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Math-3020 · mathematics · Topology

Curtis stable mod-two Hurewicz image

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is the image of positive-degree stable sphere homotopy in $H*(Q_{0}S^{0};F_{2})$ generated only by the Hurewicz images of the Hopf-invariant-one elements $\eta ,\nu ,\sigma $ and the Kervaire-invariant-one elements that exist?

Short English statement

Is the image of positive-degree stable sphere homotopy in $H*(Q_{0}S^{0};F_{2})$ generated only by the Hurewicz images of the Hopf-invariant-one elements $\eta ,\nu ,\sigma $ and the Kervaire-invariant-one elements that exist?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-3020-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-3020

Manuscript 1 (family 316, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Curtis stable mod-two Hurewicz image

OpenAI's claim

Curtis's mod-two stable Hurewicz image is claimed to contain only the span generated by eta, nu, sigma and existing Kervaire classes, with Eccles's assertion following for all positive-dimensional spheres. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $Q_0S^0$ be the base component of the infinite loop space of the sphere, and $h_d:\pi_d^S\to H_d(Q_0S^0;\mathbb F_2)$ the stable mod-two Hurewicz map. OpenAI claims \[ \operatorname{im}\!\left(\bigoplus_{d>0}h_d\right) =\operatorname{span}_{\mathbb F_2}\{h(\eta),h(\nu),h(\sigma),h(\theta_j):\theta_j\text{ exists}\}. \] The Hopf-invariant-one degrees are $1,3,7$, and $\theta_j$ is a Kervaire-invariant-one class of degree $2^{j+1}-2$, $j\ge1$. The theorem describes the Hurewicz image in every positive degree; it does not additionally assert which Kervaire classes exist.

Outline of the argument

Manuscript 1 (family 316, 2026-09-25): Iterating suspension until a spherical homology image first disappears leaves a preceding square with a constrained primitive root. Mapping cones turn its detection into Steenrod-module extensions, analyzed through Dickson modules and Cartier halving. Positive terminal levels force Hopf detection; at level zero, weight and Adams-filtration bounds force Kervaire detection, yielding the claimed span.

Formal verification

Family 316 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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