Memoirs of the American Mathematical Society 2000; 96 pp; softcover Volume: 145 ISBN10: 0821820699 ISBN13: 9780821820698 List Price: US$51 Individual Members: US$30.60 Institutional Members: US$40.80 Order Code: MEMO/145/691
 We formulate and prove a geometric version of the Fundamental Theorem of Algebraic KTheory which relates the Ktheory of the Laurent polynomial extension of a ring to the Ktheory 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 KTheory. 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  Introduction and statement of results
 Moduli spaces of manifolds and maps
 Wrappingup and unwrapping as simplicial maps
 Relaxation as a simplicial map
 The Whitehead spaces
 Torsion and a higher sum theorem
 Nil as a geometrically defined simplicial set
 Transfers
 Completion of the proof
 Comparison with the lower algebraic nil groups
 Appendix A. Controlled homotopies on mapping tori
 Bibliography
