Euler characteristic

In the mathematical sub-area of topology, the Euler characteristic is a key figure / topological invariant for topological spaces, for example for closed surfaces . Usually used as a term . ${\displaystyle \chi }$

It is named after the mathematician Leonhard Euler , who proved in 1758 that the relationship applies to the number of corners, the number of edges and the number of faces of a convex polyhedron . This special statement is called Euler's polyhedron substitution . The Euler characteristic , i.e. the number , can also be defined more generally for CW complexes . This generalization is also called the Euler-Poincaré characteristic , which is intended to indicate the mathematician Henri Poincaré . Areas that are considered to be the same from a topological point of view have the same Euler characteristic. It is therefore an integer topological invariant . The Euler characteristic is an important object in the Gauss-Bonnet theorem . This namely establishes a connection between the Gaussian curvature and the Euler characteristic. ${\displaystyle E}$ ${\displaystyle K}$ ${\displaystyle F}$ ${\displaystyle E-K+F=2}$ ${\displaystyle E-K+F}$

definition

For surfaces

A closed surface can always be triangulated , that is, it can always be covered with a finite triangular grid. The Euler characteristic is then defined as ${\displaystyle S}$ ${\displaystyle \chi }$

${\displaystyle \chi (S):=E-K+F.}$

where the number of corners, the number of edges and the number of triangles in the triangulation are meant. ${\displaystyle E}$ ${\displaystyle K}$ ${\displaystyle F}$

For CW complexes

Let be a topological space that is a finite dimensional CW complex . With the number of cells of dimension'll designated and is the dimension of the CW complex. Then the Euler characteristic is given by the alternating sum ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle k_{i}}$ ${\displaystyle i}$ ${\displaystyle n}$

${\displaystyle \chi (X):=\chi (T)=\sum _{i=0}^{n}(-1)^{i}k_{i}}$

Are defined. This Euler characteristic for CW complexes is also called the Euler-Poincaré characteristic. If the space is broken down into simplices instead of cells , then the Euler characteristic can also be defined analogously by the simplicial complex thus obtained . The following applies to the Euler characteristic ${\displaystyle C}$

${\displaystyle \chi (X):=\chi (C)=\sum _{i=0}^{n}(-1)^{i}f_{i}}$

where is the number of -dimensional simplices of . For a simplicial complex of a two-dimensional space, one obtains with , and the definition of the Euler characteristic on surfaces. The value of the characteristic is independent of the type of calculation. ${\displaystyle f_{i}}$ ${\displaystyle i}$ ${\displaystyle C}$ ${\displaystyle E=f_{0}}$ ${\displaystyle K=f_{1}}$ ${\displaystyle F=f_{2}}$

Definition by means of singular homology

Be a topological space again. The rank of the -th singular homology groups is called the -th Betti number and is denoted by. If the singular homology groups have finite rank and only finitely many Betti numbers are not equal to zero, then the Euler characteristic of is through ${\displaystyle X}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle b_{i}}$ ${\displaystyle X}$

${\displaystyle \chi (X):=\sum _{i=0}^{n}(-1)^{i}b_{i}=\sum _{i=0}^{n}(-1)^{i}\dim(H_{i}(X))}$

Are defined. If is a CW complex, then this definition gives the same value as in the definition for CW complexes. For example, a closed , orientable, differentiable manifold fulfills the requirements for singular homology. ${\displaystyle X}$

properties

Well-definition

An important observation is that the given definition is independent of the triangular lattice chosen. This can be shown by going over to a common refinement of given lattices without changing the Euler characteristic.

Since homeomorphisms are triangulated, the Euler characteristic also only depends on the topological type. Conversely, it follows from a different Euler characteristic of two surfaces that they must be topologically different. Therefore it is called a topological invariant .

Relationship to the gender of the area

The Euler characteristic and the gender of the surface are related. If the surface can be oriented , then the relationship applies ${\displaystyle \chi }$ ${\displaystyle g}$ ${\displaystyle S}$ ${\displaystyle S}$

${\displaystyle \chi (S)=2-2g,}$

if the surface cannot be oriented, the equation applies

${\displaystyle \chi (S)=2-g.}$

This formula for orientable areas results as follows: We start with a 2- sphere , i.e. an area of ​​gender 0 and Euler characteristic 2. An area of ​​gender is obtained by multiplying the connected sum with a torus. The connected sum can be set up in such a way that the gluing takes place along a triangle of the triangulation. The following balance results for each bond: ${\displaystyle g}$ ${\displaystyle g}$

• Surfaces: (the two adhesive surfaces ) ${\displaystyle F'=F-2}$
• Edges: (each 3 edges are glued, they then only count once) ${\displaystyle K'=K-3}$
• Corners: (each 3 corners are glued, they also only count once) ${\displaystyle E'=E-3}$

so overall . Each of the tori reduces the Euler characteristic by 2. ${\displaystyle \chi '=\chi -3+3-2=\chi -2}$ ${\displaystyle g}$

Connection with Euler's polyhedron substitution

Let be a convex polyhedron that can be embedded in the interior of a 2- sphere . Now the corners, edges and outer surfaces of this polyhedron can be viewed as cells of a CW complex. The singular homology groups of the complex are also finite dimensional. Since the polyhedron is orientable and gender has 0, it follows from the above section that the Euler characteristic has the value 2. Overall, the formula results ${\displaystyle S}$ ${\displaystyle \mathbb {S} ^{2}}$ ${\displaystyle S}$

${\displaystyle E-K+F=2}$,

where the number of corners describes the edges and the number of faces. This formula is called Euler's polyhedron formula . ${\displaystyle E}$ ${\displaystyle K}$ ${\displaystyle F}$

Examples

• The 2-sphere has the Euler characteristic 2. ${\displaystyle S^{2}}$
• The real projective plane cannot be oriented and has the Euler characteristic 1. ${\displaystyle \mathbb {R} P^{2}}$
• The torus has the Euler characteristic 0.
• Every odd-dimensional closed manifold has Euler characteristic 0. (This follows from Poincaré duality .)
• The Euler characteristic of even-dimensional closed manifolds can be calculated using their curvature, see Chern-Gauss-Bonnet theorem .

Connection to the Euler class

For closed , orientable , differentiable manifolds with tangent bundles and fundamental classes, the Euler characteristic of can also be defined equivalently by , where the Euler class of is. ${\displaystyle M}$ ${\displaystyle TM}$ ${\displaystyle \left[M\right]}$ ${\displaystyle M}$ ${\displaystyle \langle e(TM),\left[M\right]\rangle =\chi (M)}$ ${\displaystyle e(TM)}$ ${\displaystyle TM}$

literature

• H. Graham Flegg: From Geometry to Topology. Dover, Mineola NY 2001, ISBN 0-486-41961-4 , p. 40 ff.
• SV Matveev: Euler characteristics . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Web links

• Eric W. Weisstein : Euler Characteristic . In: MathWorld (English).