Better Understanding of Betti Numbers

Institute: SAMSI     February 2021

A simplicial complex is a set formed from points, line segments, triangles, and their higher-dimensional analogs. Betti numbers classify topological spaces according to the connectivity of simplicial complexes. The kth Betti number is the rank of the kth homology group and is the maximum number of cuts that can be made before separating a surface into two pieces. For some value n, Betti numbers are zero for all k > n and n are less than or equal to the dimension of the space.

Historically, Betti numbers were named by Henri Poincare after the Italian mathematician Enrico Betti. The modern formulation was given by Emmy Noether. Heuristically, the kth Betti number is the number of k-dimensional holes in a surface. Specifically, the zeroth Betti number counts the number of connected components, β 1 is the number of one-dimensional holes, β 2 is the number of two-dimensional holes, and so forth.

phasefig

The figure above provides a summary of what is known about scaling limits and percolation of Betti numbers of random simplicial complexes on manifolds. It identifies three regimes, and the supercritical phase is a subject of active study by a working group in SAMSI’s 2021 program on Combinatorial Probability.