Editing Betti number (section)

== Properties ==

=== Euler characteristic ===
For a finite CW-complex ''K'' we have
: <math>\chi(K) = \sum_{i=0}^\infty(-1)^i b_i(K, F), \,</math>
where <math>\chi(K)</math> denotes [[Euler characteristic]] of ''K'' and any field&nbsp;''F''.

=== Cartesian product ===
For any two spaces ''X'' and ''Y'' we have
: <math>P_{X\times Y} = P_X P_Y ,</math>
where <math>P_X</math> denotes the '''Poincaré polynomial''' of ''X'', (more generally, the [[Hilbert–Poincaré series]], for infinite-dimensional spaces), i.e., the [[generating function]] of the Betti numbers of ''X'':
: <math>P_X(z) = b_0(X) + b_1(X)z + b_2(X)z^2 + \cdots , \,\!</math>
see  [[Künneth theorem]].

=== Symmetry ===
If ''X'' is ''n''-dimensional manifold, there is symmetry interchanging <math>k</math> and <math>n - k</math>, for any <math>k</math>:
: <math>b_k(X) = b_{n-k}(X),</math>
under conditions (a ''closed'' and ''oriented'' manifold); see [[Poincaré duality]].

=== Different coefficients ===
The dependence on the field ''F'' is only through its [[characteristic (field)|characteristic]]. If the homology groups are [[torsion (algebra)|torsion-free]], the Betti numbers are independent of ''F''. The connection of ''p''-torsion and the Betti number for [[characteristic p|characteristic&nbsp;''p'']], for ''p'' a prime number, is given in detail by the [[universal coefficient theorem]] (based on [[Tor functor]]s, but in a simple case).