Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Lior Aronshtam; Nathan Linial; Tomasz Łuczak; Roy Meshulam

doi:10.1007/s00454-012-9483-8

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Lior Aronshtam, Nathan Linial^*, Tomasz Łuczak, Roy Meshulam

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

48 Scopus citations

Abstract

Let Δ_n-1 denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δ_n-1 obtained by starting with the full (d - 1)-dimensional skeleton of Δ_n-1 and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γ_d, with γ₂ ≃ 2. 45, γ₃ ≃ 3.5, and γ_d = Θ (log d) as d → ∞, so that for c < γ_d such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δ_d+1, the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant γ_d < c_d < d + 1 such that for any c > c_d and a fixed field F, asymptotically almost surely H_d(Y;F) ≠ 0.

Original language	English
Pages (from-to)	317-334
Number of pages	18
Journal	Discrete and Computational Geometry
Volume	49
Issue number	2
DOIs	https://doi.org/10.1007/s00454-012-9483-8
State	Published - Mar 2013

Bibliographical note

Funding Information:
N. Linial was supported by ISF and BSF grants; T. Łuczak was supported by the Foundation for Polish Science; and R. Meshulam was supported by ISF grant with additional partial support from ERC Advanced Research Grant no. 267165 (DISCONV).

Keywords

Collapsibility
Random complexes
Simplicial homology

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10.1007/s00454-012-9483-8

Cite this

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title = "Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes",

abstract = "Let Δn-1 denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δn-1 obtained by starting with the full (d - 1)-dimensional skeleton of Δn-1 and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γd, with γ2 ≃ 2. 45, γ3 ≃ 3.5, and γd = Θ (log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δd+1, the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd(Y;F) ≠ 0.",

keywords = "Collapsibility, Random complexes, Simplicial homology",

author = "Lior Aronshtam and Nathan Linial and Tomasz {\L}uczak and Roy Meshulam",

note = "Funding Information: N. Linial was supported by ISF and BSF grants; T. {\L}uczak was supported by the Foundation for Polish Science; and R. Meshulam was supported by ISF grant with additional partial support from ERC Advanced Research Grant no. 267165 (DISCONV).",

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AU - Łuczak, Tomasz

AU - Meshulam, Roy

N1 - Funding Information: N. Linial was supported by ISF and BSF grants; T. Łuczak was supported by the Foundation for Polish Science; and R. Meshulam was supported by ISF grant with additional partial support from ERC Advanced Research Grant no. 267165 (DISCONV).

PY - 2013/3

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N2 - Let Δn-1 denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δn-1 obtained by starting with the full (d - 1)-dimensional skeleton of Δn-1 and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γd, with γ2 ≃ 2. 45, γ3 ≃ 3.5, and γd = Θ (log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δd+1, the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd(Y;F) ≠ 0.

AB - Let Δn-1 denote the (n - 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of Δn-1 obtained by starting with the full (d - 1)-dimensional skeleton of Δn-1 and then adding each d-simplex independently with probability p = c/n. We compute an explicit constant γd, with γ2 ≃ 2. 45, γ3 ≃ 3.5, and γd = Θ (log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d - 1)-dimensional subcomplex or it contains ∂ Δd+1, the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd(Y;F) ≠ 0.

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KW - Random complexes

KW - Simplicial homology

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Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

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