The making of the book
Exotic Smoothness and Physics

Differential Topology and Spacetime Models

Torsten Aßelmeyer[*]

Carl H. Brans[*]

Last Modified: 2/5/2007

We are very happy to announce that this book has now been published by World Scientific/Imperial Press. See:


We present the following excerpts to provide some overall view of the work:


chapter 1(10/18/06)

Purpose of book: There has been a profound revolution in differential topology over the last 20 years or so, especially regarding the discovery of non-standard, ``exotic'', smoothness structures on otherwise topologically trivial manifolds such as ${\mathbb R}^4$ . These exciting new structures are strikingly counter-intuitive and open vast areas of previously unexplored mathematics. However, they may very well have equally important significance for physics. First, because the mathematical tools used in discovering and exploring them have drawn heavily from physics: the key importance of dimension 4, the use of Yang-Mills, gauge theory, moduli spaces, Seiberg-Witten equations, etc. Secondly, almost all contemporary physics makes use of a spacetime model which is at least a smooth manifold. Thus physics makes use of structures which can be schematically represented by

\begin{displaymath}\mbox{Point set}\rightarrow\mbox{TOP}\rightarrow\mbox{DIFF}\rightarrow\mbox{BUNDLE}\rightarrow\cdots.\end{displaymath}

Until recently, the middle structure, DIFF, smooth manifold structure, was thought to be relatively trivial. However, we now know that it is not, that the same TOP manifold can support an infinity of non-diffeomorphic, thus physically inequivalent structures. This opens the door to a potentially rich storeroom of tools for use in physical theories. Thus, just as Einstein's General Relativity teaches us that geometry is not physically trivial, so perhaps may non-trivial smoothness carry physical significance.

Readership: We would expect the book to be useful to workers in mathematical and theoretical physics at the graduate and research levels, especially those working in areas related to general relativity, as well as to certain mathematicians. While many mathematical books are available on various aspects of exotic smoothness, there is nothing aimed explicitly at introducing the topic to physicists or the general mathematical audience.

The Authors: Torsten Aßelmeyer (FIRST, Berlin, e-mail) and Carl H. Brans(Loyola, New Orleans, e-mail) have been actively engaged in research on these topics for since the early 1990's. They have published several papers on the topic, including most recently Cosmological Anomalies and Exotic Smoothness Structures, Gen. Rel. Grav. 34, 1767(2002).