Fundamental Groups of Random Clique Complexes
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Fundamental Groups of Random Clique Complexes

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https://arxiv.org/pdf/1207.5028.pdf
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Abstract

Clique complexes of Erd\H{o}s-R {e}nyi random graphs with edge probability between $n^{-{1\over 3}}$ and $n^{-{1\over 2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex for which every subcomplex has fewer than three times as many edges as vertices must have the homotopy type of a wedge of circles, two spheres and real projective planes. Note that $n^{-{1\over 3}}$ is a threshold for simple connectivity and $n^{-{1\over 2}}$ is one for vanishing first $\F_2$ homology.

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