Massey products <y,x,x,...,x,x,y> in Galois cohomology via rational points

Abstract

For $x$ an element of a field other than $0$ or $1$, we compute the order $n$ Massey products $$⟨(1-x{)}^{-1},x^{-1},\dots ,x^{-1},(1-x{)}^{-1}⟩$$ of $n-2$ factors of $x^{-1}$ and two factors of $(1-x{)}^{-1}$ by embedding ${\mathbb{P}}^1-\{0,1,\infty \}$ into its Picard variety and constructing $\mathrm{Gal}(k^s/k)$ equivariant maps from ${\pi }_1$ applied to this embedding to unipotent matrix groups. This method produces obstructions to ${\pi }_1$-sections of ${\mathbb{P}}^1-\{0,1,\infty \}$, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products $$⟨x^{-1},(-x{)}^{-1},\dots ,(-x{)}^{-1},x^{-1}⟩.$$

arXiv commentTo appear in the Journal of Pure and Applied Algebra's special issue in honor of Chuck Weibel, edited by Guillermo Cortiñas, Eric Friedlander, and Christian Haesemeyer. 26 pages. This paper continues arXiv:1011.0655 and arXiv:1107.1790

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