Four-manifolds, geometries and knots

TL;DR

This book provides an algebraic characterization of certain 4-manifolds and knots, focusing on their topological and geometric properties, invariants, and classifications, with significant results on manifolds fibering over surfaces and infrasolvmanifolds.

Contribution

It offers new algebraic and topological classifications of 4-manifolds and knots, including explicit invariant descriptions and characterizations of manifolds with specific fibrations and geometries.

Findings

Complete invariants for homotopy types of certain 4-manifolds

Characterizations of manifolds fibering over surfaces or S^1

Determination of 2-knots with poly-Z groups up to Gluck reconstruction

Abstract

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 and again in November 2022. For details see the end of the preface.