Hardcover ISBN:  9780821852316 
Product Code:  SURV/166 
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Electronic ISBN:  9781470413934 
Product Code:  SURV/166.E 
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Book DetailsMathematical Surveys and MonographsVolume: 166; 2010; 396 ppMSC: Primary 55; 57; Secondary 53; 58;
Seifert fiberings extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Away from the singular fibers, the fibering is an ordinary bundle with fiber a fixed homogeneous space. The singular fibers are quotients of this homogeneous space by distinguished groups of homeomorphisms. These fiberings are ubiquitous and important in mathematics. This book describes in a unified way their structure, how they arise, and how they are classified and used in applications. Manifolds possessing such fiber structures are discussed and range from the classical threedimensional Seifert manifolds to higher dimensional analogues encompassing, for example, flat manifolds, infranilmanifolds, space forms, and their moduli spaces. The necessary tools not covered in basic graduate courses are treated in considerable detail. These include transformation groups, cohomology of groups, and needed Lie theory. Inclusion of the Bieberbach theorems, existence, uniqueness, and rigidity of Seifert fiberings, aspherical manifolds, symmetric spaces, toral rank of spherical space forms, equivariant cohomology, polynomial structures on solvmanifolds, fixed point theory, and other examples, exercises and applications attest to the breadth of these fiberings. This is the first time the scattered literature on singular fiberings is brought together in a unified approach. The new methods and tools employed should be valuable to researchers and students interested in geometry and topology.
ReadershipGraduate students and research mathematicians interested in topology (transformation groups, manifolds, singular fiberings, and differential geometry).

Table of Contents

Chapters

1. Transformation groups

2. Group actions and the fundamental group

3. Actions of compact Lie groups on manifolds

4. Definition of Seifert fibering

5. Group cohomology

6. Lie groups

7. Seifert fiber space construction for $G\times W$

8. Generalization of Bieberbach’s theorems

9. Seifert manifolds with $\Gamma \setminus G/K$fiber

10. Locally injective Seifert fiberings with torus fibers

11. Applications

12. Seifert fiberings with compact connected $Q$

13. Deformation spaces

14. $S^1$actions on 3dimensional manifolds

15. Classification of Seifert 3manifolds via equivariant cohomology


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Seifert fiberings extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Away from the singular fibers, the fibering is an ordinary bundle with fiber a fixed homogeneous space. The singular fibers are quotients of this homogeneous space by distinguished groups of homeomorphisms. These fiberings are ubiquitous and important in mathematics. This book describes in a unified way their structure, how they arise, and how they are classified and used in applications. Manifolds possessing such fiber structures are discussed and range from the classical threedimensional Seifert manifolds to higher dimensional analogues encompassing, for example, flat manifolds, infranilmanifolds, space forms, and their moduli spaces. The necessary tools not covered in basic graduate courses are treated in considerable detail. These include transformation groups, cohomology of groups, and needed Lie theory. Inclusion of the Bieberbach theorems, existence, uniqueness, and rigidity of Seifert fiberings, aspherical manifolds, symmetric spaces, toral rank of spherical space forms, equivariant cohomology, polynomial structures on solvmanifolds, fixed point theory, and other examples, exercises and applications attest to the breadth of these fiberings. This is the first time the scattered literature on singular fiberings is brought together in a unified approach. The new methods and tools employed should be valuable to researchers and students interested in geometry and topology.
Graduate students and research mathematicians interested in topology (transformation groups, manifolds, singular fiberings, and differential geometry).

Chapters

1. Transformation groups

2. Group actions and the fundamental group

3. Actions of compact Lie groups on manifolds

4. Definition of Seifert fibering

5. Group cohomology

6. Lie groups

7. Seifert fiber space construction for $G\times W$

8. Generalization of Bieberbach’s theorems

9. Seifert manifolds with $\Gamma \setminus G/K$fiber

10. Locally injective Seifert fiberings with torus fibers

11. Applications

12. Seifert fiberings with compact connected $Q$

13. Deformation spaces

14. $S^1$actions on 3dimensional manifolds

15. Classification of Seifert 3manifolds via equivariant cohomology