On Expansion and Topological Overlap

Dotterrer, Dominic; Kaufman, Tali; Wagner, Uli

doi:10.48550/arxiv.1506.04558

Cited by 3 publications

(7 citation statements)

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“…This definition of expansion is more natural to the world of computer science as it can be viewed as a property testing question, where the property is whether a given set is a cocycle (see [6] for more on high dimensional expansion and property testing). Moreover, this definition of expansion implies a property called the topological overlapping of a complex, which was heavily studied [5,3]. The first known explicit bounded degree high dimensional simplicial complexes according to this definition have been constructed in [11,4].…”

Section: -Dimensional Expandersmentioning

confidence: 99%

Walking on the Edge and Cosystolic Expansion

Kaufman¹,

Mass²

2016

Preprint

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Random walks on regular bounded degree expander graphs have numerous applications.A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional simplicial complexes, which are the high dimensional analogues of graphs, calls for the natural generalization of random walks to higher dimensions. In particular, a high order random walk on a 2-dimensional simplicial complex moves at random between neighboring edges of the complex, where two edges are considered neighbors if they share a common triangle. We show that if a regular 2-dimensional simplicial complex is a cosystolic expander and the underlying graph of the complex has a spectral gap larger than 1/2, then the random walk on the edges of the complex converges rapidly to the uniform distribution on the edges.

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Section: -Dimensional Expandersmentioning

confidence: 99%

Walking on the Edge and Cosystolic Expansion

Kaufman¹,

Mass²

2016

Preprint

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“…As mentioned in the introduction there are several competing definitions for high-dimensional expansion. Without going into details, our model yields expanders with respect to toplogical expansion (see [Gro10,DKW15]), spectral expansion (c.f. [Eck45, Gar73, GW14, KR15]) as well as the Cheeger type expansion defined in [PRT15,Par13].…”

Section: Concluding Remarks and Open Questionsmentioning

confidence: 99%

“…Being coboundary expanders, the complexes are also topological expanders, i.e. they satisfy Gromov's topological overlapping property (see [Gro10,DKW15]). These are the first known coboundary expanders of dimension d ≥ 3 (for d = 2 see [LM15]) of upper bounded degree, i.e., complexes in which the codimension 1 cells have a uniformly bounded degree.…”

Section: Introductionmentioning

confidence: 99%

Random Steiner systems and bounded degree coboundary expanders of every dimension

Lubotzky¹,

Luria²,

Rosenthal³

2015

Preprint

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We introduce a new model of random d-dimensional simplicial complexes, for d ≥ 2, whose (d − 1)-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs [Kee14], and the proof of the expansion uses techniques developed by Evra and Kaufman in [EK15]. This gives a full solution to a question raised in [DK12], which was solved in the two-dimensional case by Lubotzky and Meshulam [LM15].

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“…The boundary of a general k-chain is defined by linearity, ∂(σ Finally, we need to state Gromov's overlap theorem [10] slightly more precisely. We follow the exposition in [8]. A k-cochain in X is a F 2 -linear function on C k (X).…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 6 (Gromov's overlap theorem, following [8]). Suppose X is a finite simplicial complex of dimension d, and let • be defined by (2).…”

Section: Introductionmentioning

confidence: 99%

On a topological version of Pach's overlap theorem

Bukh

Hubard

2019

Bull. London Math. Soc.

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Pach showed that every d + 1 sets of points Q1, . . . , Q d+1 ⊂ R d contain linearly sized subsets Pi ⊂ Qi such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C(log n) 1/(d−1) . We show that this is tight in dimension 2, for all surfaces other than S 2 . Surprisingly, the optimal bound for S 2 in the topological version of Pach's theorem is of the order (log n) 1/2 . We conjecture that, among higher dimensional manifolds, spheres are similarly distinguished. This improves upon the results of Bárány, Meshulam, Nevo and Tancer.

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On Expansion and Topological Overlap

Cited by 3 publications

References 0 publications

Walking on the Edge and Cosystolic Expansion

Walking on the Edge and Cosystolic Expansion

Random Steiner systems and bounded degree coboundary expanders of every dimension

On a topological version of Pach's overlap theorem

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