Introduction

Since their introduction 20 years ago, the evaluation of Quillen's higher algebraic K-groups of rings has remained a difficult problem in homotopy theory. One does have Quillen's explicit evaluation of the algebraic K-theory of finite fields and Suslin's theorem about the K-theory of algebraically closed fields. In addition, Quillen's original paper gave a number of useful formal properties of the algebraic if-groups: localization sequences, homotopy property for the K-theory of polynomial rings, reduction by resolution, and reduction by “devissage”. These tools provide, for a large class of commutative rings, a fairly effective procedure which reduces the description of the K-theory of the commutative ring to that of fields. The K-theory of a general field remains an intractable problem due to the lack of a good Galois descent spectral sequence, although by work of Thomason one can understand its so-called Bott-periodic localization. In the case of non-commutative rings, the formal properties of Quillen are not nearly as successful as in the commutative case.

A central theme in the subject has been the relationship with properties of manifolds which are not homotopy invariant; here the ring in question is usually the group ring ℤ[Γ], with Γ the fundamental group of a manifold. Examples include Wall's finiteness obstruction, the s-cobordism theorem of Barden–Mazur–Stallings, Hatcher-Wagoner's work on the connection between K2 and the homotopy type of the pseudoisotopy space, and finally Waldhausen's description of the pseudoisotopy space in terms of the K-theory of “rings up to homotopy“. Since the rings ℤ[Γ] are typically not commutative, the reduction methods of are not adequate for the description of the if-theory of ℤ[Γ].