Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula
to higher dimensional spaces. The 
th Betti number is the rational rank
of the 
th
homology group of a topological
space, computed with integer coefficients.
The first Betti number of a graph is commonly known as its circuit rank (or cyclomatic number).
The following table gives the first Betti number of some common surfaces (Hatcher 2002, p. 141). The lamina is assumed to be simply connected.
| surface | first Betti number |
| cross-cap | 1 |
| cylinder | 1 |
| Klein bottle | 1 |
| lamina | 0 |
| Möbius strip | 1 |
| projective plane | 0 |
| sphere | 0 |
| torus | 2 |
Let 
be the rational
rank of the homology group 
of a topological space 
. For a closed, connected orientable surface of genus 
, the Betti numbers are 
, 
, and 
. For a closed, connected nonorientable surface with 
cross-caps, the Betti numbers
are 
, 
, and 
.
The Betti number of a finitely generated Abelian group 
is the (uniquely determined) number 
such that
 |
where 
,
..., 
are finite cyclic groups
(see Kronecker decomposition theorem).
The Betti numbers of a finitely generated module 
over a commutative Noetherian local unit ring 
are the minimal numbers 
for which there exists a long
exact sequence
 |
which is called a minimal free resolution of 
. The Betti numbers are uniquely determined
by requiring that 
be the minimal number of generators of 
for all 
. These Betti numbers are defined in the same way for
finitely generated positively graded
modules over 
if 
is a polynomial
ring over a field.
