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Topology of random 2-dimensional cubical complexes

Ponente: Erika Roldán
Institución: Technische Universität München
Tipo de Evento: Investigación
Cuándo 04/02/2022
de 12:00 a 13:00
Dónde ZOOM ID 882 9372 3602
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We study a natural model of random 2-dimensional cubical complexes which are subcomplexes 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 the dimension n goes to infinity. This is a 2-dimensional analogue of the Burtin and Erdős-Spencer theorems characterizing 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 behaviors. 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. This is joint work with Matthew Kahle and Elliot Paquette.