Spaces of algebra structures and cohomology of operads

The aim of this paper is two-fold. First, we compare two notions of a "space" of algebra structures over an operad A: (1) the classification space, which is the nerve of the category of weak equivalences of A-algebras, and (2) the moduli space $A\{X\},$ which is the space of maps from A to the endomorphism operad of an object X.

We show that under certain hypotheses the moduli space of A-algebra structures on X is the homotopy fiber of a map between classification spaces.

Second, we address the problem of computing the homotopy type of the moduli space $A\{X\}.$ Because this is a mapping space, there is a spectral sequence computing its homotopy groups with $E\sb2$-term described by the Quillen cohomology of the operad A in coefficients which depend on X. We show that this Quillen cohomology is essentially the same, up to a dimension shift, as the Hochschild cohomology of A, and that the Hochschild cohomology may be computed using a "bar construction". (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)