Syzygies and homotopy theory
type: Item
instance of: scholarly article
title: Syzygies and homotopy theory
zbMATH Open document ID: 1233.13001
DOI: 10.1007/978-1-4471-2294-4
published in: Algebra and Applications
publication date: 2011-10-27T00:00:00Z
review text: 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 It is known from the unpublished work of W.\ Browning that finite abelian groups have the \(D(2)\)-property.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 \t…
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 55P15
zbMATH DE Number: 5964729
zbMATH Keywords: \(D(2)\)-problem
zbMATH Keywords: generalized Swan module
zbMATH Keywords: Milnor squares
zbMATH Keywords: \(R(2)\)-problem
zbMATH Keywords: stable module
zbMATH Keywords: syzygy
author: F. E. A. Johnson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4471-2294-4
OpenAlex ID: W1481794959
description: scientific article; zbMATH DE number 5964729