The Fundamental Group of 2-Dimensional Random Cubical Complexes

Explore the fundamental group of 2-dimensional random cubical complexes in this 27-minute lecture by Érika Roldán Roa. Delve into the study of cubical complexes containing the complete 1-skeleton of the d-dimensional cube, where 2-dimensional square faces are added independently with probability p. Examine these structures as cubical analogues of Linial–Meshulam random simplicial complexes and 2-dimensional versions of bond percolation on the hypercube. Discover the main result showing that for p ≤ 1/2, the fundamental group of a random cubical complex is likely nontrivial, while for p < 1/2, it is likely trivial. Learn about the implications for homology with any coefficient ring and investigate the structure of the fundamental group below the transition point, focusing on its free factorization.