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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.
The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}: As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms.
Let ρ be the negative of the Euler characteristic (so that ρ = 2m − 2 for the sphere with m holes). Then {(),} (), This is meaningful only when ρ(Y) > 0, for example when Y is a sphere with three (or more) holes. In this case, the result can be considered as a generalization of the property b) of coverings.
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.
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
Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V = 1 vertex, E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0.
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.
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).