---
title: "The Morita classes are nonzero"
authors:
  - "Alexander Kupers"
  - "Jeremy Miller"
  - "Peter Patzt"
year: 2026
venue: "arXiv:math.AT"
arxiv: "2609.12951"
doi: null
source: "https://arxiv.org/abs/2609.12951"
topics:
  - "curiosity"
  - "maths-foundations"
  - "algebraic-topology"
  - "quantum-algebra"
seed_rank: 1829
seed_batch: "curiosity-2026-09-15"
reviewed: "2026-09-15"
pool: "maths-foundations"
relevance_score: 10
lineage: morita-classes-kontsevich
cites:
  - title: "The Morita classes are nonzero"
    url: "https://arxiv.org/abs/2609.12951"
    year: 2026
    arxiv: "2609.12951"
    doi: null
see:
  - "1616-koszul-duality-and-morita-categories"
---

# The Morita classes are nonzero

## One-sentence takeaway

All Morita classes μ_k in H_{4k}(Aut(F_{2k+2}); ℚ) are nonzero — disproving a case of Kontsevich's finiteness conjecture via Steinberg homology's Hopf structure.

## Why it's lovely

Why you might love this: Kontsevich asked whether these Morita classes might vanish for large k; they do not. A novel Hopf-algebra description of Steinberg homology turns a finiteness rumour into an infinite family of nonzero stable-homology classes — spectator-sport algebra with a conjecture corpse on the table.

## Key ideas

- Morita classes μ_k ∈ H_{4k}(Aut(F_{2k+2}); ℚ) are shown nonzero for all k.
- Proof uses a new description of the Hopf algebra structure on Steinberg homology.
- Analogous nonvanishing with twisted coefficients.
- Disproves a case of Kontsevich's finiteness conjecture on these classes.

## Caveats

Homology of automorphism groups of free groups; does not by itself compute mapping-class or Out(F_n) phenomena Broadside builds with. Pair with Morita-category card 1616 for naming, not for the same theorem.

## Links

- arXiv abs: https://arxiv.org/abs/2609.12951
- PDF: https://arxiv.org/pdf/2609.12951.pdf