Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

Gigantic random simplicial complexes

Pages: 297 – 318

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a17

Authors

Jens Grygierek (Institut für Mathematik,Universität Osnabrück, Germany)

Martina Juhnke-Kubitzke (Institut für Mathematik,Universität Osnabrück, Germany)

Matthias Reitzner (Institut für Mathematik,Universität Osnabrück, Germany)

Tim Römer (Institut für Mathematik,Universität Osnabrück, Germany)

Oliver Röndigs (Institut für Mathematik,Universität Osnabrück, Germany)

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.

Keywords

random simplicial complex, Betti numbers, Poisson point process

2010 Mathematics Subject Classification

05E45, 55U10, 60B99, 60D05

Received 9 November 2018

Received revised 4 July 2019

Accepted 10 July 2019

Published 11 December 2019