Euler characteristic
The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.
It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico.
[1] Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant.
The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron.
The surfaces of nonconvex polyhedra can have various Euler characteristics: For regular polyhedra, Arthur Cayley derived a modified form of Euler's formula using the density D, vertex figure density
This is easily proved by induction on the number of faces determined by G, starting with a tree as the base case.
(The assumption that the polyhedral surface is homeomorphic to the sphere at the beginning is what makes this possible.)
Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a simple cycle: These transformations eventually reduce the planar graph to a single triangle.
[6] The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes.
In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum where kn denotes the number of cells of dimension n in the complex.
More generally still, for any topological space, we can define the nth Betti number bn as the rank of the n-th singular homology group.
The Euler characteristic can then be defined as the alternating sum This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n0.
The Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.
For example, any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0.
This explains why the surface of a convex polyhedron has Euler characteristic 2.
A counterexample is given by taking X to be the real line, M a subset consisting of one point and N the complement of M. For two connected closed n-manifolds
is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:[10] This includes product spaces and covering spaces as special cases, and can be proven by the Serre spectral sequence on homology of a fibration.
is multiplication by the Euler class of the fiber:[11] The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface (that is, a description as a CW-complex) and using the above definitions.
The n dimensional real projective space is the quotient of the n sphere by the antipodal map.
This property applies more generally to any compact stratified space all of whose strata have odd dimension.
It also applies to closed odd-dimensional non-orientable manifolds, via the two-to-one orientable double cover.
The Euler characteristic of a closed orientable surface can be calculated from its genus g (the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus k (the number of real projective planes in a connected sum decomposition of the surface) as For closed smooth 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 the two-dimensional case and the generalized Gauss–Bonnet theorem for the general case.
Hadwiger's theorem characterizes the Euler characteristic as the unique (up to scalar multiplication) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on finite unions of compact convex sets in ℝn that is "homogeneous of degree 0".
In particular, the Euler characteristic of a finite set is simply its cardinality, and the Euler characteristic of a graph is the number of vertices minus the number of edges.
[15]) More generally, one can define the Euler characteristic of any chain complex to be the alternating sum of the ranks of the homology groups of the chain complex, assuming that all these ranks are finite.
For example, the teardrop orbifold has Euler characteristic 1 + 1/ p , where p is a prime number corresponding to the cone angle 2π / p .
The concept of Euler characteristic of the reduced homology of a bounded finite poset is another generalization, important in combinatorics.
This can be further generalized by defining a rational valued Euler characteristic for certain finite categories, a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above.
In this setting, the Euler characteristic of a finite group or monoid G is 1/ | G | , and the Euler characteristic of a finite groupoid is the sum of 1/ |Gi |, where we picked one representative group Gi for each connected component of the groupoid.
