a Betti number

Strip a homology group down to its simplest numerical content and you get a Betti number: the count of independent n-dimensional holes in a space, ignoring torsion. b_0 counts connected components, b_1 counts independent loops, b_2 counts independent cavities, and so on. They are the oldest and most tangible numerical invariants of a space, named for Enrico Betti who studied them before homology was an algebraic theory.

Precisely, the n-th Betti number b_n is the rank of the n-th homology group: write H_n(X) as a finitely generated abelian group, which decomposes as a free part Z^{b_n} plus a finite torsion part; then b_n is the number of Z-summands, the free rank. Equivalently, b_n is the dimension of the rational homology vector space H_n(X; Q) — passing to rational coefficients kills the torsion and leaves a vector space whose dimension is exactly b_n. So 'Betti number' means 'how many free generators', deliberately throwing away the torsion that distinguishes, say, RP^2 from a point in degree 1.

Betti numbers assemble into the Euler characteristic by an alternating sum: chi(X) = b_0 - b_1 + b_2 - b_3 + ..., a single integer that is itself a homotopy invariant and that for a surface equals 2 - 2g (genus g) or relates to the number of cells by V - E + F. This bridge — the alternating sum of Betti numbers equals the alternating sum of cell counts — is one of the most useful sanity checks in all of topology, since the cell counts are easy and the Betti numbers are the goal.

The honest caveat is precisely what Betti numbers discard: torsion. Two spaces can have identical Betti numbers in every degree yet different integral homology, because one carries torsion the Betti numbers cannot see. The Klein bottle and the torus differ in H_1's torsion but share b_1 = 1 issues only after you look carefully — never claim Betti numbers determine homology; they determine only its free rank.