0
$\begingroup$

Euler's equation for polyhedral, Euler's polyhedral formula, V – E + F = 2, where V, E, and F, are the number of points, edges and faces, was discovered by Leonhard Euler in 1752. However, the basic idea - not the equation - was revealed much earlier by Descartes & others, and later generalized by Lhuilier, as follow: V – E + F = 2 – 2g, where g is genus, the number of holes or handles. Later on, Schläfli and Poincare also generalized the formula to the higher dimensional n-polytopes. We talk about Euler-Poincare formula and Euler-Poincare characteristic (X) for combinatorial cell complexes or polyhedral solids: X = N1 – N2 + N3 – N4 + … +/- Nk, where k is the dimension of the complex Nk and X = 2, if k is odd, or 0, if k is even.

My question is: Are there any other/later generalizations of this Euler-Poincare characteristic of a cell complex?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ The short answer is that there are too many generalizations to list. One very general example is the Grothendieck group of a triangulated category. $\endgroup$
    – Dan Petersen
    Commented May 30, 2012 at 12:44
  • $\begingroup$ Thank you. I'm mostly interested about the generalizations with genus. $\endgroup$
    – Jorma Kyppö
    Commented May 30, 2012 at 13:22
  • 2
    $\begingroup$ The genus computation is a special case of the fact that the Euler characteristic of a polytope is the alternating sum of the dimensions of its homology groups. This generalizes to bounded chain complexes of finite-dimensional vector spaces. $\endgroup$
    – Qiaochu Yuan
    Commented May 30, 2012 at 20:55

1 Answer 1

Reset to default
1
$\begingroup$

The Wikipedia article discusses (and provides some references for) several generalizations:

  • To a chain complex, when the Euler characteristic is the alternating sum of the ranks of the homology groups of the chain complex.
  • To a sheaf on a projective scheme.
  • To an orbifold, which may have a fractional Euler characteristic.
  • To a bounded finite poset.
  • To a finite group or monoid.
Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you for your answer. However, I did already study this interesting Wikipedia article. Probably I didn't focus my question enough (first time in mathoverflow). I'm mostly interested about the generalizations, where also the value of genus is aproximated. $\endgroup$
    – user24105
    Commented May 30, 2012 at 20:29

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .