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Proximal planar shapes. Correspondence between triangulated shapes and nerve complexes. (English) Zbl 1400.54041

In this paper triangulated planar shapes are considered. They represent a collection of shape nerves that have non-empty intersection. A homotopy-equivalence of a planar shape nerve complex and the union of its nerve sub-complexes is given. This important result offers a proximal computational topology approach to shapes. Here, computational topology means the combination of geometry, topology and algorithms in the study of topological structures and is going back to Edelsbrunner and Harer. At the end of this paper 5 open problems in shape theory are listed.

MSC:

54E05 Proximity structures and generalizations
54C56 Shape theory in general topology
05B45 Combinatorial aspects of tessellation and tiling problems
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
55P10 Homotopy equivalences in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
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