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  2. Euler characteristic - Wikipedia

    en.wikipedia.org/wiki/Euler_characteristic

    This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .

  3. Euler characteristic of an orbifold - Wikipedia

    en.wikipedia.org/wiki/Euler_characteristic_of_an...

    In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes contributions coming from nontrivial automorphisms.

  4. Genus (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Genus_(mathematics)

    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.

  5. Local Euler characteristic formula - Wikipedia

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

    In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galois group G K of a non-archimedean local field K.

  6. 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.

  7. Grothendieck–Riemann–Roch theorem - Wikipedia

    en.wikipedia.org/wiki/Grothendieck–Riemann...

    For example, when f is a smooth morphism, is simply a vector bundle, known as the tangent bundle along the fibers of f. Using A 1 -homotopy theory , the Grothendieck–Riemann–Roch theorem has been extended by Navarro & Navarro (2017) to the situation where f is a proper map between two smooth schemes.

  8. Eulerian number - Wikipedia

    en.wikipedia.org/wiki/Eulerian_number

    Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis. The polynomials presently known as Eulerian polynomials in Euler's work from 1755, Institutiones calculi differentialis, part 2, p. 485/6. The coefficients of these polynomials are known as Eulerian numbers.

  9. Euler class - Wikipedia

    en.wikipedia.org/wiki/Euler_class

    This can be seen intuitively in that the Euler class is a class whose degree depends on the dimension of the bundle (or manifold, if the tangent bundle): the Euler class is an element of () where is the dimension of the bundle, while the other classes have a fixed dimension (e.g., the first Stiefel-Whitney class is an element of ()).

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