Curtis’s conjecture

1 · The essentials

The manuscript claims to settle Curtis’s conjecture, a question about which features of spheres remain visible through a particular mathematical measurement. The measurement uses arithmetic in which only even or odd matters. The claim concerns every positive degree, meaning every level above zero in the setting considered. It says that all the visible results can be assembled from the images of three named classes, $η$ , $ν$ , and $σ$ , together with the Kervaire-invariant-one classes that exist. This does not assert that every possible Kervaire-invariant-one class exists. The abstract also states that Eccles’s conjecture consequently holds for every sphere $S^{n}$ with $n > 0$ . The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a scanner that records only certain features of objects. The claim is like saying that every record the scanner can produce is assembled from a specified collection of basic records—not that the scanner captures everything about the objects.

Why it matters

The claimed result gives a complete description of the positive-degree image in question, rather than merely listing some elements in it. The abstract also identifies a consequence for another conjecture: Eccles’s conjecture for every positive-dimensional sphere. The supplied source does not describe further consequences or applications.

2 · Going further

The precise claim concerns the positive-degree mod-two stable Hurewicz image of the sphere. In every positive degree, the manuscript claims that this image is spanned by the Hurewicz images of the Hopf-invariant-one classes $η$ , $ν$ , and $σ$ , together with the Kervaire-invariant-one classes that exist. Here, “spanned” means that every element of the image is a linear combination, with mod-two coefficients, of those specified images.

The source identifies this statement as Curtis’s conjecture. Its formulation distinguishes the named Hopf-invariant-one classes from the Kervaire-invariant-one classes, which are included only when they exist. The abstract states that the result implies Eccles’s conjecture for every sphere $S^{n}$ with $n > 0$ ; it does not state a version of that consequence for other spaces.

The supplied abstract announces the result but gives no main idea of the argument, intermediate steps, or technical method. Only one manuscript, “The Stable Hurewicz Image of the Sphere at Two,” is supplied, so no relationship to companion manuscripts can be described. The main result has not been formalized in Lean or formally verified.

Useful notions

Spheres and their dimensions
Stable homotopy classes
Homology with mod-two coefficients
The Hurewicz map and its image
Linear combinations and spanning

What was known before / what is new

Before

The source identifies the target statement as Curtis’s conjecture and names the Hopf-invariant-one and Kervaire-invariant-one classes appearing in it. It supplies no account of earlier partial results or the history of the conjecture. Eccles’s conjecture is named as the subject of a consequence, without further background.

New

The manuscript claims a complete spanning description of the mod-two stable Hurewicz image of the sphere in every positive degree. It further states that Eccles’s conjecture follows for every sphere of positive dimension. The abstract does not provide details of the argument supporting either statement.

Limits and assumptions

The main claim is restricted to the sphere, mod-two coefficients, and positive degrees. The Kervaire-invariant-one contribution includes only classes that exist; the supplied text does not specify which ones exist. The stated consequence for Eccles’s conjecture concerns spheres $S^{n}$ with $n > 0$ . No argument or account of remaining open questions is supplied, and the main result is not formalized in Lean and has not been formally verified.

Mini glossary

Mod-two coefficients: Coefficients using arithmetic in which only parity matters, so $1 + 1 = 0$ .
Stable Hurewicz image: The collection of homology classes reached by applying the Hurewicz map in the stable setting.
Spanned: A collection is spanned by specified elements when every element in it is a linear combination of those elements.
Hopf-invariant-one classes: The classes labeled by Hopf invariant one, listed in the source as $η$ , $ν$ , and $σ$ .
Kervaire-invariant-one classes: The classes labeled by Kervaire invariant one, whose images are included in the claimed description whenever those classes exist.

Keywords : Curtis’s conjecture · Stable Hurewicz image · Spheres

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The Stable Hurewicz Image of the Sphere at Two

25/09/2026

To be verified

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.

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