arXiv:math/0306406v1 [math.AT] 28 Jun 2003 ANDR ´ E-QUILLEN COHOMOLOGY AND RATIONAL HOMOTOPY OF FUNCTION SPACES J.BLOCK & A. LAZAREV Abstract. We develop a simple theory of Andr´ e-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent CW -complexes. This puts certain results of Sullivan in a more conceptual framework. 1. Introduction Andr´ e-Quillen cohomology is a cohomology theory for commutative algebras originally intro- duced in [1] and [15]. It was subsequently generalized to cover simplicial algebras over operads, [8], differential graded E ∞ -algebras, [14] and commutative S -algebras, [3]. One of the purposes of the present paper is to give a simple and direct treatment of the Andr´ e- Quillen cohomology in the category of commutative differential graded algebras (dga’s for short) over a field of characteristic zero. This is done in Section 2. Our initial definition of Andr´ e-Quillen cohomology of a dga A with coefficients in the differential graded (dg) module M over A is via an explicit cochain complex C ∗ AQ (A, M ) similar to the one introduced in [9]. We then produce various equivalent characterizations of Andr´ e-Quillen cohomology, introduce the Gerstenhaber bracket on C ∗ AQ (A, A) and show its homotopy invariance. In this connection we mention the recent paper [6] where analogues of some of our results were proved in the context of Hochschild cohomology. In Section 3 we apply the developed techniques to computing the homotopy groups of function spaces. (We are dealing with unpointed spaces, however our machinery could be easily adapted to the pointed situation as well.) In particular we are concerned with the group hAut(X) of homotopy classes of homotopy self-equivalences of a nilpotent CW -complex X. A well-known theorem of Sullivan [18] and Wilkerson [19] asserts that under suitable finiteness assumptions hAut(X) is an arithmetic group, that is, commensurable to the group of integer points of some algebraic group over Q. An important step is to show that the group hAut(X Q ) is isomorphic to the group of Q-points of an algebraic group. Here X Q denotes the rationalization of the space X, i.e. its localization with respect to the homology theory H ∗ (−, Q). We reprove this result and identify the Lie algebra of this algebraic group. It turns out to be isomorphic to H 0 AQ (A ∗ (X),A ∗ (X)), the zeroth Andr´ e-Quillen cohomology of the Sullivan-de Rham algebra of X with coefficients in itself. The Lie bracket corresponds to the Gerstenhaber bracket on H ∗ AQ (A ∗ (X),A ∗ (X)). We also consider the question of computing the higher homotopy groups of a function space F (X, Y ) for two rational spaces X and Y . The answer is again formulated in terms of Andr´ e- Quillen cohomology associated to the Sullivan-de Rham models of X and Y . This result was hinted at in [13]. 2000 Mathematics Subject Classification. 55P62, 13D03, 16E45. Key words and phrases. Closed model category, differential graded algebra, derivation, function space. J.B. was partially supported by the NSF grant DMS-0204558. A.L. was partially supported by the EPSRC grant No. GR/R84276/01. 1