The present book is directly inspired by the recent interactions between mathematics and physics. Starting from a detailed study of Dirac’s magnetic monopole, one reaches the construction of the complex Hopf bundle \(S^1\to S^3\to S^2\), the natural connection of this bundle and the curvature defined by this connection. Thus the reader is led to the physical gauge theories (gauge potential and gauge field strength) in a clear way. With a bigger degree of difficulty, starting from the Yang-Mills equations (a nonlinear generalization of Maxwell’s equations), the quaternionic Hopf bundle \(S^3\to S^7\to S^4\) is constructed, and accepting (without proof) the Atiyah-Hitchin-Singer theorem, the natural connection on this bundle and its curvature are defined. Then a sketch of the construction of the Donaldson theory is given and the most recent Seiberg-Witten theory is mentioned.
With the aim to formulate mathematically the physical gauge theories, a great part of the book is dedicated to establish the minimal results of topology and geometry that are necessary to reach these objectives. Thus, in Chapter 1 (Topological Spaces) only the results of general topology are developed that are necessary to establish the properties of the topological spaces that will be used; in Chapter 2 (Homotopy Groups) the construction of the fundamental group and of the homotopy groups of higher-order of a topological space are given and the homotopy groups to be used are calculated. The main result of the chapter is the homotopy lifting theorem for locally trivial bundles. In Chapter 3 (Principal Bundles) the continuous principal bundles are constructed and the classification theorem of principal bundles on spheres is proved. Chapter 4 (Differentiable Manifolds and Matrix Lie Groups) contains a detailed introduction to differentiable manifolds, Lie groups and differentiable forms. After the physical and geometrical motivations given in Chapter 0 (Physical and Geometrical Motivations), in Chapter 5 (Gauge Fields and Instantons) the mathematical formulation of the physical gauge theories is presented.
The book contains a great number of exercises, a large part of them treats fragments of proofs of theorems and propositions. With this technique and by accepting some results without proofs (exact sequence of homotopy of a fiber bundle, the Atiyah-Hitchin-Singer theorem), the author achieves that the book does not exceed 400 pages.
The author announces in the preface of the book a subsequent volume, entitled “Topology, Geometry and Gauge Fields: Interactions” that will be dedicated to the study of the geometry and topology of the interaction between gauge fields and matter fields.
The reviewer considers that this book should be very interesting for mathematicians and physicists (as well as other scientists) that are concerned with gauge theories, for the following reasons: 1. for physicists, because it presents the minimal mathematical tools that are necessary to construct the mathematical models of gauge theories; 2. for mathematicians, because they can find satisfaction in the fact that such abstract parts of mathematics have applications in the construction of models of events in the real world; 3. for physicists and mathematicians, because books like this contribute to remove the speculative mathematics that has arisen in the recent years by interactions with physics.