Star clusters in clique complexes and the Vietoris-Rips complex of planar sets

BARMAK, JONATHAN ARIEL

Seminario; AATRN Vietoris-Rips seminar; 2021

Abstract: The star cluster of a simplex in a simplicial complex K is the union of the stars of its vertices. When K is clique, star clusters are contractible. We will recall applications of this notion to the study of homotopy invariants of independence complexes of graphs.If X is a planar set, its Vietoris-Rips complex R(X) has free fundamental group (Chambers-de Silva-Erickson-Ghrist) and it is conjectured to be up to homotopy a disjoint union of wedges of spheres.We will show how to use star clusters to obtain some homotopy information of R(X). This is part of an old (2011), modest, unpublished result in collaboration with Martin Tancer.