McClure-Smith cosimplicial machinery and the cacti operad (Q492231)

In [Algebr. Geom. Topol. 9, No. 1, 237--264 (2009; Zbl 1159.18304)], \textit{P. Salvatore} proved a cyclic version of the topological Deligne conjecture, loosely following the proof of [\textit{J. McClure} and \textit{J. Smith}, Am. J. Math. 126, No. 5, 1109--1153 (2004; Zbl 1064.55008)] in the non-cyclic setting. Given a topological (cyclic) operad \(O\) with multiplication, there is an associated cosimplicial (resp. cocyclic) space \(O^\bullet\). The topological Deligne conjecture says that if \(O\) is an operad with multiplication, then the totalization \(\text{Tot}(O^\bullet)\) admits an action of an operad equivalent to the little 2-disks operad. The cyclic version says that if you begin with a cyclic operad with multiplication, then the resulting space has an action of an operad equivalent to the framed little 2-disks operad. A key step in Salvatore's work is to give a geometric description (here called \(MS\)) of the McClure-Smith operad \(\mathcal D_2\), in a way that can be modified for the cyclic case. Salvatore shows both that the operad \(MS\) is isomorphic to \(\mathcal D_2\) and that \(MS\) acts on \(\text{Tot}(O^\bullet)\) (thus reproving the non-cyclic topological Deligne conjecture). While Salvatore shows that the two operads are isomorphic, he does not show that his action and the McClure-Smith action on \(\text{Tot}(O^\bullet)\) are the same. That is the topic of the present article. The main theorem of this paper that the two actions are compatible with the isomorphism \(MS \cong \mathcal D_2\). All of the relevant definitions are included, and several proofs are reproduced in order to provide explicit formulas for the actions in question. In particular, Salvatore's action is rewritten without using trees. The author supplies many examples to illustrate the concepts contained within.