Homotopy transfer and formality
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2079-2116.

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.

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.

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DOI: 10.5802/aif.3444
Classification: 55U99,  18D50,  52C35
Keywords: formality, homotopy transfer theorem, algebraic operads, configuration spaces. hyperplane arrangements, toric arrangements
Drummond-Cole, Gabriel C. 1; Horel, Geoffroy 2

1 Center for Geometry and Physics Institute for Basic Science (IBS) 77 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do 37673 (Republic of Korea)
2 Université Sorbonne Paris Nord Institut Galilée 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Drummond-Cole, Gabriel C.; Horel, Geoffroy. Homotopy transfer and formality. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2079-2116. doi : 10.5802/aif.3444. https://aif.centre-mersenne.org/articles/10.5802/aif.3444/

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