Homological connectivity of random \(k\)-dimensional complexes.

*(English)*Zbl 1177.55011The authors study the question of homological \((k-1)\)-connectivity of random \(k\)-dimensional complexes for a general \(k\). The one-dimensional case (that is, connectivity of random graphs) was considered in [P. Erdös and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci., Ser. A 5, 17–61 (1960; Zbl 0103.16301)], and the two-dimensinal case was considered in [N. Linial and R. Meshulam, Combinatorica 26, No. 4, 475–487 (2006; Zbl 1121.55013)].

To describe the result obtained in the paper under review, let \(\Delta_{n-1}\) denote the \((n-1)\)-dimensional simplex. The authors consider the space \(Y_k(n,p)\) of complexes \(\Delta_{n-1}^{(k-1)}\subset Y\subset \Delta_{n-1}^{(k)}\) equipped with some adapted probability measure. A \((k-1)\)-simplex \(\sigma\in \Delta_{n-1}(k-1)\) is said to be isolated in \(Y\) if it is not contained in any of the \((k-1)\)-simplices of \(Y\). For such an isolated simplex \(\sigma\), its indicator function is a non-trivial \((k-1)\)-cocycle of \(Y\), hence \(H^{k-1}(Y)\not=0\). The main result of the paper says that the threshold probability for the vanishing of \(H^{k-1}(Y)\) coincides with the threshold for the non-existence of isolated \((k-1)\)-simplices in \(Y\).

To describe the result obtained in the paper under review, let \(\Delta_{n-1}\) denote the \((n-1)\)-dimensional simplex. The authors consider the space \(Y_k(n,p)\) of complexes \(\Delta_{n-1}^{(k-1)}\subset Y\subset \Delta_{n-1}^{(k)}\) equipped with some adapted probability measure. A \((k-1)\)-simplex \(\sigma\in \Delta_{n-1}(k-1)\) is said to be isolated in \(Y\) if it is not contained in any of the \((k-1)\)-simplices of \(Y\). For such an isolated simplex \(\sigma\), its indicator function is a non-trivial \((k-1)\)-cocycle of \(Y\), hence \(H^{k-1}(Y)\not=0\). The main result of the paper says that the threshold probability for the vanishing of \(H^{k-1}(Y)\) coincides with the threshold for the non-existence of isolated \((k-1)\)-simplices in \(Y\).

Reviewer: Athanase Papadopoulos (Strasbourg)

##### MSC:

55U10 | Simplicial sets and complexes in algebraic topology |

60B99 | Probability theory on algebraic and topological structures |

05C80 | Random graphs (graph-theoretic aspects) |

##### References:

[1] | Alon, The probabilistic method (2000) |

[2] | Dunfield, Finite covers of random 3-manifolds, Invent Math 166 pp 457– (2006) · Zbl 1111.57013 |

[3] | Erdös, On the evolution of random graphs, Publ Math Inst Hungar Acad Sci 5 pp 17– (1960) |

[4] | Linial, Homological connectivity of random 2-complexes, Combinatorica 26 pp 475– (2006) · Zbl 1121.55013 |

[5] | Munkres, Elements of algebraic topology (1984) |

[6] | Pippenger, Topological characteristics of random triangulated surfaces, Random Struct Algorithms 28 pp 247– (2006) · Zbl 1145.52009 |

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