Stable Modules and the D(2)-Problem
The D(n)-problem can be formulated for every integer n ≥ 1. Its solution for every n≠2 is known, which explains the attention the author pays to the D(2)-problem: “Let X be a finite connected cell complex of geometrical dimension 3, and suppose that H3(X;Z)=H3(X;B)=0 for all coefficient systems B on X, where X denotes the universal covering of X. Is it true that X is homotopy equivalent to a finite complex of dimension 2?” Let us denote G to be the fundamental group of X. It is necessary to mention that the D(2)-problem is closely connected with the realization problem. The author has proved: The D(2)-property holds for a finite group G if and only if each algebraic 2-complex over G is geometrically realizable. It is not appropriate to mention too many details here, but we can say that the author successfully attacks the realization problem and obtains in many cases the solution of the D(2)-problem. He uses two important tools, namely the Yoneda’s theory of module extensions and the Swan-Jacobinski cancellation theory. The author prepares the reader for the main part of the book. In chapters 1-3 we find necessary results from the module theory and the representation theory of finite groups. Chapters 4-7 deal with the group cohomology and the module extension theory. The author explains here Yoneda’s theory and specializes it to modules over group rings. He makes the reader familiar with the k-invariant method and with the structure of groups of periodic cohomology. Chapters 8-11 are devoted to algebraic and 2-dimensional geometric homotopy theory, to the D(2)-problems and to the realization problem. Generally we can say that the book is very well and attractively written and will be indispensable for the specialists in this field. It can also be strongly recommended to postgraduate students. They will be able to orientate themselves in the subject and will find hints for further reading and research.