Felix Klein developed the idea that a geometry was defined in terms of its invariant properties. 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.

Consider now an elastic object, say a 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 bending of the sheet changes it shape, but not its size. Uniform expansion changes it size but not its shape. 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 smooth deformation (no tearing apart or glueing together is permitted). Such processes are defined as homeomorphisms. The key idea is that such processes that preserve topological properties are continous, and have an inverse which is continuous.

The great bulk of classical physics is restricted to the study of geometric properties. Classic Tensor analysis assumes a domain that is covered by diffeomorphisms. Almost all of classical mechanics studies systems of invariant topology. In almost every case, the 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; where the process does not have a inverse. If the scientific objective is to understand the real irreversible world, then

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.

Point Set Topology Notes 1

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