Contents of the Book

• Introduction and background
• The interaction between Physics and Mathematics
              We start with a short summary of the strong interaction between physics and mathemtics in the last 30 years.
                It is also a short introduction into the history of the  theory of manifolds and the increasing interest of physicists
                on this problem.
 
• Mathematical concepts in physics
• Manifolds and Smooth Structures
               We review the notion of topological, then smooth manifolds with special concern for the corresponding equivalences: TOP and DIFF.
                The members of the latter correspond to the physical notion of ``coordinate transformation,'' so smooth manifold theory really lies
                at the foundation of the general theory of relativity.
 
• Fiber Bundles and Geometry
                The idea of a fiber bundle over a manifold generalizes the concepts inherent in tangent vector fields to objects that can serve as models
                 for physical fields. The principal bundle of frames serves as the natural framework for differential geometry (general relativity) and its
                 principal bundle generalizations (with their natural bundle equivalences) definegauge theories, so central in contemporary physics.
 
• Different versus non-diffeomorphic Smoothness Structures
• Complex structures on  as a toy model
                The distinction between different and non-diffeomorphic smoothness structures can be a very difficult concept to grasp, so we start with
                a more easily constructed model of inequivalent complex structures on . It is even possible to tie this in to plane vacuum electrostatics as
                a toy physical model in which different ``structures,'' (complex in this case) play a physical role.
 
• First Statements
                We begin with some short examples of different smooth structures on  with n=1,2,3, and point out the triviality of the problem in these cases.
 
• Exotic examples in topology
• Weierstraß function
                An early example of ``smooth exotica'' is provided by this function, which has the highly counterintuitive property of being everywhere
                continuous but nowhere differentiable.
 
• Whitehead Continuum
                Another classical ``exotic'' object is the Whitehead continuum. That is a contractible, non-compact space M where  is homeomorphic
                to   but M is not homeomorphic to  Admittedly this is a topological rather than smooth effect, but it does serve to open the reader
                to unexpected possibilities.
• Gauge theory and Moduli space
• Connections and curvature
                Building on the bundle foundation laid in 1.2.2 we proceed to the notions of ``connections'' and ``curvature.'' After briefly reviewing Einstein theory,
                we focus on Maxwell's theory in the U(1) formulation, thus pointing out the identification of curvature with field strength.
 
• Classification of Principal Fiber Bundles
                This classic problem pulls together diverse elements from homotopy and homology theory, with culmination in the crucial ``classifying spaces''
                and ``characteristic classes,''next section.
 
• Characteristic Classes
                Here we review the various important characteristic classes: Whitney, Pontrjagin, and Chern and their significance.
 
• Yang-Mills theories
                Starting again with the easily described electromagnetic field, we generalize to the class of non-linear gauge theories generally denoted as ``Yang-Mills,''
                including the role of topology in the action and tools such as instantons and self and anti-self duality.
 
• The Concept of a Moduli space
                The basic idea here is to understand how the set of solutions to certain field equations, can, when gauge is factored out, become a finite dimensional manifold
                (perhaps with singularities). An early and still interesting idea of such moduli space is provided by Wheeler's ``superspace,'' the space of three geometries
                mod diffeomorpishms. We explain the connection of this space to the eight model geometries of Thurston.
 
• Topological techniques
• Handle bodies and surgery
• The general technique
                Handle body tools play a central role in the construction of non-trivial manifolds. Here we provide background necessary to define the concept
                of surgery.
 
• 3-manifolds: Surgery along a knot
                This section will deal with the crucial techniques needed to generate every 3-manifold by gluing 2-handle bodies along knots or links.
 
• Smooth 4-manifolds: Kirby-calculus
                 Here we deal with the important question of determining which handle body decompositions leave invariant the smooth structure of a 4-manifold.
 
• Topological 4-manifolds: Casson handles
                As a prelude to a discussion of Freedman's work we explain the concept of Casson handles used in the construction of topological 4-manifolds.
 
• Morse theory and Cobordism
                Here we review the classical notions of cobordism, Morse functions and their critical points. These tools are important in relating properties of
                co-bordant manifolds.
 
• The h-cobordism theorem
                Continuing with the study of cobordism, we describe the techniques used to classify the smooth structures in dimensions greater than 4,
                including Whitney's trick.
 
• The Failure in dimension 4
                Here we discuss in some detail the reasons for the failure of the higher-dimensional techniques when applied to the physically significant dimension four.
 
• The Intersection form
                If we embed two surfaces in a 4-manifold then we can choose the embedding in such a manner that the surfaces intersect transversely in a countable
                number of points. If these embedded surfaces generate elements of H₂ , the algebraic sum of these (oriented) intersection numbers leads to the intersection
                form, an important topological invariant of a 4-manifold.
 
