**Memoirs of the American Mathematical Society**

2000;
96 pp;
Softcover

MSC: Primary 19; 57; 55;

Print ISBN: 978-0-8218-2069-8

Product Code: MEMO/145/691

List Price: $51.00

AMS Member Price: $30.60

MAA Member Price: $45.90

**Electronic ISBN: 978-1-4704-0282-2
Product Code: MEMO/145/691.E**

List Price: $51.00

AMS Member Price: $30.60

MAA Member Price: $45.90

# Control and Relaxation over the Circle

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*Bruce Hughes; Stratos Prassidis*

We formulate and prove a geometric version of the Fundamental Theorem of Algebraic K-Theory which relates the K-theory of the Laurent polynomial extension of a ring to the K-theory of the ring. The geometric version relates the higher simple homotopy theory of the product of a finite complex and a circle with that of the complex. By using methods of controlled topology, we also obtain a geometric version of the Fundamental Theorem of Lower Algebraic K-Theory. The main new innovation is a geometrically defined Nil space.

#### Readership

Graduate students and research mathematicians interested in algebraic topology and algebraic \(K\)-theory.

#### Table of Contents

# Table of Contents

## Control and Relaxation over the Circle

- Contents vii8 free
- Chapter 1. Introduction and Statement of Results 112 free
- Chapter 2. Moduli Spaces of Manifolds and Maps 920
- Chapter 3. Wrapping–up and Unwrapping as Simplcial Maps 1829
- Chapter 4. Relaxation as a Simplcial Map 3041
- Chapter 5. The Whitehead Spaces 3243
- 5.1. The Controlled Whitehead Space 3243
- 5.2. Abelian Monoid–like Structures 3344
- 5.3. The Various SimpHcial Maps 3546
- 5.4. The Forget Control Map 3647
- 5.5. The Unwrapping Map 3748
- 5.6. The Wrapping Up Map 3748
- 5.7. Classifying Space Interpretation 3849
- 5.8. Delooping the Whitehead Space 4051
- 5.9. The Relaxation Map 4051
- 5.10. The Boundedness Condition on the Infinite Cyclic Cover 4152
- 5.11. Bounded Whitehead and Pseudoisotopy Spaces 4253

- Chapter 6. Torsion and a Higher Sum Theorem 4455
- Chapter 7. Nil as a Geometrically Defined Simplcial Set 4859
- Chapter 8. Transfers 6374
- Chapter 9. Completion of the Proof 7283
- Chapter 10. Comparison with the Lower Algebraic Nil Groups 8697
- Appendix A. Controlled Homotopies on Mapping Tori 92103
- Bibliography 95106