Result 310, Topology

Quillen's conjecture in rational homology

Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.

Proof

The bigger picture

Why it matters

A finite group's algebraic structure can force a geometric shape built from its subgroups to resist shrinking to a point. The unreviewed manuscript reports a universal criterion that detects this obstruction using rational homology.

What changes?

The claim covers every finite group G and every prime p, assuming G has no nontrivial normal subgroup whose order is a power of p. It considers nontrivial elementary abelian p-subgroups, products of cyclic groups of order p, ordered by inclusion. Nested chains of these subgroups form a geometric complex. The manuscript reports nonzero augmented reduced rational homology, a topological invariant computed with rational coefficients. Consequently, the complex is not contractible: it cannot be continuously shrunk to a point.

What does that help mathematicians do?

The rational-homology conclusion gives more information than noncontractibility alone: the obstruction is visible to an algebraic invariant of the complex. Read in reverse, the result says that vanishing augmented reduced rational homology forces the group to contain a nontrivial normal p-subgroup. Researchers can therefore use information about the topology of subgroup chains to deduce a specific structural feature of the original finite group.

Are there practical applications?

Its immediate value is foundational, connecting finite-group structure with computable topological invariants. The reported theorem supplies a consistency check for calculations of these subgroup complexes and rules out attempts to contract them under its hypothesis. The supplied abstract establishes no runtime guarantees or practical application; its stated contribution is a general mathematical constraint.

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

Rational homology and Quillen's conjecture

September 24, 2026 63 pages

We prove Quillen's conjecture for all finite groups and all primes. More precisely, if a finite group G has trivial largest normal p-subgroup Op(G)O_p(G), then the poset of nontrivial elementary abelian p-subgroups of G has nonzero augmented reduced rational homology. This establishes the stronger rational-homology form of the conjecture.

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Cite (BibTeX) 
@misc{OAI:Rational-homology-and-Quillens-conjecture-September-24-2026,
  author = {{OpenAI}},
  title = {{Rational homology and Quillen's conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Rational-homology-and-Quillens-conjecture-September-24-2026/paper.pdf}{OAI:Rational-homology-and-Quillens-conjecture-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.