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The Stable Hurewicz Image of the Sphere at Two
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Abstract
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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Possibly relevant literature (found by the agent, not checked)
- Hadi Zare, On the Bott periodicity, J-homomorphisms, transfer maps and H_Q_0S^{-n}* (arXiv:0911.4478, 2009–2010). Shows that Curtis's conjecture holds when restricted to the factor J in Sullivan's splitting Q_0S^0 = J × cok J, a partial case of Theorem 1.1.
- Hadi Zare, On the Hurewicz homomorphism on the extensions of ideals in π_^s and spherical classes in H_Q_0S^0 (arXiv:1504.06752, 2015). Gives partial results toward Curtis's conjecture, including non-decomposability of certain elements with nonzero Hurewicz image and the passage of the conjecture from a set to the ideal it generates, in the setting of Theorem 1.1.
- Hadi Zare, Generalised geometric weak conjecture on spherical classes and non-factorisation of Kervaire invariant one elements (arXiv:1512.02040, 2015). Shows that the Hurewicz image on products of positive-degree elements is determined by η², ν², σ², and proves Curtis's conjecture on the image of the n-fold transfer for n > 1; both are special cases of Theorem 1.1.
- Hadi Zare, Freudenthal theorem and spherical classes in $H_*QS^0$ (arXiv:1801.06427, 2018-01-18; revised 2018-01-22). Its Hurewicz-vanishing results for stable classes factoring through certain finite spectra and its spherical-class calculations for certain finite loop spaces of spheres give prior partial results relevant to Theorem 1.1 and the Eccles consequence in Corollary 1.2.
- Hadi Zare, Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions (arXiv:1801.00735, 2018-01-02; revised 2018-01-03). Notes that finiteness of the set of spherical classes in H_*Q_0S^0 follows from Curtis's conjecture and proves a filtered finiteness result for Hurewicz images from finite loop spaces, for comparison with Corollary 1.3.
Digests and expositions
- Latent Seminar, The stable Hurewicz image: last suspensions and cone detection (2026-10-07). Explains the manuscript's last nonzero homology suspension, mapping-cone detection, Dickson-module obstruction, and Adams-filtration argument; the post states that it covers selected proof mechanisms rather than independently verifying the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 316
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 316: 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.
This is the only paper in this family.