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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.
The odd–even condition follows from Euler's formula. Any simplicial generalized homology sphere is an Eulerian lattice. Let L be a regular cell complex such that | L | is a manifold with the same Euler characteristic as the sphere of the same dimension (this condition is vacuous if the dimension is odd).
The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere, and so is also true, for example, of any polyhedron which is star-shaped with respect to some interior point.
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.
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
The Euler characteristic of a sphere, triangulated like an icosahedron, is V − E + F = 12 − 30 + 20 = 2 Main article: Gauss–Bonnet theorem Since every compact oriented 2-manifold M can be triangulated by small geodesic triangles, it follows that
These effects add as vectors to make the overall molecule polar. A polar molecule has a net dipole as a result of the opposing charges (i.e. having partial positive and partial negative charges) from polar bonds arranged asymmetrically. Water (H 2 O) is an example of a polar molecule since it has a slight positive charge on one side and a ...
Traditionally this is called the Euler class. This is [] = [] = since the class of can be represented by five points (by Bézout's theorem). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of by using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem.