Poincaré duality in dimension 3. In memory of Charles B. Thomas (Q2219664)

The two main objects considered in the present book are Poincaré duality complexes and Poincaré duality groups, concentrating mainly on the ``critical dimension'' 3. Poincaré duality complexes first appeared in a paper by C.T.C. Wall from 1967 and model the homotopy types of closed manifolds, satisfying Poincaré duality; the notion of Poincaré duality groups on the other hand was introduced by F.E.A. Johnson and Wall in 1972 as an algebraic analogue of the notion of closed \textit{aspherical} manifold: a group \(\pi\) is an \(n\)-dimensional Poincaré duality group (a \(\mathrm{PD}_n\)-group for short) if and only if its classifying space (or Eilenberg-MacLane space, or \(K(\pi,1)\), i.e. aspherical (contractible universal covering) and with fundamental group \(\pi\)) satisfies Poincaré duality of dimension \(n\); an algebraic definition, using homological algebra, was given shortly thereafter by \textit{R. Bieri} [Homological dimension of discrete groups. London: Queen Mary College, University of London (1976; Zbl 0357.20027)] which, together with the book of [\textit{K. S. Brown}, Cohomology of groups. New York, NY: Springer (1982; Zbl 0584.20036)], is one of the classical texts on the cohomology of groups. Concerning dimension 2, by results of Eckmann, Müller and Linnell around 1980, every \(\mathrm{PD}_2\)-group is isomorphic to the fundamental group of a closed surface. On the other hand, in dimension 4 and higher dimensions there is a plethora of exotic examples, e.g. for every \(n \ge 4\) there is a finitely generated \(\mathrm{PD}_n\)-group which is not finitely presented and hence not isomorphic to the fundamental group of a closed aspherical \(n\)-manifold, and there are uncountably many \(\mathrm{PD}_4\)-groups (Wall conjectured that every \textit{finitly presented} \(\mathrm{PD}_n\)-groups is the fundamental group of a closed, aspherical \(n\)-manifold). So this directs attention to the critical dimension 3 where it is not known whether every \(\mathrm{PD}_3\)-group is isomorphic to the fundamental group of a closed aspherical 3-manifold. The fundamental group plays a crucial role for aspherical 3-manifolds: after a long history and in particular the geometrization of 3-manifolds, closed aspherical 3-manifolds are well understood and determined by their fundamental groups; however, no purely group-theoretical characterization of their fundamental groups is known, so the groups themselves remain somewhat mysterious. On the other hand, \(\mathrm{PD}_3\)-groups is a rather sophisticated notion, so one would like to find also some more basic and concrete group-theoretical properties which determine 3-manifold groups; such properties and their possible generalizations to \(\mathrm{PD}_3\)-groups are also an important point of the present book. An account on algebraic properties of 3-manifold groups, after the solution of almost all of Thurston's famous problems on 3-manifolds and Kleinian groups, can be found in a book by \textit{M. Aschenbrenner} et al. [3-manifold groups. Zürich: European Mathematical Society (EMS) (2015; Zbl 1326.57001)] (an interesting property of 3-manifold groups is e.g. coherence, i.e. every finitely generated subgroup is finitely presented, and coherent groups are discussed in detail in a survey by [\textit{D. T. Wise}, Ann. Math. Stud. 205, 326--414 (2020; Zbl 1452.57019)], as a class of groups close in nature to the fundamental groups of 3-manifolds). Concerning 3-dimensional Poincaré duality complexes instead, every orientable \(\mathrm{PD}_3\)-complex is a connected sum of indecomposables (with respect to internal connected sums and boundary-connected sums, algebraically with respect to free products), and the indecomposables are either aspherical or have virtually free fundamental group, isomorphic to the fundamental groups of certain finite graphs of finite groups (trees with nontrivial cyclic edge groups and finite vertex groups of cohomological period dividing 4, e.g. dihedral groups which occur as fundamental groups of finite \(\mathrm{PD}_3\)-complexes but not of 3-manifolds). Explicit examples discussed in the book have as fundamental group the dihedral or symmetric group \(S_3\) of order 6, and the free products with amalgamation \(S_3 *_{\mathbb Z/2\mathbb Z}S_3\) and \(S_3 *_{\mathbb Z/2\mathbb Z}\mathbb Z/4\mathbb Z\) which are fundamental groups of \(\mathrm{PD}_3\)-complexes but not of 3-manifolds. ``This book shall give an account of the reduction to indecomposables for \(\mathrm{PD}_3\)-complexes, and what is presently known about them'' (following work of Hendriks, Swarup, C.B. Thomas, Turaev, Crisp and the present author). This is the content of the first seven chapters of the book (containing also a discussion of the finite groups with periodic cohomology, related to the spherical space form problem). However, ``the primary interest is in the aspherical case'', and in particular in a discussion of possible approaches to prove that every \(\mathrm{PD}_3\)-group is a 3-manifold group. The book has about 160 pages, a final appendix with 64 open questions and seven pages of references; the titles of the 12 chapters as follows: 1. Generalities; 2. Classification, Realization and Splitting; 3. The relative case; 4. The Centralizer Condition; 5. Orientable \(\mathrm{PD}_3\)-complexes with \(\pi\) virtually cyclic; 6. Indecomposable orientable \(\mathrm{PD}_3\)-complexes; 7. Nonorientable \(\mathrm{PD}_3\)-complexes; 8. Asphericity and 3-manifolds; 9. Centralizers, normalizers and ascendant subgroups; 10. Splitting along \(\mathrm{PD}_2\)-subgroups; 11. The Tits Alternative; 12. Homomorphisms of nonzero degree. ``Our general approach is to prove most assertions which are specific about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results.'' The book is located in the rich and fascinating field of intersections of low-dimensional topology, 3-manifold topology, group theory, homological algebra and cohomology of groups. Some of the main topics discussed are various group-theoretical properties of 3-manifold groups and their possible generalizations to \(\mathrm{PD}_3\)-groups. The book is densely written (a reader needs a solid background in cohomology of groups), contains a lot of information and is stimulating to read (maybe starting spontaneously with one of the many subsections of some chapter); it is the first book treating \(\mathrm{PD}_3\)-complexes and \(\mathrm{PD}_3\)-groups in a systematic way.