In keeping with the general aim of the "D.M.V.-Seminar" series, this book is princi pally a report on a group of lectures held at Blaubeuren by Professors H. J. Baues, S. Halperin and J.-M. Lemaire, from October 30 to November 7, 1988. These lec tures were devoted to providing an introduction to the theory of models in algebraic homotopy. The three lecturers acted in concert to produce a coherent exposition of the theory. Commencing from a common starting point, each of them then proceeded naturally to his own subject of research. The reader who is already familiar with their scientific work will certainly give the lecturers their due. Having been asked by the speakers to take on the responsibility of writing down the notes, it seemed to me that the material elucidated in the short span of fifteen hours was too dense to appear, unedited, in book form. Some amplification was necessary. Of course I submitted to them the final version of this book, which received their approval. I thank them for this token of confidence. I am also grateful to all three for their help and advice in writing this book. I am particularly indebted to J.-M. Lemaire who was indeed very often consulted. For basic notions (in particular those concerning homotopy groups, CW complexes, (co)homology and homological algebra) the reader is advised to refer to the fundamental books written by E. H. Spanier , R. M. Switzer  and G. Whitehead .
Table of Contents
1: Basic Homotopy Theory.- §1. Homotopy.- §2. Cofibrations and fibrations.- 2: Homology and Homotopy Decomposition of Simply Connected Spaces.- §1. Eckmann-Hilton duality.- §2. Homology and homotopy decompositions.- §3. Application: Classification of 2-stage spaces.- 3: Cofibration Categories.- §1. Basic definitions.- §2. Homotopy in a cofibration category.- §3. Properties of cofibration categories.- §4. Properties of cofibrant models.- §5. The homotopy category as a localization.- 4: Algebraic Examples of Cofibration Categories.- §1. The category CDA.- §2. The category Chain+.- §3. The category DA.- §4. The category DL.- 5: The Rational Homotopy Category of Simply Connected Spaces.- §1. The category of rational spaces.- §2. Quillen’s model category.- §3. Sullivan’s model theory.- §4. Some easy applications.- Appendix: Relations between the Various Models of a Space.- A.1. A functor between DL and CDA.- A.2. Models over/p?.- A.3. Sullivan Models.- 6: Attaching Cells in Topology and Algebra.- §1. Algebraic models of spaces with a cell attached.- §2. Inertia.- 7: Elliptic Spaces.- §1. Finiteness of the formal dimension.- §2. Elliptic models.- §3. Some equalities and inequalities.- §4. Topological interpretation.- 8: Non Elliptic Finite C.W.-Complexes.- §1. Homotopy invariants of spaces.- §2. Sullivan models and the (algebraic) Lusternik-Schnirelmann category.- §3. Lie algebras of finite depth.- §4. The mapping theorem.- §5. Proof of Theorem 0.1.- 9: Towards Integral Algebraic Models of Homotopy Types.- §1. Introduction and general problem.- §2. Algebraic description of the integral homotopy types in dimension 4.- §3. Algebraic description of the integral homotopy types in dimension N.