Lower K- and L-theory. (English) Zbl 0752.57002

Ranicki, Andrew

London Mathematical Society Lecture Note Series. 178. Cambridge: Cambridge University Press. 174 p. (1992).

The main theme of the book is nicely expressed in the Introduction: “The algebraic \(K\)-groups \(K_ *(A)\) and the algebraic \(L\)-groups \(L_ *(A)\) are the obstruction groups to the existence and uniqueness of geometric structures in homotopy theory, via Whitehead torsion and the Wall finiteness and surgery obstructions. In the topological applications the ground ring \(A\) is the group ring \(\mathbb{Z}[\pi]\) of the fundamental group \(\pi\). For \(K\)-theory a geometric structure is a finite CW complex, while for \(L\)-theory it is a compact manifold. The lower \(K\)- and \(L\)-groups are the obstruction groups to imposing such a geometric structure after stabilization by forming a product with the \(i\)-fold torus \(T^ i=S^ 1\times S^ 1\times \dots\times S^ 1\), arising algebraically as the codimension \(i\) summands of the \(K\)- and \(L\)-groups of the \(i\)-fold Laurent polynomial extension of \(A\) \[ A[\pi_ 1(T^ i)]=A[z_ 1,(z_ 1)^{-1},z_ 2,(z_ 2)^{-1},\ldots,z_ i,(z_ i)^{-1}]. \] The object of this text is to provide a unified algebraic framework for lower \(K\)- and \(L\)-theory using chain complexes, leading to new computations in algebra and to further applications in topology.”
In addition to providing a systematic exposition (from a new point of view) of topics from the 1960’s and 1970’s, the book incorporates an algebraic framework recently developed for “controlled” or “bounded” topology. The extra generality needed for the geometric applications is obtained by working throughout with chain complexes over an additive category. This leads for example to a new proof using polynomial extensions of the fundamental theorem of Pedersen and Weibel expressing the lower \(K\)-theory \(K_{-i}(A)\) of an additive category as \(K_ 1\) of a suitable bounded category \(C_{i+1}(A)\).
The book is written clearly and carefully. It will be suitable both as a reference and foundation for further research work in bounded topology, and as a text accessible to beginning graduate students in this area.

Reviewer: I.Hambleton (Hamilton / Ontario)

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R67	Surgery obstructions, Wall groups
19Jxx	Obstructions from topology
19-02	Research exposition (monographs, survey articles) pertaining to \(K\)-theory

Keywords:

group ring of the fundamental group; algebraic \(K\)-groups; algebraic \(L\)- groups; existence and uniqueness of geometric structures; Whitehead torsion; Wall finiteness; surgery obstructions; product with the \(i\)-fold torus; chain complexes over an additive category; bounded category; controlled topology; bounded topology