Why Topology?

Felix Klein developed the idea that a geometry was defined in terms of its invariant properties relative to a group of transformations. Euclidean geometry has dominated classical thinking, for it is the geometry constructed from the invariants of translation and rotations. That is, Euclidean geometry is the geometry of rigid body motion. Two obvious invariants are size and shape, but there are others. Such processes of translation and rotations are defined as diffeomorphisms. The domain of a translation is unrestricted, and all points "move". The motion is called transitive. On the other hand, a rotation has a fixed point. The motion is intransitive.

Consider now an elastic object, say a thin rubber notebook sheet of paper with three holes in it. Certainly rigid motions of rotation and translation do not change the size and shape of the object. However (ignoring the finite thickness of the sheet) bending of the sheet changes its shape, but not its size. That is, a connected set of points between a point A and a point B on the object establish a length or distance that does not change when the object is bent. This idea of length is often defined in terms a quadratic form as a "metrical" idea.

Uniform expansion changes the size but not the shape of the object. The distance between point A and B changes as the object is stretched, so the metric idea of length is not constant. (It is important to remember this fact later on when a primitive concept of "Covariant" differentiation is defined in terms of preserving the line element - a length. Such a constrained differentiation process, constructed in terms of a metric, will cover bending, but cannot include stretching. Those interested in General Relativity, take note. A more general definition of "Covariant" differentiation is based upon the notion of a "Connection", and does not depended upon the existence of a metrical distance. The idea of a connection is to define a differential process such that when it operates on a tensor, the resultant object is also a tensor. A more inclusive concept of differentiation is needed to include stretching. The Lie derivative permits the concept of stretching, and yet preserves tensorial properties.) A general stretching can change both size and shape, but all of these things leave one property invariant. The number of holes does not change.

The number of holes, not their size or shape, is a topological property. A topological property is an invariant of a special class of processes defined as homeomorphisms. A special subclass of such processes are smooth deformations (no tearing apart or cutting is permitted, but glueing together or pasting is OK). The key idea is that homeomorphic processes that preserve topological properties are continous, and have an inverse which is continuous. (Be alert to the fact that is meant by "continuous" in topology is a bit more subtle than the intuitive idea of simple deformation. For physical purposes, perhaps the best physical definition of continuity is based on the idea that a "process" or map is continuous iff the limit points of the domain permute into the closure of the range. )

The great bulk of classical physics is restricted to the study of geometric properties. Classic Tensor analysis limits the maps of interest to diffeomorphisms, which are special types of constrained homoeomorphisms. Almost all of classical mechanics studies systems and processes of invariant topology. In almost every case, these highly developed theories are REVERSIBLE. The parts of science that are controversial involve explanations of events where the topology of the initial state is not the same as the topology of the final state -- the real irreversible world. then the question is not one of geometry, but instead can be phrased as:

WHAT ARE THE LAWS OF TOPOLOGICAL EVOLUTION?

Cartan's theory of exterior differential forms can give part of the answer, for it appears that Cartan's methods can be applied to problems of continuous topological evolution. Such problems do not have unique continuous inverses. Yet, by using the methods of functional substitution and the pullback (RETRODICTION) some headway can be made in the understanding of irreversible phenomena. For example, Cartan's methods may be used to say something about the decay of turbulence, as a continuous irreversible process (think glueing together). The creation of turbulence (think discontinuous punctures or tearing into parts) is as of yet beyond current knowledge.

The idea is to learn about topology and topological properties, for when it is recognized that topology has changed during a process, then a signal has been given that such a process is irreversible in a thermodynamic sense. Irreversibility, up to now, has eluded physical theories, except in a statistical sense.

 Topology and Physics

 Continuous Topological Evolution

 As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.