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## Über dieses Buch

This is a book on topology and geometry and, like any books on subjects as vast as these, it has a point-of-view that guided the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. We ask the reader to come to us with some vague notion of what an electromagnetic field might be, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. Iwould go over both volumes thoroughly and make some minor changes in terminology and notation and correct any errors I find. In this new edition, a chapter on Singular Homology will be added as well as minor changes in notation and terminology throughout and some sections have been rewritten or omitted. Reviews of First Edition: “It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style." (NZMS Newletter) "...this book should be very interesting for mathematicians and physicists (as well as other scientists) who are concerned with gauge theories." (Zentralblatt Fuer Mathematik)

## Inhaltsverzeichnis

### Chapter 0. Physical and Geometrical Motivation

It sometimes transpires that mathematics and physics, pursuing quite different agendas, find that their intellectual wanderings have converged upon the same fundamental idea and that, once it is recognized that this has occurred, each breathes new life into the other. The classic example is the symbiosis between General Relativity and amply attest, the results of such an interaction can be spectacular. The story we have to tell is of another such confluence of ideas, more recent and perhaps even more profound. Our purpose in this preliminary chapter is to trace the physical and geometrical origins of the notion of a “gauge field” (known to mathematicians as a “connection on a principal bundle”). We will not be much concerned yet with rigorously defining the terms we use, nor will we bother to prove most of our assertions. Indeed, much of the remainder of the book is devoted to these very tasks. We hope only to offer something in the way of motivation.
Gregory L. Naber

### Chapter 1. Topological Spaces

We begin by recording a few items from real analysis (our canonical reference for this material is [Sp1],Chap.​ 1Chap.​ 3, which should be consulted for details as the need arises). For any positive integer n, Euclidean n-space $${\mathbb{R}}^{n} =\{ ({x}^{1},\ldots,{x}^{n}) :\ {x}^{i} \in\mathbb{R},\ i = 1,\ldots,n\}$$ is the set of all ordered n-tuples of real numbers with its usual vector space structure (x + y = (x 1, , x n ) + (y 1, , y n ) = (x 1 + y 1, , x n + y n ) and ax = a(x 1, , x n ) = (ax 1, , ax n )) and norm $$(\|x\| = {({({x}^{1})}^{2} + \cdots+ {({x}^{n})}^{2})}^{1/2}).$$
Gregory L. Naber

### Chapter 2. Homotopy Groups

The real line ℝ is not homeomorphic to the plane ℝ2, but this fact is not quite the triviality one might hope. Perhaps the most elementary proof goes as follows: Suppose there were a homeomorphism h of ℝ onto ℝ2. Select some point x 0 ∈ ℝ. The restriction of h to ℝ − {x 0} would then carry it homeomorphically onto ℝ2 − {h(x 0)}. However, $$\mathbb{R} -\{ {x}_{0}\} = (-\infty,{x}_{0}) \cup({x}_{0},\infty )$$ is not connected, whereas ℝ2 − {h(x 0)} certainly is connected (indeed, pathwise connected). Since connectedness is a topological property, this cannot be and we have our contradiction.
Gregory L. Naber

### Chapter 3. Homology Groups

The homotopy groups πn(X, x0) of a space are relatively easy to define, clearly topological invariants and, indeed, invariants of homotopy type, but are also extraordinarily difficult to compute even for quite simple spaces. Now we intend to provide a brief introduction to another set of invariants for which this situation is reversed. The singular homology groups require some work to define and their homotopy invariance is not so obvious, but once some basic tools are developed their computation is comparatively straightforward.
Gregory L. Naber

### Chapter 4. Principal Bundles

In this chapter we meld together locally trivial bundles and group actions to arrive at the notion of a C 0 (continuous) principal bundle (smoothness hypotheses are added in Chapter 5). The source of our interest in these structures was discussed at some length in Chapter 0, where we also suggested that principal bundles over spheres were of particular significance. Our goal here is to use the homotopy-theoretic information assembled in Chapter 2 to classify the principal bundles over S n .
Gregory L. Naber

### Chapter 5. Differentiable Manifolds and Matrix Lie Groups

If X is a topological manifold and (U 1, φ 1) and (U 2, φ 2) are two charts on X with $${U}_{1} \cap{U}_{2}\neq \emptyset$$, then the overlap functions $${\varphi }_{1} \circ{\varphi }_{2}^{-1} : {\varphi }_{2}({U}_{1} \cap{U}_{2}) \rightarrow{\varphi }_{1}({U}_{1} \cap{U}_{2})$$ and $${\varphi }_{2} \circ{\varphi }_{1}^{-1} : {\varphi }_{1}({U}_{1} \cap{U}_{2}) \rightarrow{\varphi }_{2}({U}_{1} \cap{U}_{2})$$ are necessarily homeomorphisms between open sets in some Euclidean space. In the examples that we have encountered thus far (most notably, spheres and projective spaces) these maps actually satisfy the much stronger condition of being C , that is, their coordinate functions have continuous partial derivatives of all orders and types (see Exercise 1.1.8 and (1.2.4)).
Gregory L. Naber

### Chapter 6. Gauge Fields and Instantons

The Im ℍ-valued 1-form $$\omega = \mathrm{Im}\,(\bar{{q}}^{1}{\mathit{dq}}^{1} +\bar{ {q}}^{2}{\mathit{dq}}^{2})$$ will occupy center stage for much of the remainder of our story. We begin by adopting its two most important properties ((5.9.10) and (5.9.11)) as the defining conditions for a connection on a principal bundle.
Gregory L. Naber

### Appendix A. SU (2) and SO(3)

The special unitary group SU(2) and its alter ego Sp(1) have figured prominently in our story and we have, at several points (e.g., Sections 0.5 and 6.3), intimated that the reasons lie in physics. To understand something of this one must be made aware of a remarkable relationship between SU(2) and the rotation group SO(3). It is this service that we hope to perform for our reader here.
Gregory L. Naber

### Appendix B. Donaldson’s Theorem

The moduli space $$\mathcal{M}$$ of anti-self-dual connections on the Hopf bundle SU(2) → S7 → S4 is a rather complicated object, but we have constructed a remarkably simple picture of it. We have identified $$\mathcal{M}$$ with the open 5-dimensional unit ball B5 in $${\mathbb{R}}^{6}$$. The gauge equivalence class [ω] of the natural connection sets at the center. Moving radially out from [ω] one encounters gauge equivalence classes of connections with field strengths that concentrate more and more at a single point of S4. Adjoining these points at the ends of the radial segments gives the closed 5-dimensional disc D5 which we view as a compactification of the moduli space in which the boundary is a copy of the base manifold S4. In this way the topologies of $$\mathcal{M}$$ and S4 are inextricably tied together. In these final sections we will attempt a very broad sketch of how Donaldson [Don] generalized this picture to prove an extraordinary theorem about the topology of smooth 4-manifolds. The details are quite beyond the modest means we have at our disposal and even a bare statement of the facts is possible only if we appeal to a substantial menu of results from topology, geometry and analysis that lie in greater depths than those we have plumbed here. What follows then is nothing more than a roadmap. Those intrigued enough to explore the territory in earnest will want to move on next to [FU] and [Law].
Gregory L. Naber

### Backmatter

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