Splitting varieties for triple Massey products

Koszulity of cohomology = K(π,1)-ness + quasi-formality

“…In the paper [38], this conjecture was proven for all the (one-dimensional) local and global fields. On the other hand, there is a series of recent papers [16,27,8,28] discussing and partially proving the conjecture that tuple Massey products of degree-one elements vanish in the cohomology algebra H * (G F , Z/l). The results above in this section and the discussion below show that the Koszulity conjecture implies vanishing of the tensor Massey products in H * (G F , Z/l), but may have no direct implications concerning the problem of vanishing of the tuple Massey products.…”

“…The following naïve attempt to construct a commutative version of Example 6.3 illustrates one of the intricacies of relations sets (2). Consider, instead of the (m + 2)-variable Lie relation in Example 6.4, a two-variable Lie relation (16) [x, y] + q 3 (x, y) + q 4 (x, y) + q 5 (x, y) + · · · = 0, where q n are homogeneous Lie expressions of degree n in the variables x and y. We claim that the relation (16) is always equivalent to the relation [x, y] = 0 in the world of (Lie or associative) formal power series in x and y, so the conilpotent (coenveloping) coalgebra C defined by (16) is in fact cocommutative and isomorphic to gr N C. Indeed, the innermost bracket in any Lie monomial in x and y is always ±[x, y].…”

“…We claim that the relation (16) is always equivalent to the relation [x, y] = 0 in the world of (Lie or associative) formal power series in x and y, so the conilpotent (coenveloping) coalgebra C defined by (16) is in fact cocommutative and isomorphic to gr N C. Indeed, the innermost bracket in any Lie monomial in x and y is always ±[x, y]. Substituting the expression for [x, y] obtained from (16) in place of the innermost bracket in every term of degree 3 in (16), one deduces from (16) a new Lie relation in x and y with every term of degree n 3 replaced by an (infinite) linear combination of terms of degrees higher than n. Continuing in this fashion and passing to the limit in the formal power series topology, one concludes that the relation (16) In fact, the exterior algebra in two variables of degree 1, which is the cohomology algebra H * (C) of the coalgebra C defined by (16), is a free (super)commutative graded algebra, so it is intrinsically formal as a commutative graded algebra (i. e., any commutative DG-algebra with such cohomology algebra is formal). Example 6.3 shows that a noncommutative DG-algebra with such cohomology algebra does not have to be formal, while Example 6.4 provides a (super)commutative Koszul graded algebra that is not intrinsically formal in the commutative world already.…”

“…It was conjectured in the papers [32,37] that the cohomology algebra H * (G F , Z/l) is Koszul for the absolute Galois group G F of any field F containing a primitive l-root of unity; and the question was asked in the paper [16] whether the cochain DG-algebra of the group G F with coefficients in Z/l is formal. As the group H in our counterexample is the maximal quotient pro-l-group of the absolute Galois group G F of an appropriate p-adic field F containing a primitive l-root of unity, our results provide a negative answer to this question of Hopkins and Wickelgren.…”

“…Section 9.11, where these ideas were first put in writing, was inserted into the paper [37] shortly before its journal publication under the influence of a question asked by P. Deligne during the author's talk at the "Christmas mathematical meetings" in the Independent University of Moscow in January 2011. I learned that the question about formality of the cochain DG-algebras of absolute Galois groups was asked in the paper [16] from a conversation with I. Efrat while visiting Ben Gurion University of the Negev in Be'er Sheva in the Summer of 2014. The decision to write this paper was stimulated by the correspondence with P. Schneider in the Summer of 2015.…”