Topological Shape Theory

Abstract: Kendall's Similarity Shape Theory for constellations of points in the carrier space $\mathbb{R}^n$ was developed for use in Probability and Statistics. It was subsequently shown to reside within (Classical and Quantum) Mechanics' Shape-and-Scale Theory, in which the points are interpreted as particles and the carrier space plays the role of absolute space. In other more recent work, Kendall's Similarity Shape Theory has been generalized to affine, projective, conformal and supersymmetric versions, as well as to $\mathbb{T}^n$, $\mathbb{S}^n$, $\mathbb{RP}^n$ and Minkowski spacetime carrier spaces. This has created a sizeable field of study: generalized Kendall-type Geometrical Shape(-and-Scale) Theory. Aside from offering a wider range of shape-statistical applications, this field of study is an exposition of models of Background Independence of relevance to the Absolute versus Relational Motion Debate, and the Foundations and Dynamics of General Relativity and Quantum Gravity. In the current article, we consider a simpler type of Shape Theory, comprising relatively few types of behaviour: the Topological Shape Theory of rubber shapes. This underlies the above much greater diversity of more structured Shape Theories; in contrast with the latter's (stratified) manifolds shape spaces, the former's are just graphs. We give examples of these graphs for the smallest nontrivial point-or-particle numbers, and outline how these feature within a wider range of Geometric Shape Theories' shape spaces, and are furthermore straightforward to do Statistics, Dynamics and Quantization with.