Memoirs of the American Mathematical Society 2000; 149 pp; softcover Volume: 143 ISBN10: 0821819208 ISBN13: 9780821819203 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/143/682
 Abstract. The main goal of this paper is to prove the following conjecture of Baues and Lemaire: the differential graded Lie algebra associated with the Sullivan model of a space is homotopy equivalent to its Quillen model. In addition we show the same for the cellular Lie algebra model which we build from the simplicial analog of the classical AdamsHilton model. It turns out that this cellular Lie algebra model is one link in a chain of models connecting the models of Quillen and Sullivan. The key result which makes all this possible is Anick's correspondence between differential graded Lie algebras and Hopf algebras up to homotopy. In addition we show that the Quillen model is a rational homotopical equivalence, and we conclude the same for the other models using our main result. The construction of the three models is given in detail. The background from homotopy theory, differential algebra, and algebra is presented in great generality. Readership Graduate students and research mathematicians interested in algebraic topology. Table of Contents  Introduction
 Homotopy theory
 Differential algebra
 Complete algebra
 Three models for spaces
 Notations
 Bibliography