• Early Exotic Manifolds
• Milnor's Exotic 7-Spheres
                In 1957 Milnor gave the first example of exotic manifolds, a class of topological 7-spheres none of which are diffeomorphic to S⁷ with standard smoothness.
• Geometrical Consequences
                Gromoll, Meyer and Rigas studied the geometric properties of Milnor's exotic S⁷'s, placing restrictions on their sectional curvature. This is an important
                example of potential links between exotic smoothness and curvature properties.
 
• The  higher-dimensional case
                Here we study the properties of manifolds with structures TOP, PL and DIFF for dimensions greater than 5. In the first part of this section we give an
                overview of the classification of differential structures on sphere according to Kervaire and Milnor. Then we generalize with the work of Kirby and Siebenman
                 to the general case.
 
• The First results in dimension 4
• Freedman's work on the topology of 4-manifolds
                In 1981 Freedman classified all simply-connected, compact, topological 4-manifolds by using the Casson handles. We give a short introduction to these results.
 
• Donaldson theory
                Soon after Freedman's work, Donaldson used the gauge-theoretic techniques of anti-selfdual connections to prove certain non-existence results for smooth
                structures on certain 4-manifolds. This beautiful result is explained in some detail because of its crucial importance in the discoveries of  ,
                the exotic   In particular, we focus on the Donaldson polynomials which provide diffeomorphism-invariant information for a 4-manifolds.
 
• The Infinite Proliferation of Exotic 
                We review the work of Gompf and others generating infinite numbers of non-diffeomorphic smooth structures. From the theory of Taubes on manifolds with
                 periodic ends, the cardinality of the set of exotic  increases i.e. there is an uncountable number of exotic . In fact, Freedman and Taylor have
                 constructed a ``universal''  containing all others.
 
• Explicit descriptions
                With the help of Casson handles it is now possible to construct explicit (infinite) handle body decompositions for exotic . We relate these constructions
                  to (infinite) coordinate patch presentations.
 
• Seiberg-Witten theory: the modern approach
• Seiberg-Witten invariants
                The work of Seiberg and Witten on certain N=2 supersymmetric 4-dimensional gauge theories has resulted in an unexpected revolution of differential topology
                in dimension 4. We state and briefly review these equations and the properties of their smoothness invariants.
 
• Gluing formulas
                These are techniques in moduli spaces which correspond to surgery on the underlying 4-manifold.
 
• Changing of smooth structures by surgery
• Elliptic surfaces: Logarithmic transform
                Here we discuss the additional work of Gompf constructing an infinite number of non-diffeomorphic smooth structures on elliptic surfaces using a
                special surgery known as logarithmic transform.
 
• Rational blow-up
                Continuing the logarithmic transform work, we discuss techniques of Fintushel and Stern. We suggest an explicit relationship between the
                Seiberg-Witten invariant and Donaldson polynomials.
 
• The general case: Surgery along knots and links
                Continuing the discussion in 3.1.2, we describe recent results of Fintushel and Stern providing for nearly every simply-connected 4-manifold a homeomorphic
                but non-diffeomorphic 4-manifold using a knot or link.
 
• Physical implications
• The Principle of Relativity
                The underlying principle of general relativity is that the form of the laws of physics must be observer independent. The standard interpretation of coordinate
                transformation (going from one observer to another) is that of a diffeomorphism. Consequently, two manifolds which are not diffeomorphic represent truly
                distinct physical models. If certain smooth fields on standard   could be transferred (with singularities) to those on a  the implications for
                physics would be non-trivial.
 
• Exotic cosmology
                Puncturing a   gives rise to an exotic product  ,  which provides the basis for considering exotic modifications of standard FRW cosmologies.
 
• Extension of Metrics
                Even lacking explicit finite coordinate patch presentations, it is possible to prove certain results about metrics on 's, including their extension from
                standard to global regions and the possibility of localizing the exoticness to a timelike product of time with a compact space-like region. We also look into
                questions of global hyperbolicity, and standard singularity theorems.
 
• Global Anomaly Cancellation of Witten
                Another, early, physical applications was Witten's 1985 paper using exotic S¹⁰'s to cancel global gravitational anomalies in superstring theories.
 
• Speculations

               The physical applications of exotic smoothness are largely unexplored. We speculate on certain possibilities

• Exotic smooth structures and source terms in GRT
                We conjecture that the transition from one manifold to another homeomorphic but not diffeomorphic one can lead to a transfer of one connection to another,
                with singularities, which could correspond to a possible exotic generation of (singular) mass terms in Einstein theory.
 
• Exotic gauge theories
                We investigate the possible interpretation of Milnor's exotic seven-spheres as bundles corresponding to exotic Yang-Mills gauge theories over S⁴,
                 compactified standard .
 
• Appendix
• An historical timeline