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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.
Given a CW complex S containing one vertex, one edge, one face, and generally exactly one cell of every dimension, Euler's formula V − E + F − · · · for the Euler characteristic of S returns 1 − 1 + 1 − · · ·. There are a few motivations for defining a generalized Euler characteristic for such a space that turns out to be 1/2.
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.
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 ()).
Euler's identity therefore states that the limit, as n approaches infinity, of (+) is equal to −1. This limit is illustrated in the animation to the right. Euler's formula for a general angle. Euler's identity is a special case of Euler's formula, which states that for any real number x,
Among the main theorems are results on the vanishing of cohomology in various situations, results on finite-dimensionality of cohomology, comparisons between coherent sheaf cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem.
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.
However, it is a mistake to write that the Euler characteristic is or equals the Euler formula V – E + F for these shapes. There is an Euler formula that looks like that, but it doesn't apply to the given vertices, edges, and faces, because they don't form a valid system of faces in the sense of Klain and Rota.