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Its Euler characteristic is 0, by the product property. More generally, any compact parallelizable manifold, including any compact Lie group, has Euler characteristic 0. [12] The Euler characteristic of any closed odd-dimensional manifold is also 0. [13] The case for orientable examples is a corollary of Poincaré duality.
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.
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
This constant, χ, is the Euler characteristic of the plane. The study and generalization of this equation, specially by Cauchy [9] and Lhuillier, [10] is at the origin of topology. Euler characteristic, which may be generalized to any topological space as the alternating sum of the Betti numbers, naturally arises from homology.
the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology , in this case.
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula + = for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press , with a paperback edition in 2012.
There is an approach to intersection number, introduced by Snapper in 1959-60 and developed later by Cartier and Kleiman, that defines an intersection number as an Euler characteristic. Let X be a scheme over a scheme S , Pic( X ) the Picard group of X and G the Grothendieck group of the category of coherent sheaves on X whose support is proper ...