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28. In this section, the ground ring k is either torsion free or a Z/pZ-algebra. Moreover all k-modules are assumed to be flat. The (signed) shuffle bialgebra T (sR) has a canonical augmentation T (sR) → k ⊕ sR. We wrote I(sR) for its augmentation ideal. There is a decreasing filtration · · · ⊂ I(sR)n ⊂ I(sR)n−1 ⊂ · · · ⊂ I(sR)1 ⊂ I(sR)0 = T (sR). This filtration induces a filtration of Hochschild (co)chain spaces · · · ⊂ M ⊗I(sR)n ⊂ M ⊗I(sR)n−1 ⊂ · · · ⊂ M ⊗I(sR)1 ⊂ M ⊗I(sR)0 = C∗ (R, M ), 1 ⊥ C ∗ (R, M ) =CoDer(I(sR)0 , A⊥ R (M )) → CoDer(I(sR) , AR (M )) → .

The following Lemma is taken from an appendix of Sullivan [Su] together with an application of Tradler and Zeinalian [TZ]. We write C ∗ (X), C∗ (X) for the simplicial complexes associated to the triangulation of a space. Hopefully, the context should always makes clear if we are working with singular chains or the ones from a triangulation. We denote by d : C∗ (X) → C∗−1 (X) the differential and by Δ : C∗ (X) → C∗ (X) ⊗ C∗ (X) the diagonal. We also write respectively d, ∪ for the differential and the cup-product on C ∗ (X) (induced by Δ).

It follows that the coalgebra map F : A⊥ (S) → A⊥ (R) is also a map of algebras (for the shuffle product). 19, A⊥ R (M ) is a differential module over the shuffle algebra A⊥ (R). Thus A⊥ S (M ) is a differential module over the shuffle algebra A⊥ (S). Hence M is a C∞ -bimodule over S. 10. Let (R, D) be an A∞ -algebra, (M, DM ) an R-bimodule and (S, B) be a A∞ -algebras. • If there is an A∞ -map F : (S, B) → (R, D), then there is a linear map F ∗ : HH ∗ (R, M ) → HH ∗ (S, M ) which is an isomorphism if F1 : (S, dS ) → (R, dR ) is a quasi-isomorphism.

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