Mináč–Tăn's vanishing conjecture for higher Massey products

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Let FF be a field, let pp be a prime, let n≥3n\geq 3, and let χ1,…,χn∈H1(F,Z/pZ)\chi_1,\dots,\chi_n\in H^1(F,\mathbb Z/p\mathbb Z). The Massey product ⟨χ1,…,χn⟩⊂H2(F,Z/pZ)\left\langle\chi_1,\dots,\chi_n\right\rangle\subset H^2(F,\mathbb Z/p\mathbb Z) is defined when it is non-empty, and it vanishes when it contains 00.

Mináč–Tăn's conjecture. For every field FF, every prime pp, every n≥3n\geq 3, and all χ1,…,χn∈H1(F,Z/pZ)\chi_1,\dots,\chi_n\in H^1(F,\mathbb Z/p\mathbb Z), if the Massey product

⟨χ1,…,χn⟩⊂H2(F,Z/pZ)\left\langle\chi_1,\dots,\chi_n\right\rangle\subset H^2(F,\mathbb Z/p\mathbb Z)

is defined, then it vanishes.

This conjecture proposes that every defined higher mod-pp Massey product in the cohomology of an absolute Galois group contains zero. The triple-product case is known for arbitrary fields, while the general case remains open in the stated formulation.

References

Primary source

Alexander Merkurjev and Federico Scavia, “Degenerate fourfold Massey products over arbitrary fields”, arXiv:2208.13011 (2023).

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