Let \(G\) be a finite group and \(J\rightarrowtail F_2\to F_1\to F_0\twoheadrightarrow\mathbb{Z}\) an exact complex of finitely generated free modules \((F_i)\) over \(\mathbb{Z}[G]\). Here \(\mathbb{Z}\) has trivial \(G\)-structure and \(J\) may be defined as the kernel of the map \(F_2\to F_1\). A classical construction shows that such a complex is defined by the chains of the universal cover of a 2-dimensional CW-complex \(K\) with fundamental group isomorphic to \(G\) \((J= \pi_2(k))\), and the question of geometric realization is whether every algebraic complex is chain homotopy equivalent to one obtained in this way. A related question is the \(D(2)\)-problem: if \(X\) is a 3-dimensional complex such that \(H^3(X;{\mathcal B})= 0\) for all local coefficient systems \({\mathcal B}\), is \(X\) homotopy equivalent to a 2-dimensional complex? One of the author’s main results is that the \(D(2)\)-problem has a positive answer if and only if every algebraic 2-complex is geometrically realizable. This is proved towards the end of the book in Chapter 10, with an analogue for a large class of infinite discrete groups given in Appendix B. The bulk of the book is thus devoted to geometric realization. The cleanest result is probably that if \(G\) admits a free resolution of period 4, and \(G\) has the free cancellation property, then \(G\) satisfies the \(D(2)\)-property if and only if \(G\) admits a balanced presentation (Theorem VI, Chapter XI). Reminders: period 4 groups include the dihedral groups of order \(2p\) (\(p=\text{odd}\) prime) and finite 3-manifold groups, in a balanced presentation the numbers of generators equals the number of relations, and the free cancellation property holds provided that a stably free module is free. Underpinning much of the argument is Wesley Brown’s work on a derived category approach to cancellation.
This book appears at just the right moment in the development of low-dimensional geometric topology. If all 3-manifolds with finite fundamental group are elliptic, i.e. admit a geometry modelled on the natural geometry of \(S^3\), then the theory of 3-dimensional Poincaré complexes is strikingly different from that of manifolds. This is a statement about finite groups; for \(\text{PD}^3\)-groups without torsion the parallel with 3-manifold groups is equally striking. This is illustrated by realization theorems recently obtained by the author for such groups.
One particularly attractive feature of the book is its attention to detail, and the background chapters on orders, representations and periodic groups may well appeal to an audience wider than that of specialists.