Examples
- The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
- the Poincaré polynomial is
- .
- the Poincaré polynomial is
- The Betti number sequence for a two-torus is 1, 2, 1, 0, 0, 0, ...;
- the Poincaré polynomial is
- .
- the Poincaré polynomial is
- The Betti number sequence for a three-torus is 1, 3, 3, 1, 0, 0, 0, ... .
- the Poincaré polynomial is
- .
- the Poincaré polynomial is
- Similarly, for an n-torus,
- the Poincaré polynomial is
- (by the Künneth theorem), so the Betti numbers are the binomial coefficients.
- the Poincaré polynomial is
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:
Read more about this topic: Betti Number
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