Realizing operadic plus-constructions as nullifications

In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded operads to the framework of cofibrant operads over an arbitrary ring (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. We construct a universal H*Q-acyclic algebra U and we define A→A+ as the U-nullification of the algebra A. This map induces an isomorphism in Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plus-constructions of gl(R) in the categories of homotopy Lie and homotopy Leibniz algebras. This gives rise to two new homology theories for associative algebras, namely homotopy cyclic and homotopy Hochschild homologies. Over the rationals these theories coincide with the classical cyclic and Hochschild homologies. © 2004 Kluwer Academic Publishers.