Result 309, Topology

The Kervaire invariant problem at the prime three

Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.

Proof

The bigger picture

Why it matters

The manuscript reports that only three members of a standard family of algebraic candidates represent genuine topological elements at the prime three. This pinpoints where a recurring algebraic pattern reflects topology and where it does not.

What changes?

The mod-three Adams spectral sequence is a staged algebraic calculation of stable homotopy groups, which describe maps between spheres after repeated suspension. For the standard Kervaire classes, the manuscript reports survival exactly at indices 0, 2, and 3, in stems, or degrees, 10, 106, and 322. Each surviving class detects a set of possible elements, called a detection coset. Each such coset contains an element of exact additive order three: three copies sum to zero, but one does not.

What does that help mathematicians do?

The classification rules out every other index in this standard family as a source of surviving Kervaire classes at the prime three. For the surviving indices, the order statement gives additional information about the elements detected, not just their existence. In particular, the manuscript identifies an order-three Kervaire element in degree 322, providing a specific target for further work on the structure of these groups.

Are there practical applications?

Its immediate value is foundational, in understanding the three-primary part of stable homotopy theory, where elements whose orders are powers of three are studied. The claimed result gives further calculations a precise existence and nonexistence pattern to respect. It is not a general method for computing all stable homotopy groups.

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 Kervaire invariant problem at the prime three

September 24, 2026 90 pages

We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026,
  author = {{OpenAI}},
  title = {{The Kervaire invariant problem at the prime three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf}{OAI:The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

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.