$K$-théorie algébrique et représentations de groupes

Abstract

This is a fundamental paper on higher algebraic K-theory, both for its new ideas and results, mainly concerning multiplicative structures, but also for the coherent and careful presentation it furnishes of the basic constructions, notably Quillen's "plus construction''. The latter is a homology isomorphism i:X→X+ attached to a cell complex X and normal perfect subgroup N of π1(X) such that π1(i) is surjective with kernel N. For a ring A Quillen takes X=BGL(A), N=N=N=N=N=N=N=\tex, and defines Kn(A)=πn(BGL(A)+) for n≥1. The author gives the details of this construction, and establishes the isomorphisms K2(A)=H2( (of Kervaire and Steinberg) and K3(A)=H3(St(A),Z) (of Gersten). He further shows that Kn(SA)=Kn−1(A), where SA denotes the (Karoubi-Villamayor) suspension of A. This leads to the Gersten-Wagoner spectrum (K−−−−A)n=Ω−n(K0(A)×BGL(A)+) if n<0, (K−−−−A)n=K0(SnA)×BGL(SnA)+ if n≥0, with nth homotopy Kn(A). In Chapter II the author introduces multiplicative structures in K-theory, with all of the desired formal properties. These involve products ★:Kp(A)×Kq(A′)→Kp+q(A⊗A′), and a structure of a graded anti-commutative ring on K∗(A) when A is commutative. The product for p=q=1 is shown to coincide (up to sign) with Milnor's, and a very interesting calculation of it in homological terms is given for p=1, q=2. There is a canonical homorphism Z[t,t−1]→SZ, sending {t}∈K1(A[t,t−1]) to an element τ∈K1(SZ). The author shows that the isomorphism Kn(A)→Kn+1(SA) is x↦x★τ; this confirms a conjecture of Gersten. It also furnishes a split injection Kn(A)→Kn+1(A[t,t−1]), x↦x★{t}, which implies a decomposition Kn+1(A[t,t−1])=Kn+1(A)⊕Kn(A)⊕?. The third chapter develops analogous multiplicative structures in Hermitian K-theory. These furnish, in particular, the final ingredient in the contingent periodicity theorems of Karoubi. Chapter IV is concerned with the groups Kn(ZG), and the groups Whn(G) introduced by Wagoner, Hatcher, and Volodin. Using results of Waldhausen the author shows that Wh2(G)=0 for a class of groups containing torsion free groups with one relation, fundamental groups of submanifolds of S3, and stable under amalgamated free products and HNN constructions. The final Chapter V relates the groups K∗(A[G]), h∗(BG,K−−−−A), and KGn(A)=KG0(SnA), where G is a group, A a ring, and KG0(A) denotes the Grothendieck group of representations of G in the category P(A) of finitely generated projective A-modules. The latter groups have quadratic analogues obtained by replacing P(A) with one of the categories of Hermitian K-theory. In this setting the author formulates a conjecture, generalizing the Hirzebruch signature formula, whose validity would partially confirm Novikov's conjecture on the homotopy invariance of his higher signatures.