---
abstract: 'In this note, we show that the normalized Hochschild co–chains of an associative algebra with a non–degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber’s bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the little discs operad in terms of spineless cacti with natural chain models for $(S^1)^{\times n}$ adapted to cacti.'
address: 'University of Connecticut, Department of Mathematics, Storrs, CT 06269'
author:
- 'Ralph M. Kaufmann'
title: 'A proof of a cyclic version of Deligne’s conjecture via Cacti'
---

Introduction {#introduction .unnumbered}
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In this note, we expand our chain model of the little discs operad which we gave in terms of spineless cacti to a chain model for the framed little discs operad in terms of normalized cacti. Extending the philosophy of [@del], we then show that the 