Volume 5 (2002)

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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)

Part I: Manifolds and PD–complexes

1 Group theoretic preliminaries


2 2–Complexes and PD3–complexes


3 Homotopy invariants of PD4–complexes


4 Mapping tori and circle bundles


5 Surface bundles


6 Simple homotopy type and surgery

Part II: 4–dimensional Geometries

7 Geometries and decompositions


8 Solvable Lie geometries


9 The other aspherical geometries


10 Manifolds covered by S²×R²


11 Manifolds covered by S3×R


12 Geometries with compact models


13 Geometric decompositions of bundle spaces

Part III: 2–knots

14 Knots and links


15 Restrained normal subgroups


16 Abelian normal subgroups of rank ≥2


17 Knot manifolds and geometries


18 Reflexivity