Poincaré Duality in Dimension 3

Abstract

Contents viiPreface xi Chapter 1. Generalities 1.1. Group theoretic preliminaries 1.2. Group rings and finiteness conditions 1.3. Projective homotopy of modules 1.4. Graphs of groups 1.5. Covering spaces 1.6. Poincaré duality 1.7. Poincaré duality groups 1.8. Poincaré duality pairs 1.9. Poincaré duality complexes of dimension 1 or 2 1.10. Infinite cyclic covers of 4-manifolds Chapter 2. Classification, Realization and Splitting 2.1. A low-dimensional simplification 2.2. The Classification Theorem 2.3. Homological triple versus algebraic 2-type 2.4. The Group Realization Theorem 2.5. Statement of the Realization Theorem of Turaev 2.6. Connected sum decompositions 2.7. Uniqueness of factorization 2.8. The finiteness obstruction 2.9. π 2 as a ‫[ޚ‬π ]-module Chapter 3. The relative case 3.1. The basic invariants 3.2. A Realization Theorem for certain pairs 3.3. Connected sums 3.4. The Algebraic Loop Theorem 3.5. Cubes with handles vii viii JONATHAN A. HILLMAN Chapter 4. The Centralizer Condition 4.1. Trees with ∞-vertices and the invariant ξ 4.2. The action of cyclic subgroups on a G-tree module 4.3. Consequences for PD 3 -complexes Chapter 5. Orientable PD 3 -complexes with π virtually cyclic 5.1. Finite fundamental groups 5.2. Finite groups with periodic cohomology 5.3. Finite groups of cohomological period dividing 4 5.4. A finite complex with fundamental group S 3 5.5. Self homotopy equivalences 5.6. Finiteness, standardness, and ‫ޓ‬ 3 -manifolds 5.7. Fundamental group with two ends Chapter 6. Indecomposable orientable PD 3 -complexes 6.1. A finite complex with group S 3 * ‫ޚ2/ޚ‬ S 3 6.2. Other PD 3 -complexes with this group 6.3. Vertex groups have periodic cohomology 6.4. Eliminating groups with cohomological period > 4 6.5. Groups with cohomological period 1, 2 or 4 6.6. Extending the construction of §1 6.7. Homotopy types with given fundamental group Chapter 7. Nonorientable PD 3 -complexes 7.1. Nonorientable PD 3 -complexes with π virtually free 7.2. Indecomposable nonorientable 3-manifolds 7.3. Indecomposable nonorientable PD 3 -complexes Chapter 8. Asphericity and 3-manifolds 8.1. Some properties of 3-manifold groups 8.2. Are PD 3 -groups 3-manifold groups? 8.3. Some reductions 8.4. A compendium of frequently cited theorems 8.5. Mapping tori and Seifert fibred 3-manifolds 8.6. The groups of aspherical geometric 3-manifolds Chapter 9. Centralizers, normalizers and ascendant subgroups 9.1. Centralizers 9.2. The maximum condition on centralizers 9.3. Normalizers of rank 1 abelian subgroups 9.4. Normalizers of other FP 2 subgroups CONTENTS ix 9.5. Ascendant FP 2 subgroups 9.6. Extending Elkalla's results? 9.7. Splitting PD 2 -subgroups with nontrivial normalizers Chapter 10. Splitting along PD 2 -subgroups 10.1. Kropholler-Roller splitting theorems 10.2. Commensurators of PD 2 -subgroups 10.3. Kropholler's JSJ Decomposition Theorem 10.4. Castel's Centralizer Theorem 10.5. No nontrivial Baumslag-Solitar relations Chapter 11. The Tits Alternative 11.1. The virtually Haken case 11.2. Profinite and pro-p completio...

Around 3-manifold groups

“…It is worth remarking that by the Sphere Theorem 3.3 and Perelman's Theorem 3.10 a PD(3)-group can only be the fundamental group of a closed, aspherical 3-manifold. Results similar to some of the fundamental results in 3-manifold theory, have been established in the setting of PD(3)-groups, see [Hil19], [Wa03,Wa04], [Tho95]. They give some evidence towards Wall's conjecture in dimension 3.…”

“…They give some evidence towards Wall's conjecture in dimension 3. For a detailled exposition of results in this direction, we refer to J. Hillman's books [Hil02,Hil19]. B. Bowditch [Bow04] verified Wall Conjecture for a PD(3) groups which contains a non-trivial, normal cyclic subgroup, see also [Hil85] for the case of positive first Betti number.…”

“…See [DuSw00] and [Wa03,Theorem 10.8] or [Wa04, Theorem 4.2] for the case of finitely presentable PD(3) groups, and [Cas07] or [Kro90] and [Hil06] for the general case which avoids the finitely presentable assumption (cf. [Hil19] for more details).…”

“…This would imply that an atoroidal PD(3)group always contain a non-abelian free group by the following result due to M. Kapovich This, in turn, would be sufficient to show that PD(3)-groups verfy the Tits alternative, since by the torus theorem [DuSw00] a non atoroidal PD(3)-group is a Seifert fibered 3-manifold group or splits over a Z ⊕ Z subgroup. The Tits alternative for PD(3)-groups is still an open question, see [BoB19], [Hil03], [Hil19] for a detailed discussion of this question.…”

Centralizers of torsion in 3-manifold groups

“…One possible approach to this question might be to apply the Turaev Criterion and its consequences [9]. (See also [5,Corollary 2.4…”