Gigantic random simplicial complexes

TL;DR

This paper constructs a gigantic random simplicial complex from a Poisson point process, which almost surely contains infinitely many copies of every compact topological manifold, highlighting its vast topological diversity.

Contribution

It introduces a new model of random simplicial complexes with unprecedented topological richness, including infinite copies of all compact manifolds.

Findings

Contains infinitely many copies of every compact manifold

Almost surely exhibits complex topological structures

Demonstrates the vastness of the constructed complex

Abstract

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.