experts confirm it.

What it doesn't show. This does not say which of the nine possible degrees actually carry a nonzero image, or settle Eccles's conjecture for spaces other than spheres.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if correct. The paper claims a full proof of Curtis's conjecture, but no Lean check covers it and no specialist review is published.

Words used

Stable stem: The group of ways a sphere of dimension n + d can wrap around a sphere of dimension n, once n is large enough that the answer stops changing.
Homology mod 2: A way of counting the holes of a space, keeping only whether each count is even or odd.
Mod-2 Hurewicz map: The rule that sends a map from a sphere to the mod-2 homology class it hits.
Q₀S⁰: A large space whose homotopy groups are the stable stems. Its homology is known explicitly.
η, ν, σ: The Hopf-invariant-one classes, in degrees 1, 3 and 7. They come from the Hopf maps.
Kervaire-invariant-one class θⱼ: A class in degree 2^(j+1) − 2 whose matching framed manifold has a certain invariant equal to one.
Primitive class: A homology class that splits in the simplest possible way under the diagonal map of the space.
Steenrod operations: Standard operations that act on mod-2 homology and cohomology of every space.

The original paper

The Stable Hurewicz Image of the Sphere at Two (PDF, opens in a new tab)
September 25, 2026

All quoted lines (12)

Curtis proposed the Hopf–Kervaire description in his study of the Dyer–Lashof and lambda algebras.
The Stable Hurewicz Image of the Sphere at Two Back to text
May's subsequent account records a gap found by Wellington in the proposed proof.
The Stable Hurewicz Image of the Sphere at Two Back to text
The image of $\bigoplus_{d>0}h_d$ is spanned over $\F_2$ by the Hurewicz images of the Hopf-invariant-one classes $\eta,\nu,\sigma$ , in degrees $1,3,7$ , and the Kervaire-invariant-one classes $\theta_j$ , in degrees $2^{j+1}-2$ , $j\ge1$ , whenever they exist.
The Stable Hurewicz Image of the Sphere at Two Back to text
In particular, Curtis's conjecture is true in all positive degrees.
The Stable Hurewicz Image of the Sphere at Two Back to text
Hill, Hopkins, and Ravenel prove that $\theta_j$ can exist only for $j\le6$ .
The Stable Hurewicz Image of the Sphere at Two Back to text
In particular, the positive stable mod-2 Hurewicz image vanishes above degree $126$ .
The Stable Hurewicz Image of the Sphere at Two Back to text
The support in Corollary … is an upper bound; it does not assert nonvanishing in every listed degree or an exact dimension of the image.
The Stable Hurewicz Image of the Sphere at Two Back to text
In their preprint, Lin, Wang, and Xu prove existence in that dimension.
The Stable Hurewicz Image of the Sphere at Two Back to text
A spherical class must be primitive and annihilated by positive Steenrod operations, but these necessary conditions alone do not establish its realization.
The Stable Hurewicz Image of the Sphere at Two Back to text
Section … proves the exclusion of fourth powers by spectrum pairings and integral characteristic classes; the conclusion itself is already present in that prior work.
The Stable Hurewicz Image of the Sphere at Two Back to text
Kuhn's theorem bounds its Adams filtration by two, and Browder's identification gives Kervaire detection.
The Stable Hurewicz Image of the Sphere at Two Back to text
Consequently Eccles's conjecture holds for every sphere $S^n$ , $n>0$ : the alternatives are the bottom stably spherical class and Hopf-invariant-one classes.
The Stable Hurewicz Image of the Sphere at Two Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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