Bachelor Thesis in Mathematics

A Vietoris–Rips complex R is the clique complex of a proximity graph. This graph can arise from a finite set of points in Euclidean n-space, and then there is a natural projection onto its n-dimensional shadow; p : R æ S. An article by Chambers et al.[Cha+10] demonstrates that p induces an isomorphism of fundamental groups for n = 1, 2, while the induced homomorphism fails to be surjective for n Ø 4. In the case of n = 3, neither injectivity nor surjectivity are known. The existing proof of surjectivity for n = 2 proceeds by lifting paths from S to R. In this report we propose a new proof of surjectivity for n = 2. We proceed by induction on the vertex set, thus we examine the geometry of the shadow in the neighborhood of a single vertex. At this vertex we can make decompositions of R and S to apply the Seifert–van Kampen theorem. This yields a diagram of fundamental groups, from which surjectivity follows by a category-theoretic argument.