Temporal Proximity of 1-cycles in CW Spaces, Time-Varying Cell Complexes

Abstract

This paper introduces approximate temporal proximities of 1-cycle cell complexes in a spacetime view of a planar Whitehead CW space. Divergence of the vector field of a 1-cycle provides a natural basis for an approximate Temporal Proximity (aTP) of time-varying 1-cycles useful in the detection, characterization, analysis, and measurement of the closeness of changing geometric realizations of simplicial complexes in a J.H.C Whitehead CW topological space. A practical application of aTP is given in terms of the temporal closeness of 1-cycle shapes in sequences of video frames. A main result in this paper is that every pair of cell complexes with the same descriptions over the same temporal interval have two properties, namely, (i) persistence and (ii) approximate temporal closeness.