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:

Read more about this topic:  Betti Number

Famous quotes containing the word examples:

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)