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ISSN (electronic): 1464-8997
ISSN (print): 1464-8989

Four-manifolds, geometries and knots

Jonathan Hillman

The goal of this book is to characterize algebraically the closed 4–manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2–knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2–6), geometries and geometric decompositions (Chapters 7–13), and 2–knots (Chapters 14–18). In many cases the Euler characteristic, fundamental group and Stiefel–Whitney classes together form a complete system of invariants for the homotopy type of such manifolds, and the possible values of the invariants can be described explicitly. The strongest results are characterizations of manifolds which fibre homotopically over S¹ or an aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to homeomorphism). As a consequence 2–knots whose groups are poly–Z are determined up to Gluck reconstruction and change of orientations by their groups alone.

This book arose out of two earlier books: 2–Knots and their Groups and The Algebraic Characterization of Geometric 4–Manifolds, published by Cambridge University Press for the Australian Mathematical Society and for the London Mathematical Society, respectively. About a quarter of the present text has been taken from these books, and I thank Cambridge University Press for their permission to use this material. The arguments have been improved and the results strengthened, notably in using Bowditch's homological criterion for virtual surface groups to streamline the results on surface bundles, using L² methods instead of localization, completing the characterization of mapping tori, relaxing the hypotheses on torsion or on abelian normal subgroups in the fundamental group and in deriving the results on 2–knot groups from the work on 4–manifolds. The main tools used are cohomology of groups, equivariant Poincare duality and (to a lesser extent) L²–cohomology, 3–manifold theory and surgery.

The book has been revised in March 2007 and July 2014. For details see the end of the preface.

Jonathan Hillman, December 2002

Geometry & Topology Monographs 5 (2002)

References
Part I: Manifolds and PD–complexes

1 Group theoretic preliminaries

3

2 2–Complexes and PD3–complexes

25

3 Homotopy invariants of PD4–complexes

47

4 Mapping tori and circle bundles

69

5 Surface bundles

89

6 Simple homotopy type and surgery

111
Part II: 4–dimensional Geometries

7 Geometries and decompositions

131

8 Solvable Lie geometries

151

9 The other aspherical geometries

179

10 Manifolds covered by S²×R²

195

11 Manifolds covered by S3×R

217

12 Geometries with compact models

233

13 Geometric decompositions of bundle spaces

251
Part III: 2–knots

14 Knots and links

269

15 Restrained normal subgroups

295

16 Abelian normal subgroups of rank ≥2

311

17 Knot manifolds and geometries

327

18 Reflexivity

341

Bibliography

363

Index

389