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Control and Relaxation over the Circle
Bruce Hughes and Stratos Prassidis, Vanderbilt University, Nashville, TN

Memoirs of the American Mathematical Society
2000; 96 pp; softcover
Volume: 145
ISBN-10: 0-8218-2069-9
ISBN-13: 978-0-8218-2069-8
List Price: US$51
Individual Members: US$30.60
Institutional Members: US$40.80
Order Code: MEMO/145/691
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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.


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
  • Wrapping-up 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
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