Homotopy transfer and formality

Drummond-Cole, Gabriel C.; Horel, Geoffroy

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 2079-2116.

Dans cet article nous montrons que, sous certaines hypothèses, la $n$-formalité d’une structure algébrique différentielle graduée est équivalente à l’existence d’un automorphisme au niveau des chaînes relevant un isomorphisme gradué tordant relatif à une unité d’ordre plus grand que $n$. Un résultat similaire sous des hypothèse légèrement différentes avait été prouvé par Joana Cirici et le second auteur. Nous utilisons le théorème de transfert homotopique et une procédure récursive explicite pour tuer les opérations supérieures. Comme application de ce résultat, nous prouvons la formalité à coefficients dans les entiers $p$-adiques de certaines dg-algèbres venant d’arrangements d’hyperplans ou d’arrangements toriques ainsi que des espaces de configurations.

In this paper, we prove that, given appropriate hypotheses, $n$-formality of a differential graded algebraic structure is equivalent to the existence of a chain-level automorphism lifting a degree twisting isomorphism relative to a unit of order greater than $n$. A similar result with slightly different hypothesis was proved by Joana Cirici and the second author. We use the homotopy transfer theorem and an explicit inductive procedure in order to kill the higher operations. As an application of our result, we prove formality with coefficients in the $p$-adic integers of certain dg-algebras coming from hyperplane and toric arrangements and configuration spaces.

Classification : 55U99, 18D50, 52C35
Keywords: formality, homotopy transfer theorem, algebraic operads, configuration spaces. hyperplane arrangements, toric arrangements
Mot clés : formalité, théorème de transfert homotopique, opérades algébriques, espaces de configurations, arrangements d’hyperplans, arrangements toriques

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