Book contents
- Frontmatter
- Contents
- Preface
- I Algebraic Preliminaries
- II General results on the homotopy type of 4-manifolds
- III Mapping tori and circle bundles
- IV Surface bundles
- V Simple homotopy type, s-cobordism and homeomorphism
- VI Aspherical geometries
- VII Manifolds covered by S2 × R2
- VIII Manifolds covered by S3 × R
- IX Geometries with compact models
- X Applications to 2-knots and complex surfaces
- Appendix: 3-dimensional Poincaré duality complexes
- Problems
- References
- Index
Preface
Published online by Cambridge University Press: 16 September 2009
- Frontmatter
- Contents
- Preface
- I Algebraic Preliminaries
- II General results on the homotopy type of 4-manifolds
- III Mapping tori and circle bundles
- IV Surface bundles
- V Simple homotopy type, s-cobordism and homeomorphism
- VI Aspherical geometries
- VII Manifolds covered by S2 × R2
- VIII Manifolds covered by S3 × R
- IX Geometries with compact models
- X Applications to 2-knots and complex surfaces
- Appendix: 3-dimensional Poincaré duality complexes
- Problems
- References
- Index
Summary
It is well known that every closed surface admits a geometry of constant curvature, and that such surfaces may be classified up to homeomorphism either by their fundamental group or by their Euler characteristic and orientation character. Much current research in dimension 3 is guided by the expectation that all closed 3-manifolds have decompositions into geometric pieces, and that (lens spaces aside) the homeomorphism type is essentially determined by the fundamental group. (Here the Euler characteristic is always 0).
In dimension 4 there is no reason to expect that every closed 4-manifold may have a geometric decomposition, and the Euler characteristic and fundamental group are independent invariants. Nevertheless the closed 4-manifolds which admit geometries or fibre over a geometric base with geometric fibre form a large and interesting class. In these notes we shall attempt to characterize algebraically such 4-manifolds (up to homotopy equivalence or homeomorphism). This task has three main parts: finding complete invariants for the homotopy type, determining which systems of invariants are realizable and applying surgery (where possible) to obtain s-cobordisms. In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together form a complete system of invariants for the manifold and the possible invariants can be described explicitly.
- Type
- Chapter
- Information
- The Algebraic Characterization of Geometric 4-Manifolds , pp. vii - xPublisher: Cambridge University PressPrint publication year: 1994