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  2. Euler characteristic - Wikipedia

    en.wikipedia.org/wiki/Euler_characteristic

    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.

  3. Riemann–Hurwitz formula - Wikipedia

    en.wikipedia.org/wiki/Riemann–Hurwitz_formula

    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.

  4. Planar graph - Wikipedia

    en.wikipedia.org/wiki/Planar_graph

    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.

  5. Local Euler characteristic formula - Wikipedia

    en.wikipedia.org/wiki/Local_Euler_characteristic...

    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

  6. Contributions of Leonhard Euler to mathematics - Wikipedia

    en.wikipedia.org/wiki/Contributions_of_Leonhard...

    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.

  7. Geometric function theory - Wikipedia

    en.wikipedia.org/wiki/Geometric_function_theory

    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.

  8. Euler's Gem - Wikipedia

    en.wikipedia.org/wiki/Euler's_Gem

    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.

  9. Intersection number - Wikipedia

    en.wikipedia.org/wiki/Intersection_number

    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 ...