Characteristic of Euler

The characteristic of Euler - or Euler-Poincaré - is a numerical invariant, a number which describes an aspect of a form of the topological Espace or structure. It is commonly noted by $\ chi \,$ .

The characteristic of Euler was defined at the origin for the Polyèdre S and was used to show various theorems with regard to them, including the classification of the solid of Plato. Leonhard Euler, by which the concept had its name, was responsible for much in this work for pioneer. In more modern mathematics the characteristic of Euler appears in the cohomologic homology and methods . It is given in general by the alternate sum of dimensions of the groups of cohomology considered:

\ chi= \ sum_ {i=0} ^ {\ infty} (- 1) ^i \ mathrm {dim} (H^i)

Polyhedrons

The characteristic of Euler $\ chi \,$ was defined in a traditional way for the polyhedrons, according to the formula

$\ chi = S - has + F, \, \!$

where S , has and F is respectively the numbers of tops (corners), edges and faces in a given polyhedron. For an unspecified polyhedron homeomorphic with a Sphere, the characteristic of Euler proves to be

$\ chi = S - has + F = 2 \, \!$ .

This result is known under the name formula of Euler .

Convex examples of polyhedrons

The surface of a convex polyhedron is homeomorphic with a sphere and consequently has a characteristic of Euler equalizes to 2, by the formula of Euler. This fact can be used to show that there exist only five solid of Plato (polyhedral regular):

Not-convex examples of polyhedrons

The not-convex polyhedrons can have various characteristics of Euler:

Formal definition

The polyhedrons discussed above are, in modern language, of the complex CW finished with two dimensions. (when only the triangular faces are used, they are called complex simpliciaux finished with two dimensions). In general, for unspecified finished complex-CW, the characteristic of Euler can be defined as the alternated sum, in dimension K:

$\ chi = k_0 - k_1 + k_2 - k_3 + \ cdots = 1 - (- 1) ^k$ (only for the regular polytopes)

where $k_n$ indicates the number of cells of dimensions $n$ in the complex.

More generally still, for a topological Space unspecified, we can define N E Nombre of Betti $b_n$ like the row of N E groups homological. The characteristic of Euler can be defined as the alternate sum

$\ chi = b_0 - b_1 + b_2 - b_3 + \ cdots$ .

This quantity is well defined if the numbers of Betti all are finished and if they are equal to zero beyond of a certain index $n_0$ . This definition includes the preceding ones.

Theory of the groups

In the case of the cohomology of the pro-p-groups, the characteristic of Euler makes it possible for example to characterize cohomologic dimension: either G a m. groups, then, G is of cohomologic size lower than N if and only if the characteristic of Euler truncated with the order N is multiplicative through the sub-groups open of G , i.e. if and only if:

\ forall U<_o G, \ sum_ {i=0} ^n (- 1) ^i \ mathrm {dim} _ {\ mathbf {F} _p} H^i (U, \ mathbf {F} _p) = (G: U) \ sum_ {i=0} ^n (- 1) ^i \ mathrm {dim} _ {\ mathbf {F} _p} H^i (G, \ mathbf {F} _p)

Algebraic topology

Definition

In algebraic Topology, the characteristic of Euler of a variety, noted C or χ (Chi), is the alternate sum of the numbers of Betti, as indicated above. In particular, C = 2 for the projective Plane and the Sphere, C = 1 for the disc of the plan and C = 0 for the Torus and the Bottle of Klein.

Properties

An unspecified space Contractible, (i.e., an equivalent homotopic at a point) has a commonplace homology, which means that the 0e number of Betti is 1 and others 0. Consequently, its characteristic of Euler is 1. This box includes the Euclidean Espace $\ unspecified R^n$ of dimension, as much as the unit solid ball in an unspecified Euclidean space — the interval at a dimension, the disc with two dimensions, the ball with three dimensions, etc

The characteristic of Euler can be calculated easily for general surfaces by a grid on surface (i.e., a description in the form of a complex CW). For an object, it represents the number of singularities necessary to net this object with its Géodésique S.

Examples

the Sphère has as a characteristic 2: it has two poles .
the Tore has a null characteristic: it is possible to net it without introducing singularity.

See too

Theorem of Descartes-Euler For the case of the polyhedrons.

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