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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.
It has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. The topological real projective plane can be constructed by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk. Alternately, the real projective plane can be constructed by ...
is called the real projective line, which is topologically equivalent to a circle. is called the real projective plane.This space cannot be embedded in .It can however be embedded in and can be immersed in (see here).
The Euler characteristic of CP n is therefore n + 1. By Poincaré duality the same is true for the ranks of the cohomology groups . In the case of cohomology, one can go further, and identify the graded ring structure, for cup product ; the generator of H 2 ( CP n , Z ) is the class associated to a hyperplane , and this is a ring generator, so ...
The classification is consistent with the Gauss–Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2g, where g is the genus of the 2-manifold, i.e. the number of "holes".
A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
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. [17]