Betti numbers are topological invariants that describe the number of holes at different dimensions in a topological space, commonly used in algebraic topology and combinatorial algebra. They provide a way to quantify the connectivity of a space, where the $i$-th Betti number counts the number of $i$-dimensional holes. In the context of monomial ideals and Stanley-Reisner rings, Betti numbers relate to the structure of these ideals and can be interpreted via the simplicial complex associated with the ring. For Gröbner bases and initial ideals, Betti numbers help analyze the syzygies of modules, reflecting the relationships among generators of these ideals.
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