Koszul A-infinity algebras and free loop space homology

A remark on Hochschild cohomology and Koszul duality

“…Another proof of this, under less stringent connectedness assumptions, is given in section 3 of Briggs-Gélinas' [2]. For Koszul A 8 -algebras, an isomorphism of the Hochschild cohomologies as weight graded A 8 -algebras (but not as Gerstenhaber algebras) is proved by Berglund-Börjeson in Theorem 3.2 of [1].…”

“…I am indebted to Wendy Lowen for a talk at Trinity College Dublin in May 2019 on the results of [8]. Many thanks to Vladimir Dotsenko for reminding me of references [4] and [2] and to Pedro Tamaroff for pointing out [1].…”

“…Our first proof of the lift to the B 8 -level is now superseded by recent work of Lowen-Van den Bergh [8], which we use in the very short proof of Theorem 2.6. Previous results relating the Hochschild cohomologies of Koszul(-Moore) dual algebras can be found in [3], [5] [4], [2] and [1] cf. remark 2.7.…”

Formality criteria in terms of higher Whitehead brackets

“…where, on the right, we have the classical cochains on L, dual of its chains (see Remark 2) and on the left, we use the isomorphism (ΛsH , δ) ♯ ∼ = (Λ(sH ) ♯ , d) [13,Lemma 23.1]. For each k ≥ 2, the bracket ℓ k on H (which defines δ and vanishes for k ≥ 3) is identified with the kth part d k of the differential d by the formula (9), see Section 3.2. In particular, d is decomposable and just quadratic, that is, (Λ(sH ) ♯ , d) is a Sullivan minimal model with quadratic differential (see [13, §12]).…”

“…From a classical point of view, one might see coformal spaces as building blocks for rational homotopy types, since every rational simply connected homotopy type can be realized as a perturbation of the corresponding coformal model ( [25]). More recently, a series of works embracing [26,6,8,9] prove very interesting results, combining Koszul duality and rational homotopy theory methods, which allow for effectively computing new results on (free and based) loop space homology and other interesting topics. In these, coformality plays a distinguished role.…”