The recent revolution in differential topology related to the discovery of non-standard (“exotic”) smoothness structures on topologically trivial manifolds such as R4 suggests many exciting opportunities for applications of potentially deep importance for the spacetime models of theoretical physics, especially general relativity. This rich panoply of new differentiable structures lies in the previously unexplored region between topology and geometry. Just as physical geometry was thought to be trivial before Einstein, physicists have continued to work under the tacit — but now shown to be incorrect — assumption that differentiability is uniquely determined by topology for simple four-manifolds. Since diffeomorphisms are the mathematical models for physical coordinate transformations, Einstein’s relativity principle requires that these models be physically inequivalent. This book provides an introductory survey of some of the relevant mathematics and presents preliminary results and suggestions for further applications to spacetime models.

Contents:
Introduction and Background
Algebraic Tools for Topology
Smooth Manifolds, Geometry
Bundles, Geometry, Gauge Theory
Gauge Theory and Moduli Space
A Guide to the Classification of Manifolds
Early Exotic Manifolds
The First Results in Dimension Four
Seiberg–Witten Theory: The Modern Approach
Physical Implications
From Differential Structures to Operator Algebras and Geometric Structures

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