Integral Homology of Random Simplicial Complexes

Łuczak, Tomasz; Peled, Yuval

doi:10.1007/s00454-017-9938-z

Cited by 29 publications

(44 citation statements)

References 16 publications

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“…Moreover, there is an apparent lack of torsion in H 1 (Y (n, c/n)) if c < c 2 . This has not been proved, but extensive experiments conducted in [9] provide evidence to support this, and [14] state the following conjecture regarding torsion in homology:…”

Section: Discussionmentioning

confidence: 99%

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Freeness of the random fundamental group

Newman

2018

J. Topol. Anal.

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Let Y (n, p) denote the probability space of random 2-dimensional simplicial complexes in the Linial-Meshulam model, and let Y ∼ Y (n, p) denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber, and Kappeler, it is shown that for p = o(1/n) with high probability π 1 (Y ) is free. Following that, a paper of Costa and Farber shows that for values of p which satisfy 3/n < p ≪ n −46/47 , with high probability π 1 (Y ) is not free. Here we improve on both of these results to show that there are explicit constants γ 2

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Section: Discussionmentioning

confidence: 99%

“…Conjecture (2-dimensional case of the conjecture from [14]). For every p = p(n) such that |np−c 2 | is bounded away from 0, H 1 (Y d (n, p); Z) is torsion-free with high probability.…”

Section: Discussionmentioning

confidence: 99%

Freeness of the random fundamental group

Newman

2018

J. Topol. Anal.

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“…This is due to Linial and Meshulam [22], for the statement with coefficients. This was extended by Meshulam and Wallach [26] to all finite coefficient rings, and finally to coefficients by Łuczak and Peled [24]; see also [28] for the extension of the -coefficients result to higher dimensions. In summary, there is a big gap between the thresholds for and , but Theorem 1.2 implies that the thresholds for and (with any coefficients) coincide.…”

Section: Introductionmentioning

confidence: 99%

Topology of random -dimensional cubical complexes

Kahle

Paquette

Roldán

2021

Forum of Mathematics, Sigma

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We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

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“…This phenomenon of enormous torsion in homology in this random setting has been observed experimentally, for example by [11] and by [6] but the reason it occurs remains unknown. Nevertheless, Table 2 provides examples of randomly generated simplicial complexes with torsion in homology coming from the Linial-Meshulam torsion burst.…”

Section: Introductionmentioning

confidence: 76%

Small Simplicial Complexes with Prescribed Torsion in Homology

Newman

2018

Discrete Comput Geom

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For d ≥ 2 and G a finite abelian group, define T d (G) to be the minimum number of vertices n so that there exists a simplicial complex X on n vertices which has the torsion part of H d−1 (X) isomorphic to G. Here we use the probabilistic method, in particular the Lovász Local Lemma, to establish an upper bound on T d (G) which matches the known lower bound up to a constant factor. That is, we prove that for every d ≥ 2 there exist constants c d and C d so that for any finite abelian group

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Integral Homology of Random Simplicial Complexes

Cited by 29 publications

References 16 publications

Freeness of the random fundamental group

Freeness of the random fundamental group

Topology of random -dimensional cubical complexes

Small Simplicial Complexes with Prescribed Torsion in Homology

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