Geometry and Topology of Random 2-complexes

2013/07/13 by A. Costa, A. E. Costa, Michael Färber +3 · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.AT #math.GT

paper · pdf · doi:10.48550/arxiv.1307.3614

37 pages

openalex publication_date 2013/07/13 · arxiv created 2014/06/23 · arxiv updated 2014/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies the Whitehead conjecture. We use inequalities for Cheeger constants and systoles of simplicial surfaces to analyse spheres and projective planes lying in random 2-complexes. Our proofs exploit the strong hyperbolicity property of random 2-complexes.

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• Fundamental groups of random clique complexes

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• The asphericity of random 2-dimensional complexes
• Topology of random 2-complexes
• Large Random Simplicial Complexes, II; the fundamental groups
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• On simple connectivity of random 2-complexes