$E_{\infty}$-structure for $Q_*(R)$

Abstract.

MacLane homology of a ring R is the Hochschild homology of the so called cubical construction $Q_*(R)$, which is a chain-algebra. If we take a commutative ring R, the Dixmier product on $Q_*(R)$ is no longer commutative. The main result of this paper is that it is commutative up to higher homotopies, i.e. that it is an $E_{\infty}$-algebra. The $E_{\infty}$-operad which acts on $Q_*(R)$ is constructed by using the analogue of the $Q_*$-complex in the context of finite sets. For the precise notation of an operad action on these complexes the definition of an $E_{\infty}$-monoidal functor is introduced.