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Homological Domination in Large Random Simplicial Complexes · Around

2015

Homological Domination in Large Random Simplicial Complexes

Costa, A., Farber, M.

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In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions.

We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres.
These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.

Built on

P. Erdős, A. Rényi, On the evolution of random graphs
1960
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N. Dunfield and W. P. Thurston, Finite covers of random 3-manifolds
2006
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N. Linial, R. Meshulam, Homological connectivity of random 2 2 -complexes
2006
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N. Pippenger and K. Schleich, Topological characteristics of random triangulated surfaces
2006
Earlier work this paper cites.
M. Farber, Topology of random linkages
2008
Earlier work this paper cites.

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Then

A. E. Costa, M. Farber, T. Kappeler, Topics of stochastic algebraic topology
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