 | Euler characteristic: Encyclopedia II - Euler characteristic - Definitions and properties
Euler characteristic - Definitions and properties
For a finite CW-complex and in particular for a finite simplicial complex, the Euler characteristic can be defined as the alternating sum
where ki denotes the number of cells of dimension i.
Then, one can define the Euler characteristic of a manifold as the Euler characteristic of a simplicial complex homeomorphic to it. For example, the circle and torus have Euler characteristic 0 and solid balls have Euler characteristic 1.
The Euler characteristic of closed orientable surfaces can be calculated using their genus g
χ = 2 − 2g.
The Euler characteristic of closed non-orientable surfaces can be calculated using their (non-orientable) genus k
χ = 2 − k.
The Euler characteristic is independent of the triangulation. The formula can also be used for decompositions into arbitrary polygons.
For the disk we have χ = 1, for the plane we have χ = 2, counting the outside as a face.
For closed manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold.
For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature--see the Gauss-Bonnet theorem for two-dimensional case and generalized Gauss-Bonnet theorem for general case. A discrete analog of the Gauss-Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).
More generally, for any topological space, we can define the nth Betti number bn as the rank of the n-th homology group. The Euler characteristic can then be defined as the alternating sum
This definition makes sense if the Betti numbers are all finite and zero beyond a certain index n0.
Two topological spaces which are homotopy equivalent have isomorphic homology groups and hence the same Euler characteristic.
From this definition and Poincaré duality, it follows that Euler characteristic of any closed odd-dimensional manifold is zero.
If M and N are topological spaces, then the Euler characteristic of their product space M × N is
.
Other related archivesAlgebraic topology, Betti number, CW-complex, Cauchy, Descartes', Euler class, Gauss-Bonnet theorem, Möbius function, Poincaré duality, Proofs and Refutations, Riemannian manifolds, Topological graph theory, algebraic topology, circle, closed manifolds, combinatorics, cube, defect (geometry), fundamental class, generalized Gauss-Bonnet theorem, genus, homeomorphic, homology, homotopy equivalent, homotopy invariant, incidence algebra, isomorphic, polyhedron, poset, product space, rank, simplicial complex, sphere, surfaces, tangent bundle, tetrahedron, topological invariant, topological space, topological spaces, torus, triangulation
 Adapted from the Wikipedia article "Definitions and properties", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
