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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?

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The short answer is that there are too many generalizations to list. One very general example is the Grothendieck group of a triangulated category. –  Dan Petersen May 30 '12 at 12:44
    
Thank you. I'm mostly interested about the generalizations with genus. –  Jorma Kyppö May 30 '12 at 13:22
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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. –  Qiaochu Yuan May 30 '12 at 20:55
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1 Answer 1

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.
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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. –  user24105 May 30 '12 at 20:29
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