Contents Chapter 1. Introduction and Statement of Results 1 1.1. Introduction 1 1.2. Statement of Results 3 Chapter 2. Moduli Spaces of Manifolds and Maps 9 2.1. The SimpHcial Set of Manifolds 9 2.2. The SimpHcial Set of Manifold Approximate Fibrations 10 2.3. The SimpHcial Set of Manifold Bands 11 2.4. The Forget Control Map 16 Chapter 3. Wrapping-up and Unwrapping as SimpHcial Maps 18 3.1. Unwrapping 18 3.2. Wrapping Up 21 3.3. Definition of w 23 3.4. Classifying Space Interpretation 24 3.5. Independence of Covering Isotopy 28 Chapter 4. Relaxation as a SimpHcial Map 30 Chapter 5. The Whitehead Spaces 32 5.1. The Controlled Whitehead Space 32 5.2. Abelian Monoid-like Structures 33 5.3. The Various SimpHcial Maps 35 5.4. The Forget Control Map 36 5.5. The Unwrapping Map 37 5.6. The Wrapping Up Map 37 5.7. Classifying Space Interpretation 38 5.8. Delooping the Whitehead Space 40 5.9. The Relaxation Map 40 5.10. The Boundedness Condition on the Infinite Cyclic Cover 41 5.11. Bounded Whitehead and Pseudoisotopy Spaces 42 Chapter 6. Torsion and a Higher Sum Theorem 44 Chapter 7. Nil as a Geometrically Defined SimpHcial Set 48 7.1. Preliminaries 48 7.2. The SimpHcial Set of Nil Simplices 49 7.3. The SimpHcial Maps : WhiXx^xS1 -+ Rn) - J\fil(XxRn - E n ) 54 7.4. The Homomorphism P* is a Split Epimorphism 58 7.5. Homotopy Inverses for Afil(XxRn -+ Rn) 61
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