Operations in A-theory

AbstractA 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

The Seifert-van Kampen theorem and generalized free products of S-algebras

In a Seifert-van Kampen situation a path-connected space Z may be written as the union of two open path-connected subspaces X and Y along a common path-connected intersection W . The fundamental group of Z is isomorphic to the colimit of the diagram of fundamental groups of the three subspaces. In case the maps of fundamental groups are all injective, the fundamental group of Z is a classical free product with amalgamation, and the integral group ring of the fundamental group of Z is also a free product with amalgamation in the category of rings. In this case relations among the K-theories of the group rings have been studied. Here we describe a generalization and stablization of this algebraic fact, where there are no injectivity hypotheses on the fundamental groups and where we work in the category of S-algebras. Some of the methods we use are classical and familiar, but the passage to S-algebras blends classical and new techniques. Our most important application is a description of the algebraic K-theory of the space Z in terms of the algebraic K-theories of the other three spaces and the algebraic K-theory of spaces Nil-term

The approximation theorem and the K-theory of generalized free products

this paper to show how to derive the main theorems of [8] as applications of results and methods of [9]. Using different methods, Pierre Vogel [7] has also reconsidered results of [8]. We first develop language so that essentially the same fibration used in [8] may be derived from a general fibration theorem (Theorem 1.6.4 of [9, page 354]) developed as part of the overall approach to abstract algebraic K-theory described in [9]. This is Proposition 2.1 below. Our main contribution is the description of how the approximation theorem (Theorem 1.6.7 of [9, page 354]) may then be used to interpret terms in the fibration. These results are Theorems 2 and 3 stated at the end of the section. We are able to replace the technical manouvering required in the original proofs with arguments that follow a standard pattern and are more conceptual. The paper also provides an introduction to a few of the ideas we will use in [5], where we generalize the situation to the case of simplicial rings. One of the goals of [5] is to set up a framework which will also allow us to handle decomposition problems in th

MacLane homology and topological Hochschild homology

Fiedorowic Z, Pirashvili T, Schwänzl R, Vogt RM, Waldhausen F. MacLane homology and topological Hochschild homology. Mathematische Annalen. 1995;303(1):149-164

Topological Hochschild Homology

Schwänzl R, Vogt RM, Waldhausen F. Topological Hochschild Homology. Journal of the London Mathematical Society (2). 2000;62(2):345-356