# Selected Applications of Geometry to Low-Dimensional Topology

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*Michael H. Freedman; Feng Luo*

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 introduction to this vital area of
current research.

#### Table of Contents

# Table of Contents

## Selected Applications of Geometry to Low-Dimensional Topology

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction ix10 free
- Chapter 1. Manifolds and Smooth Structures 114 free
- Chapter 2. The Euler Number 922
- Chapter 3. Foliations 2235
- Chapter 4. The Topological Classification of Simply-Connected 4-Manifolds 3851
- Chapter 5. Donaldson's Theory 5770
- Chapter 6. Fake R[sup(4)]'s 7285
- References 7790
- Back Cover Back Cover193