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. Informally, the Betti number is the maximum number
of cuts that can be made without dividing a surface into two separate pieces (Gardner
1984, pp. 9-10). Formally, the 
th Betti number is the rank of the 
th homology group of a topological space.
The first Betti number of a graph is commonly known as its circuit rank (or cyclomatic number).
The following table gives the Betti number of some common surfaces.
| surface | Betti number |
| cross-cap | 1 |
| cylinder | 1 |
| Klein bottle | 2 |
| lamina | 0 |
| Möbius strip | 1 |
| projective plane | 1 |
| sphere | 0 |
| torus | 2 |
Let 
be the group
rank of the homology group 
of a topological space 
. For a closed, orientable surface of
genus 
, the Betti numbers are 
, 
, and 
. For a 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 if 
is a polynomial ring over
a field.
