Selected Applications of Geometry to Low-Dimensional Topology ( University Lecture Series )

Publication series ：University Lecture Series

Author： Michael H. Freedman;Feng Luo

Publisher： American Mathematical Society‎

Publication year： 1989

E-ISBN: 9781470421540

P-ISBN(Paperback): 9780821870006

Subject： O189 topology (geometry of situation)

Keyword： Geometry and Topology

Language： ENG

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Selected Applications of Geometry to Low-Dimensional Topology

Description

This book, the inaugural volume in the University Lecture Series, is based on lectures presented at Pennsylvania State University in February 1987. The lectures attempt to give a taste of the accomplishments of manifold topology over the last 30 years. By the late 1950s, algebra and topology had produced a successful and beautiful fusion. Geometric methods and insight, now vitally important in topology, encompass analytic objects such as instantons and minimal surfaces, as well as nondifferentiable constructions. Keeping technical details to a minimum, the authors lead the reader on a fascinating exploration of several developments in geometric topology. They begin with the notions of manifold and smooth structures and the Gauss-Bonnet theorem, and proceed to the topology and geometry of foliated 3-manifolds. They also explain, in terms of general position, why four-dimensional space has special attributes, and they examine the insight Donaldson theory brings. The book ends with a chapter on exotic structures on $\mathbf R^4$, with a discussion of the two competing theories of four-dimensional manifolds, one topological and one smooth. Background material was added to clarify the discussions in the lectures, and references for more detailed study are included. Suitable for graduate students and researchers in mathematics and the physical sciences, the book requires only background in undergraduate mathematics. It should prove valuable for those wishing a not-too-technical

Chapter

Cover

Title

Contents

Introduction

Chapter 1. Manifolds and Smooth Structures

A. Definitions and Examples

B. Tangent Bundle, Cotangent Bundles, and Riemannian Metrics

C. Some Historical Remarks

Chapter 2. The Euler Number

A. The Degree Theorem

B. Hopf Index Theorem for Vector Fields and the Euler Number

C. The Morse Formula for the Euler Number

D. The Gauss-Bonnet Theorem

Chapter 3. Foliations

A. Definition and Examples of Foliations

B. Sullivan's Theorem

C. Novikov's Theorem

D. Dissecting Foliated 3-Manifolds

Chapter 4. The Topological Classification of Simply-Connected 4-Manifolds

A. Intersection Pairing and the Statement of the Classification Theorem

B. A Review of Classical h-Cobordism Theory in Dimension ≥ 5

C. The Disk Theorems

Chapter 5. Donaldson's Theory

A. Vector Bundles and Connections

B. The Self-Dual Equation, Topological Invariants, and Yang-Mills Theory over a Complex Line Bundle

C. A Glimpse of Donaldson's Proof and Fintushel and Stern's Work

Chapter 6. Fake R[sup(4)]'s

A. Kirby's Picture

B. Some other Fake R[sup(4)]'s

C. Uncountably Many Exotic Smooth Structures on R[sup(4)]

References

Back Cover

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