Characteristic class of oriented, real vector bundles
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.
Throughout this article is an oriented, real vector bundle of rank over a base space .
algebraic topology, the Eulerclass is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted"...
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant...
numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class [ M ] ∈ H n ( M ) {\displaystyle...
Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Eulerclass of...
mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique...
bundle is partially characterized by its Eulerclass, which is a degree n + 1 {\displaystyle n+1} cohomology class in the total space of the bundle. In the...
derivative is the virtual dimension. The localized Eulerclass of the pair (E,s) is a homology class with closed support on the zero set of the section...
cohomology class e called the Eulerclass of the bundle. Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented...
resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving...
e(E)} denotes the Eulerclass of E {\displaystyle E} , and ⌣ {\displaystyle \smile } denotes the cup product of cohomology classes. As was shown by Shiing-Shen...
vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Eulerclass. A complex vector bundle...
space X determines a cohomology class on X, the Eulerclass χ(E) ∈ Hr(X,Z). Informally, the Eulerclass is the class of the zero set of a general section...
posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has even degree....
invariant of an oriented bundle is the Eulerclass. The multiplication (that is, cup product) by the Eulerclass of an oriented bundle gives rise to a...
An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining...
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k...
Poincaré duality, Eulerclass of Sphere bundles, Thom classes and Thom isomorphism, and more. Milnor, John. Characteristic classes. is another standard...
as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler (Letters to a German Princess) in 1768. The idea was popularised by Venn...
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e...
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