Geometry & Topology Monographs

Volume 5: 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 S^1 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 L^2 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) L^2-cohomology, 3-manifold theory and surgery.

Jonathan Hillman
December 2002

AMS subject classification. Primary: 57N16, 57M50. Secondary: 57N13, 57Q45.

E-print: arXiv:math.GT/0212142

Submitted to GT on 23 July 2002. Accepted 8 October 2002. Published 9 December 2002.

Complete book (379 + xiv pages)

Part I : Frontmatter and Chapters 1 to 6 (pages i-xiv, 1-128) Part II : Chapters 7 to 13 (pages 129-264) Part III : Chapters 14 to 18, biblio and index (pages 265-379) Notes on file formats

The University of Sydney

GTM home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to