Let k be a field, let p be a prime, and let n≥ 3. For cohomology classes alpha_0,…,alpha_n-1∈ H^1(k,mathbf F_p), consider the n-fold Massey product langle alpha_0,…,alpha_n-1rangle⊆ H^2(k,mathbf F_p). Massey vanishing conjecture. If this Massey product is non-empty, then it contains 0∈ H^2(k,mathbf F_p). The conjecture extends the vanishing of triple Massey products from coefficients in mathbf F_2 to all primes and all lengths, and connects arithmetic Galois cohomology with the topology of link complements. It remains open in general.

Massey vanishing conjecture for Galois cohomology

Mináč and Tân formulated the conjecture that every defined higher Massey product contains zero. The full assertion remains open beyond its established special cases.

Known results

All lengths $n \ge 3$ over number fields, for every prime $p$ (2019 paper).
Triple products for $p=2$ (Hopkins–Wickelgren).
Triple products for fields containing a primitive $p$ th root of unity (Matzri; Efrat–Matzri; Mináč–Tân, 2014).
Triple products for odd primes via generalized Bockstein maps (2020).

August 25, 2023 conditional result

Merkurjev and Scavia proved vanishing for triple products, and for some degenerate fourfold products, under a formal Hilbert $90$ hypothesis on a profinite group with character $\theta:G\to\mathbb{Z}_p^\times$ . This does not settle arbitrary fields and arbitrary $n$ .

Current status (as of September 2026): The conjecture is settled for number fields and several special cases, but the full assertion for arbitrary fields and general $n \ge 3$ remains open.

Massey vanishing conjecture for Galois cohomology

Primary source

Progress summary

Solutions 0