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.
© Walter de Gruyter
