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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. [13] The Euler characteristic of any closed odd-dimensional manifold is also 0. [14] The case for orientable examples is a corollary of Poincaré duality.
Thus the Euler class is a generalization of the Euler characteristic to vector bundles other than tangent bundles. In turn, the Euler class is the archetype for other characteristic classes of vector bundles, in that each "top" characteristic class equals the Euler class, as follows. Modding out by 2 induces a map
This isomorphism is realized by the Euler class; equivalently, it is the first Chern class of a smooth complex line bundle (essentially because a circle is homotopically equivalent to , the complex plane with the origin removed; and so a complex line bundle with the zero section removed is homotopically equivalent to a circle bundle.)
In calculating the Euler characteristic of S′ we notice the loss of e P − 1 copies of P above π(P) (that is, in the inverse image of π(P)). Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics.
Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. [16] This was followed by Riemann 's definition of genus and n -fold connectedness numerical invariants in 1857 and Betti 's proof in 1871 of the independence of "homology numbers" from the choice of basis.
K) is a Chern number and the self-intersection number of the canonical class K, and e = c 2 is the topological Euler characteristic. It can be used to replace the term χ(0) in the Riemann–Roch theorem with topological terms; this gives the Hirzebruch–Riemann–Roch theorem for surfaces.
The Euler characteristic of a coherent sheaf E can be computed from the Chern classes of E, according to the Riemann–Roch theorem and its generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem. For example, if L is a line bundle on a smooth proper geometrically connected curve X over a field k, then
Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.