Computing Interesting Topological Features

Finally, we examine a more fundamental homotopy problem in a different setting. A Rips complex is a simplicial complex defined by a set of points from some metric space where every pair of points within distance 1 is connected by an edge, and every (k + 1)-clique in that graph forms a k-simplex. We prove that the projection map which takes each k-simplex in the Rips complex to the convex hull of the original points in the plane induces an isomorphism between the fundamental groups of both spaces. Since the union of these convex hulls is a polygonal region in the plane, possibly with holes, our result implies that the fundamental group of a planar Rips complex is a free group, allowing us to design efficient algorithms to answer homotopy questions in planar Rips complexes.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Chambers, Erin Wolf

Contributors dc:contributor

• Jeff Erickson

• Computer Science

Language dc:language

eng

Identifier

(MiAaPQ)AAI3337719

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/81824