Exotic smoothness and physics. Differential topology and spacetime models.

*(English)*Zbl 1177.83001
Singapore: World Scientific (ISBN 978-981-02-4195-7/hbk). xiv, 322 p. (2007).

The second author of this book is one of the founders of the now famous Brans-Dicke theory of gravitation, see e.g. [C. H. Brans, Scalar-tensor theories of gravity: some personal history. Gravitation and cosmology. Proceedings of the third international meeting, Morelia, Michoacan, Mexico, 26–30 May 2008. Melville, NY: American Institute of Physics (AIP). AIP Conference Proceedings 1083, 34–46 (2008; Zbl 1169.83005)] and for his newer works [see “New level of relativity”, Gen. Relativ. Gravitation 40, No. 5, 1071–1086 (2008; Zbl 1140.83396)].

Publisher’s description: “The recent revolution in differential topology related to the discovery of non-standard (“exotic”) smoothness structures on topologically trivial manifolds such as \(\mathbb R^4\) 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.

Publisher’s description: “The recent revolution in differential topology related to the discovery of non-standard (“exotic”) smoothness structures on topologically trivial manifolds such as \(\mathbb R^4\) 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.

Reviewer: Hans-Jürgen Schmidt (Potsdam)

##### MSC:

83-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

57R55 | Differentiable structures in differential topology |

53C80 | Applications of global differential geometry to the sciences |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |

83E05 | Geometrodynamics and the holographic principle |