Four-manifolds, geometries and knots
By Jonathan A. Hillman. 2002
198 citations recorded by the source corpus.
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 book has been revised in March 2007. For details see the end of the preface.
View Paper: https://www.semanticscholar.org/paper/bec3f7f6af345ee18a2bce26e5981fa8a37ecbb2
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
Citations in the corpus
- Ricci flow on locally homogeneous closed 4-manifolds (2005, background): “One can find more detailed description of four dimensional locally homogeneous geometries in Part II of [ Hi ].”
- TOPOLOGICAL 4-MANIFOLDS WITH GEOMETRICALLY TWO-DIMENSIONAL FUNDAMENTAL GROUPS (2008, background): “It is attractive to see whether the results of Theorems A and B also hold for more general fundamental groups of geometric dimension 2. In a series of papers [ 17 ], [18], [19], [20], Jonathan Hillman has investigated the homotopy classication of Poincar e 4complexes under various fundamental group ”