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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 10, Issue 5 (Sep 1998)

Issues

Duality in Waldhausen Categories

Michael Weiss
  • Dept. of Mathematics, University of Notre Dame, Notre Dame in 46556, USA.
  • Email:
/ Bruce Williams
  • Dept. of Mathematics, University of Notre Dame, Notre Dame in 46556, USA.
  • Email:
Published Online: 2008-03-11 | DOI: https://doi.org/10.1515/form.10.5.533

Abstract

We develop a theory of Spanier-Whitehead duality in categories with cofibrations and weak equivalences (Waldhausen categories, for short). This includes L-theory, the involution on K-theory introduced by [Vogell, W.: The involution in the algebraic K-theory of spaces. Proc. of 1983 Rutgers Conf. on Alg. Topology. Springer Lect. Notes in Math. 1126, pp. 277–317] in a special case, and a map Ξ relating L-theory to the Tate spectrum of ℤ/2 acting on K-theory. The map Ξ is a distillation of the long exact Rothenberg sequences [Shaneson, J.: Wall's surgery obstruction groups for G × ℤ. Ann. of Math. 90 (1969), 296–334], [Ranicki, A.: Algebraic L-theory I. Foundations. Proc. Lond. Math. Soc. 27 (1973), 101–125], [Ranicki, A.: Exact sequences in the algebraic theory of surgery. Mathematical Notes, Princeton Univ. Press, Princeton, New Jersey 1981], including analogs involving higher K-groups. It goes back to [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K-theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107] in special cases. Among the examples covered here, but not in [Weiss, M., Williams, B.: Automorphisms of manifolds and algebraic K-theory, Part II. J. Pure and Appl. Algebra 62 (1988), 47–107], are categories of retractive spaces where the notion of weak equivalence involves control.

About the article


Received: 1996-10-09

Revised: 1997-07-25

Published Online: 2008-03-11

Published in Print: 1998-09-01


Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/form.10.5.533.

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