Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

Schwänzl, R.

FB Mathematik, Universität Osnabrück.

Vogt, R. M.

FB Mathematik, Universität Osnabrück.

Waldhausen, F.

FB Mathematik, Universität Bielefeld.

An un-delooped version of algebraic K-theory1992In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 79, no 3, p. 255-270Article in journal (Refereed)

Abstract [en]

Problems working with the Segal operations in algebraic K-theory of spaces-constructed by F. Waldhausen (1982)-arose from the absence of a nice groupcompletion on the category level. H. Grayson and D. Gillet (1987) introduced a combinatorial model G. for K-theory of exact categories. For dealing with K-theory of spaces we need an extension wG. of their result to the context of categories with cofibrations and weak equivalences. Our main result is that in the presence of a suspension functor-as in the case of retractive spaces-the wG. construction on the category of prespectra is an un-delooped version of the K-theory of the original category. In a sequel to this paper we show that Grayson's formula (1988) for Segal operations works as intended.

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.

Schwänzl, Roland

Fachbereich Mathematik/Informatik, Universität Osnabrück.

Operations in A-theory2002In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 174, no 3, p. 263-301Article in journal (Refereed)

Abstract [en]

A construction for Segal operations for K-theory of categories with cofibrations, weak equivalences and a biexact pairing is given and coherence properties of the operations are studied. The model for K-theory, which is used, allows coherence to be studied by means of (symmetric) monoidal functors. In the case of Waldhausen A-theory it is shown how to recover the operations used in Waldhausen (Lecture Notes in Mathematics, Vol. 967, Springer, Berlin, 1982, pp. 390-409) for the A-theory Kahn-Priddy theorem. The total Segal operation for A-theory, which assembles exterior power operations, is shown to carry a natural infinite loop map structure. The basic input is the un-delooped model for K-theory, which has been developed from a construction by Grayson and Gillet for exact categories in Gunnarsson et al. (J. Pure Appl. Algebra 79 (1992) 255), and Grayson's setup for operations in Grayson (K-theory (1989) 247). The relevant material from these sources is recollected followed by observations on equivariant objects and pairings. Grayson's conditions are then translated to the context of categories with cofibrations and weak equivalences. The power operations are shown to be well behaved w.r.t. suspension and are extended to algebraic K-theory of spaces. Staying close with the philosophy of Waldhausen (1982) Waldhausen's maps are found. The Kahn-Priddy theorem follows from splitting the "free part" off the equivariant theory. The treatment of coherence of the total operation in A-theory involves results from Laplaza (Lecture Notes in Mathematics, Vol. 281, Springer, Berlin, 1972, pp. 29-65) and restriction to spherical objects in the source of the operation.

We extend earlier work of Waldhausen which defines operations on the algebraic K-theory of the one-point space. For a connected simplicial abelian group X and symmetric groups Sigma(n) we define operations theta(n) : A(X) -> A(X x B Sigma(n)) in the algebraic K-theory of spaces. We show that our operations can be given the structure of E-infinity-maps. Let phi(n) : A(X x B Sigma(n)) -> A(X x E Sigma(n)) similar or equal to A(X) be the Sigma(n)-transfer. We also develop an inductive procedure to compute the compositions phi(n) circle theta(n) and outline some applications.