Metacyclic Groups and the D(2) Problem

The D(2) problem is a fundamental problem in low dimensional topology. In broad terms, it asks when a three-dimensional space can be continuously deformed into a two-dimensional space without changing the essential algebraic properties of the spaces involved.The problem is parametrized by the fundamental group of the spaces involved; that is, each group G has its own D(2) problem whose difficulty varies considerably with the individual nature of G.This book solves the D(2) problem for a large, possibly infinite, number of finite metacyclic groups G(p, q). Prior to this the author had solved the D(2) problem for the groups G(p, 2). However, for q > 2, the only previously known solutions were for the groups G(7, 3), G(5, 4) and G(7, 6), all done by difficult direct calculation by two of the author's students, Jonathan Remez (2011) and Jason Vittis (2019).The method employed is heavily algebraic and involves precise analysis of the integral representation theory of G(p, q). Some noteworthy features are a new cancellation theory of modules (Chapters 10 and 11) and a simplified treatment (Chapters 5 and 12) of the author's theory of Swan homomorphisms.Contents: Projective Modules and Class GroupsHomological AlgebraThe Derived Module CategoryExtension and Restriction of ScalarsSwan HomomorphismsModules Over Quasi-Triangular AlgebrasA Fibre Product DecompositionGalois ModulesThe Sequencing TheoremA Cancellation Theorem for ExtensionsCancellation of Quasi-Swan ModulesSwan Homomorphisms for Metacyclic GroupsAn Obstruction to MonogenicityThe D(2) Property
Readership: Academic mathematicians; postgraduate and higher.D(2) Problem;Integral Representations of Metacyclic Groups;Syzygies;Stable Module;Derived Module Category;Cancellation Properties of Modules;Swan Homomorphisms0Key Features:The obvious point of originality is the proof of the D(2) property for a large collection of metacyclic groups. This is to be found nowhere else in the literatureHowever, various other aspects are also original and novel; the use of the derived module category throughoutAbove all, however, the characteristic feature of the book is the systematic application of homological algebra and representation theory to solve a significant problem