Homotopy theory and models : based on lectures held at a DMV Seminar in Blaubeuren by H.J. Baues, S. Halperin, and J.-M. Lemaire

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.