Proximal Planar Shapes. Correspondence between Shapes and Nerve Complexes

TL;DR

This paper explores the relationship between proximal planar shapes and nerve complexes, demonstrating that a shape nerve complex is homotopy equivalent to the union of its sub-complexes, linking shape topology with nerve structures.

Contribution

It introduces the concept of shape nerve complexes and proves their homotopy equivalence to unions of sub-complexes, advancing shape analysis using nerve theory.

Findings

Shape nerve complexes are homotopy equivalent to their sub-complex unions.

Proximal shape nerves are collections of intersecting 2-simplexes.

The approach links shape topology with nerve complex theory.

Abstract

This article considers proximal planar shapes in terms of the proximity of shape nerves and shape nerve complexes. A shape nerve is collection of 2-simplexes with nonempty intersection on a triangulated shape space. A planar shape is a shape nerve complex, which is a collection of shape nerves that have nonempty intersection. A main result in this paper is the homotopy equivalence of a planar shape nerve complex and the union of its nerve sub-complexes.