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For P ⊂ 3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP. Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(3), with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such groups ΓP in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate the K−1 and Wh of the groups Dn and Dn × 3, and we also summarize what is known about the 0.
This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.
Projective modules are of some interest as natural generalizations of free modules, but generalizations become significant when they connect objects already under study. In Chapter 7 we find that the integral domains R whose ideals are f.g. projective R-modules are the rings of primary interest in algebraic number theory. And in Chapter 8 the rings, all of whose modules are projective, turn out to be the gateway to the matrix representations of finite groups.
In connection with these two types of ring are two abelian groups that are precursors of K0(R). Pinpointing the historical origin of K0 is like locating the source of a great river; there are many tributaries along the way, and the identity of the true source can be a subjective judgement. I propose that the source of K0 is the ideal class group, described in quite modern terms by Dedekind in 1893. We consider the class group in Chapter 7. The origin of K0 of noncommutative rings is perhaps the ring of virtual characters of finite groups. This subject has an equally long history, going back to Frobenius in the 1890s (with earlier roots due to Dirichlet, Dedekind, and Kronecker), but it recognizably enters the stream of K-theory in the 1960 paper of Swan, “Induced Representations and Projective Modules.” We develop character theory in Chapter 8.