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Forum Mathematicum

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Duality in Waldhausen Categories

Citation Information: Forum Mathematicum. Volume 10, Issue 5, Pages 533–603, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/form.10.5.533, March 2008

Publication History

Published Online:


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.

Citing Articles

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Filipp Levikov
Journal of Homotopy and Related Structures, 2015
Emanuele Dotto
Journal of the Institute of Mathematics of Jussieu, 2015, Page 1
Michael Weiss
Journal of Pure and Applied Algebra, 1999, Volume 138, Number 2, Page 185
Thomas Hüttemann, John R. Klein, Wolrad Vogell, Friedhelm Waldhausen, and Bruce Williams
Journal of Pure and Applied Algebra, 2002, Volume 167, Number 1, Page 53

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