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Forum Mathematicum

Managing Editor: Brüdern, Jörg

Editorial Board Member: Bruinier, Jan Hendrik / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Gallavotti, Giovanni / Garnier, Josselin / Neeb, Karl-Hermann / Noguchi, Junjiro / Ranicki, Andrew / Segal, Dan / Shahidi, Freydoon / Shigekawa, Ichiro / Sogge, Christopher D. / Strambach, Karl

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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

Received:
1996-10-09
Revised:
1997-07-25
Published Online:
2008-03-11

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.

Citing Articles

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[1]
Filipp Levikov
Journal of Homotopy and Related Structures, 2015
[2]
Emanuele Dotto
Journal of the Institute of Mathematics of Jussieu, 2015, Page 1
[3]
Michael Weiss
Journal of Pure and Applied Algebra, 1999, Volume 138, Number 2, Page 185
[4]
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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