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The Euler characteristic can be defined for connected plane graphs by the same + formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face. The Euler characteristic of any plane connected graph G is 2.
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
Two special cases worth singling out are the following. If the order of M is relatively prime to the characteristic of the residue field of K, then the Euler characteristic is one. If K is a finite extension of the p-adic numbers Q p, and if v p denotes the p-adic valuation, then
Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and their preimages in S' so that the restriction of is a covering. Removing a disc from a surface lowers its Euler characteristic by 1 by the formula for connected sum, so we finish by the formula for a non-ramified covering.
Euler's formula can also be proved as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers both e and f by one, leaving v – e + f constant. Repeat until the remaining graph is a tree; trees have v = e + 1 and f = 1, yielding v – e + f = 2, i. e., the Euler characteristic is 2.
Noether's formula states that = + = (.) + where χ=χ(0) is the holomorphic Euler characteristic, c 1 2 = (K. K) is a Chern number and the self-intersection number of the canonical class K, and e = c 2 is the topological Euler characteristic.
(The appearance of in the summation is the usual Euler characteristic.) [1] [2] If the action is free ... Kawasaki's Riemann–Roch formula; References
Whitney sum formula: ... In the language of characteristic numbers, the Euler characteristic is the characteristic number corresponding to the Euler class.