Topology of geometric joins

TL;DR

This paper investigates the topological properties of geometric joins in Euclidean space, proving contractibility in low dimensions and providing bounds for higher dimensions, with extensions to matroid generalizations.

Contribution

It establishes contractibility results for geometric joins in dimensions 2 and 3, and provides bounds for higher dimensions, including a matroid generalization.

Findings

Proved contractibility of geometric joins in dimensions 2 and 3.

Showed geometric join is contractible when the number of sets is quadratic in dimension.

Extended results to a matroid generalization of geometric joins.

Abstract

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case.