University Lecture Series 1989; 79 pp; softcover Volume: 1 Reprint/Revision History: reprinted with corrections 1990; third printing 1999 ISBN10: 0821870009 ISBN13: 9780821870006 List Price: US$25 Member Price: US$20 Order Code: ULECT/1
 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 GaussBonnet theorem, and proceed to the topology and geometry of foliated 3manifolds. They also explain, in terms of general position, why fourdimensional 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 fourdimensional 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 nottootechnical introduction to this vital area of current research. Readership Graduate students and research mathematicians and physical scientists. Table of Contents  Manifolds and smooth structures
 The Euler number
 Foliations
 The topological classification of simplyconnected 4manifolds
 Donaldson's theory
 Fake \(\mathbf R^4\) 's
