Betti Number - Examples

Examples

  1. The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
    the Poincaré polynomial is
    .
  2. The Betti number sequence for a two-torus is 1, 2, 1, 0, 0, 0, ...;
    the Poincaré polynomial is
    .
  3. The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... .
    the Poincaré polynomial is
    .
  4. Similarly, for an n-torus,
    the Poincaré polynomial is
    (by the Künneth theorem), so the Betti numbers are the binomial coefficients.

It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers. An example is the infinite-dimensional complex projective space, with sequence 1, 0, 1, 0, 1, ... that is periodic, with period length 2. In this case the Poincaré function is not a polynomial but rather an infinite series

,

which, being a geometric series, can be expressed as the rational function

More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above (e.g., has generating function

),

and more generally linear recursive sequences are exactly the sequences generated by rational functions; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence.

The Poincaré polynomials of the compact simple Lie groups are:

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