Betti numbers are topological invariants that describe the number of independent cycles in a topological space, capturing its connectivity features. They help in understanding the shape and structure of spaces by providing counts of holes in various dimensions: the zeroth Betti number counts connected components, the first counts loops, and higher numbers count higher-dimensional voids. These invariants play a significant role in both induced cohomomorphisms and Hodge theory, revealing deeper relationships between algebraic and geometric properties.
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