Formality of Cyclic Chains

We prove the cyclic formality conjecture for chains, raised by Tsygan [“Formality conjectures for chains.” Differential Topology, Infinite-dimensional Lie Algebras, and Applications 261–74. American Mathematical Society Translation Series 2, 194. Providence, RI: American Mathematical Society, 1999.]. It states the existence of an L∞-quasi-isomorphism of L∞-modules between the cyclic chain complex of smooth functions on a manifold and the differential forms on that manifold. Concretely, we prove that the u-linear extension of Shoikhet’s morphism of Hochschild chains [Shoikhet, B. “A proof of the Tsygan formality conjecture for chains.” Advances in Mathematics 179, no. 1 (2003): 7–37.] solves Tsygan’s conjecture.