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
III - Mapping tori and circle bundles
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
The study of 4-manifolds which fibre over the circle benefits from both 4-dimensional surgery and from 3-manifold theory, which may be applied to the fibre and the characteristic map. Unfortunately, there is as yet no satisfactory fibration theorem in this dimension. We shall give a homotopy approximation to such a theorem. By a result of Quinn, an infinite cyclic covering space of a closed n-manifold M is homotopy equivalent to a PDn-1-complex if and only if it is finitely dominated. The fundamental group v of the covering space must then be finitely generated, and χ(M) = 0. It is conceivable that these conditions may suffice when n = 4; we shall see that this is so if also either v is free or v does not contain the centre of π1(M).
In the final section we give conditions for a 4-manifold to be homotopy equivalent to the total space of an S1-bundle over a PD3-complex, and show that these conditions are sufficient if the fundamental group of the PD3-complex is torsion free but not free.
A general criterion
Let E be a connected cell complex and let f : E → S1 be a map which induces an epimorphism f* from π1(E) to Z = π1(S1) with kernel v.
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- Chapter
- Information
- The Algebraic Characterization of Geometric 4-Manifolds , pp. 32 - 49Publisher: Cambridge University PressPrint publication year: 1994