RATIONAL HOMOTOPY Contents Introduction iv §1. The Simplicial Algebra v 1 §2. The Polynomial De Rham Theory 7 §3. Multiplicative Structure and the Eilenberg-Moore Theorem 12 §4. A Quillen Homotopy Theory for DG Algebras 18 §5. Function Spaces for DG Algebras 24 §6. The Homotopy Relation for DG Algebras 31 §7. Minimal DG Algebras 37 §8, The De Rham Functors and Their Adjoints 43 §9. The Sullivan-De Rham Equivalence Theorem 50 §10. Proof of the Sullivan-De Rham Equivalence Theorem 55 §11. The Sullivan-De Rham Localization and Homotopy Theorems 63 §12. Extensions of the Sullivan-De Rham Theorems 69 Appendix §13. Polynomial De Rham Theory for Simplicial Complexes 80 §14. Another Proof of 2.2 84 §15. Sullivan Homotopies are Homotopies in DASH 88 §16. Minimal Algebras for Spaces with "Nice" Cohomology 92 References 94 iii

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