Formality of Cyclic Chains

TL;DR

This paper proves Tsygan's conjecture by establishing an L-infinity-quasiisomorphism between the cyclic chain complex of smooth functions and differential forms on a manifold, extending Shoikhet's morphism.

Contribution

It demonstrates the existence of a specific L-infinity-quasiisomorphism that confirms Tsygan's conjecture, connecting cyclic chains and differential forms.

Findings

Confirmed Tsygan's conjecture for cyclic chains

Constructed an explicit L-infinity-quasiisomorphism

Extended Shoikhet's morphism to solve the conjecture

Abstract

We prove a conjecture raised by Tsygan, namely the existence of an L-infinity-quasiisomorphism of L-infinity-modules between the cyclic chain complex of smooth functions on a manifold and the differential forms on that manifold. Concretely, we prove that the obvious u-linear extension of Shoikhet's morphism of Hochschild chains solves Tsygan's conjecture.