A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.
For a space with finite-dimensional homology over a specified field and only finitely many nonzero Betti numbers, this polynomial records their graded dimensions. Its value at one is the total Betti number, and its value at minus one is the Euler characteristic. The Künneth theorem gives over a field. For integer homology ranks, use the rational Betti numbers.
Articles by others on the same topic
In algebraic topology, Betti numbers are a sequence of integers that provide important information about the topology of a topological space. They are used to classify spaces based on their connectivity properties and to understand their shape and structure. Specifically, the \(n\)-th Betti number, denoted \(b_n\), represents the rank of the \(n\)-th homology group \(H_n(X)\) of a topological space \(X\).