Saveliev Lectures on the Topology of 3-Manifolds

An Introduction to the Casson Invariant
2. revidierte ed
ISBN: 978-3-11-025036-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 218 Seiten

Reihe: De Gruyter Textbook

A beginner's introduction to low-dimensional topology 2nd revised and extended edition New material is added to provide a link to recent research in Heegaard Floer homology and contact topology

Progress in low-dimensional topology has been very quick in the last three decades, leading to the solutions of many difficult problems. Among the earlier highlights of this period was Casson's ?-invariant that was instrumental in proving the vanishing of the Rohlin invariant of homotopy 3-spheres. The proof of the three-dimensional Poincaré conjecture has rendered this application moot but hardly made Casson's contribution less relevant: in fact, a lot of modern day topology, including a multitude of Floer homology theories, can be traced back to his ?-invariant. The principal goal of this book, now in ist second revised edition, remains providing an introduction to the low-dimensional topology and Casson's theory; it also reaches out, when appropriate, to more recent research topics. The book covers some classical material, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It then proceeds through the Kirby calculus and Rohlin's theorem to Casson's invariant and ist applications, and concludes with a brief overview of recent developments. The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic and differential topology, including the fundamental group, basic homology theory, transversality, and Poincaré duality on manifolds

Zielgruppe

Graduate Students, Lecturers, and Researchers in Mathematics and Theoretical Physic; Academic Libraries

Autoren/Hrsg.

Saveliev, Nikolai

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

Frontmatter
Preface
Contents
Introduction
Glossary
Lecture 1. Heegaard splittings
Lecture 2. Dehn surgery
Lecture 3. Kirby calculus
Lecture 4. Even surgeries
Lecture 5. Review of 4-manifolds
Lecture 6. Four-manifolds with boundary
Lecture 7. Invariants of knots and links
Lecture 8. Fibered knots
Lecture 9. The Arf-invariant
Lecture 10. Rohlin’s theorem
Lecture 11. The Rohlin invariant
Lecture 12. The Casson invariant
Lecture 13. The group SU(2)
Lecture 14. Representation spaces
Lecture 15. The local properties of representation spaces
Lecture 16. Casson’s invariant for Heegaard splittings
Lecture 17. Casson’s invariant for knots
Lecture 18. An application of the Casson invariant
Lecture 19. The Casson invariant of Seifert manifolds
Conclusion
Bibliography
Index

Über Autor(innen)

Saveliev, Nikolai
Nikolai Saveliev, University of Miami, Florida, USA.

Nikolai Saveliev, University of Miami, Florida, USA.