Syzygies and homotopy theory
The book \textit{Syzygies and Homotopy Theory} is concerned with the algebraic classification of certain finite dimensional geometric complexes with a nontrivial, finitely presented fundamental group \(G\) and is directed towards to basic problems: the \(D(2)\)- and \(R(2)\)-problem. The dimension two problem (\(D(2)\)-problem) asks whether a finite connected \(3\)-dimensional cell complex \(X\) is homotopy equivalent to a finite \(2\)-dimensional cell complex if \(X\) is cohomologically \(2\)-dimensional, that is, \(H_3(\tilde{X};\mathbb{Z})=H^3(X;\mathcal{B})=0\) for all coefficient systems \(\mathcal{B}\) on \(X\). The \(D(2)\)-property is said to hold for a finitely presented group \(G\) when the above question is answered in the affirmative for \(X\) such that \(\pi_1(X)\cong G\). A problem closely related to the \(D(2)\)-problem is the dimensional two realization problem (\(R(2)\)-problem) for a finitely presented group \(G\) which asks whether every algebraic \(2\)-complex over \(G\), that is, an exact sequence of \(\mathbb{Z}[G]\)-modules of the form \[ 0 \to J \to F_2 \to F_1 \to F_0 \to \mathbb{Z} \to 0 \] with \(F_i\) a finitely generated stably free module, is geometrically realizable, that is, homotopy equivalent in the algebraic sense, to the cellular chain complex of a finite \(2\)-dimensional cell complex \(X\). We say that \(G\) has the realization property when all algebraic \(2\)-complexes over \(G\) are geometrically realizable. In the wake of Perelman's solution of the Poincarè conjecture, the \(D(2)\)-\(R(2)\) problem is perhaps the most significant and recalcitrant one left in low-dimensional topology. It is known from the unpublished work of W.\ Browning that finite abelian groups have the \(D(2)\)-property. The \(D(2)\)-property for free groups follows from Theorem 2.4 of the paper [\textit{J. G. Ratcliffe}, Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 221--254 (1987; Zbl 0632.57003)] and \textit{H.\ Bass}'s Theorem [J. Algebra 1, 367--373 (1964; Zbl 0145.26604)] that finitely generated projective modules over a free group are free. The author's previous book [Stable Modules and the \(D(2)\)-Problem. London Mathematical Society Lecture Note Series 301. Cambridge: Cambridge University Press. (2003; Zbl 1055.57002)] presents a thorough exposition of the author's proof that the \(D(2)\)-problem is entirely equivalent to the \(R(2)\)-problem and develops algebraic techniques to show that certain finite nonabelian groups of cohomological free period \(4\) have the \(D(2)\)-property. In order to consider these problems for finitely presented groups \(G\) the foundations of this book must be re-built; in large part the aim of \textit{Syzygies and Homotopy Theory} is to do precisely that. There are two parts: Theory and Practice. The chapter titles in Part I (Theory) are: Preliminaries, The Restricted Linear Group, The Calculus of Corners and Squares, Extensions of Modules, The Derived Module Category, Finiteness Conditions, The Swan Mapping, Classification of Algebraic Complexes. Whereas, the chapter titles in Part II (Practice) are: Rings with Stably Free Cancellation, Group Rings of Cyclic Groups, Group Rings of Dihedral Groups, Group Rings of Quaternion Groups, Parametrizing \(\Omega_1(\mathbb{Z})\): Generic Case, Parametrizing \(\Omega_1(\mathbb{Z})\): Singular Case, Generalized Swan Modules, Parametrizing \(\Omega_1(\mathbb{Z})\): \(G=C_\infty\times \Phi\), Conclusion. The appendix titles are: A Proof of Dieudonné's Theorem, Change of Ring, Group Rings with Trivial Units, The Infinite Kernel Property. The first part of the book is theoretical and consists of an excellent introduction to the theory of modules over group rings. Many of arguments are formulated in terms of modules over the group ring \(\mathbb{Z}[G]\), where \(G\) is a specified finitely presented group as a fundamental group. The innovation here is to regard Hilbert's method of syzygies as stable modules rather than minimal modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. Given a \(\Lambda\)-module \(M\), write \(\Omega_n(M)\) for the stable class of any \(n\)th-syzygy of \(M\) which lives in the derived category \(\text{Der}(\Lambda)\) of the \(\Lambda\)-module category \(\text{Mod}_{\Lambda}\). Chapter 7 extends the \textit{R. G. Swan} result [Ann. Math. (2) 76, 55--61 (1962; Zbl 0112.02702)] so that, for suitable modules \(J\) over reasonably general rings \(\Lambda\), it takes the form \(S_J : \text{Aut}_{\text{tiny Der}}(J)\to\tilde{K}_0(\Lambda)\). Then, Chapter 8 classifies algebraic \(n\)-complexes over \(\mathbb{Z}[G]\) up to homotopy equivalence. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. This part of the book considers how the theory works out in detail for the specific examples \(F_n\times\Phi\), where \(F_n\) is a free group of rank \(n\) and \(\Phi\) is finite. In the applying the classification result to the original problem, one needs specific information about the syzygies \(\Omega_n(\mathbb{Z})\). In the first instance the author contents to study \(\Omega_1(\mathbb{Z})\). In the most familiar case, namely when \(G\cong F_n\times C_m\), a complete description of all odd \(\Omega_{2k+1}(\mathbb{Z})\) is presented. The book is concluded with \textit{T. Edward}'s solution [Ph.D.\ Thesis, University College London (2006)] and [Algebr. Geom. Topol. 6, 71-89 (2006; Zbl 1097.57005)] of the \(R(2)\)-problem for \(C_\infty\times C_m\) and the current status of the \(D(2)\)-\(R(2)\) problem. The theory developed within this book has potential applications in various branches of algebra. \textit{Syzygies and Homotopy Theory} is well written, nicely organized, and is a pleasure to read. One particularly attractive feature of the book is its attention to detail, and the background chapters may well appeal to an audience wider than that of specialists.
scientific article; zbMATH DE number 5964729