PrefaceThis book is about the physics of spacetime at a deep and fundamental level, encoded in the mathematical assumptions of differentiability. We understand the phrase, "physics of spacetime" in the sense that general relativity has taught us, namely that the geometry, the topology, and now the smoothness of our spacetime mathematical models have physical significance.Our aim is to introduce some of the exciting developments in the mathematics of differential topology over the last fifteen or twenty years to a wider audience than experts. In particular, we are concerned with the discoveries of "exotic" (sometimes called "fake" or "non-standard" ) smoothness (differentiable) structures on R4 and other topologically simple spaces. We hope to help physicists gain at least a superficial understanding of these results and their potential impact on physical theories involving spacetime models, i.e., all fundamental theories. Diffeomorphisms, the basic morphisms of differential topology, are the mathematical representations of the physical notion of transformations between reference frames. As we have learned from Einstein the investigation of these transformations can lead to deep insights into our physical world, as embodied for example in his General Relativistic theory of spacetime and gravity. What the mathematicians have discovered is that the global properties of diffeomorphisms are not at all trivial, even on topologically trivial spaces, such as R4. Yet in general relativity and other field theories physicists continue to assume that the global covering of such spacetime models with smooth reference frames is trivial. This is strongly reminiscent of the assumption of geometric triviality (flatness) of spacetime physics before Einstein. We hope that by presenting an overview of the mathematical discoveries, we may induce physicists to consider the possible physical significance of this newly discovered wealth Within each section we attempt to accompany the mathematical presentation with parallel narratives of related physical topics as well as more informal "physical" descriptions if feasible. The abundant cross-fertilization of physics and mathematics in recent differential topology makes this endeavor quite natural.Chapter 1 is an introduction, discussing the traditional interaction of physics and mathematics and speculations that this interaction extends to these differential topology results. We also review some "exotic" and unexpected or at least counter-intuitive facts in elementary topology and analysis to provide somewhat easier analogs to the more technically challenging "exotic" mathematics coming later. We then survey possible physical consequences of these unexpected structures. Chapter 2 begins with a review of some of the mathematical tools and techniques of algebraic topology. Chapter 3 concentrates on the notion of "smoothness" as defined by the introduction of differential structures on topological spaces. The field of differential topology is built on these constructions. In Chapter 4 we pr...

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This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic R 4 , containing this class of 3-manifolds naturally. We begin with a topological trivial space, the R 4 , and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic R 4 's generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein-Hilbert action.

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein-Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a mass-less Dirac term in the Einstein-Hilbert action. For sufficient complicated links and knots, there are "connecting tubes" (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1)xSU(2)xSU(3).Comment: 30 pages, 3 figures, svjour style, complete reworking now using Fintushel-Stern knot surgery of elliptic surfaces, discussion of Lorentz metric and global hyperbolicity for exotic 4-manifolds added, final version for publication in Gen. Rel. Grav, small typos errors fixe

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