## Abstract

The fundamental group of the 2-dimensional Linial–Meshulam random simplicial complex Y_{2}(n, p) was first studied by Babson, Hoffman, and Kahle. They proved that the threshold probability for simple connectivity of Y_{2}(n, p) is about p≈ n^{- 1 / 2}. In this paper, we show that this threshold probability is at most p≤ (γn) ^{- 1 / 2}, where γ= 4 ^{4}/ 3 ^{3}, and conjecture that this threshold is sharp. In fact, we show that p= (γn) ^{- 1 / 2} is a sharp threshold probability for the stronger property that every cycle of length 3 is the boundary of a subcomplex of Y_{2}(n, p) that is homeomorphic to a disk. Our proof uses the Poisson paradigm, and relies on a classical result of Tutte on the enumeration of planar triangulations.

Original language | American English |
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Pages (from-to) | 17-32 |

Number of pages | 16 |

Journal | Discrete and Computational Geometry |

Volume | 67 |

Issue number | 1 |

State | Published - Jan 2022 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Poisson paradigm
- Simple connectivity
- Simplicial complexes
- Threshold probability
- Triangulations