Encyclopedia
In algebraic topology, the
Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure. It is commonly denoted by .
The Euler characteristic was originally formulated for
polyhedra and used to prove various theorems about them, including the classification of the
Platonic solids.
Leonhard Euler, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology and connects to many other invariants.
Polyhedra
The
Euler characteristic was classically defined for polyhedra, according to the formula
where
V,
E, and
F are respectively the numbers of vertices , edges and faces in the given polyhedron. For any polyhedron homeomorphic to a
sphere the Euler characteristic turns out to be
This result is known as
Euler's formula. A proof is given below.
Examples of convex polyhedra
The surface of any convex polyhedron is homeomorphic to a sphere and therefore has Euler characteristic 2, by Euler's formula. This fact can be used to show that there are only five
Platonic solids :
Proof of Euler's formula
The first rigorous proof of Euler's formula, given by a 20-year-old Cauchy, is as follows.
Remove one face of the polyhedron. By pulling the edges of the missing face away from each other, deform all the rest into a planar network of points and curves, as illustrated by the first of the three graphs for the special case of the cube. After this deformation, the regular faces are generally not regular anymore — in fact, they are not even polygons. However, the numbers of vertices, edges and faces remain the same as those of the given polyhedron.
If there is a face with more than three sides, draw a diagonal — that is, a curve through the face connecting two vertices that aren't connected yet. This adds one edge and one face and does not change the number of vertices, so it does not change the quantity
V −
E +
F. Continue adding edges in this manner until all of the faces are triangular.
Apply repeatedly either of the following two transformations:
- Remove a triangle with only one edge adjacent to the exterior, as illustrated by the second graph. This decreases the number of edges and faces by one each and does not change the number of vertices, so it preserves V − E + F.
- Remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph. Each triangle removal removes a vertex, two edges and one face, so it preserves V − E + F.
Repeat these two steps, one after the other, until only one triangle remains.
At this point the lone triangle has
V = 3,
E = 3, and
F = 2 , so that
V −
E +
F = 2. This equals the original
V −
E +
F, since each transformation step has preserved this quantity. Therefore at the start of the process it was true that
V −
E +
F = 2. This proves the theorem.
For additional proofs, see . Multiple proofs, including their flaws and limitations, are used as examples in
Proofs and Refutations by Lakatos.
Examples of non-convex polyhedra
Non-convex polyhedra can have various Euler characteristics:
| Name | Image | V | E | F | Euler characteristic: V − E + F |
|---|
| Tetrahemihexahedron | | 6 | 12 | 7 | 1 |
| Octahemioctahedron | | 12 | 24 | 12 | 0 |
| Cubohemioctahedron | | 12 | 24 | 10 | −2 |
Formal definition
The polyhedra 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 denotes the number of cells of dimension in the complex.
More generally still, for any topological space, we can define the
nth Betti number as the rank of the
n-th 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 . This definition subsumes the previous ones.
Properties
Since the homology is a topological invariant , so is the Euler characteristic.
If
M and
N are any two topological spaces, then the Euler characteristic of their disjoint union is the sum of their Euler characteristics, since homology is additive under disjoint union:
More generally, if
M and
N are subspaces of a larger space
X, then so are their union and intersection. The Euler characteristic obeys a version of the
inclusion-exclusion principle:
Also, the Euler characteristic of any
product space M ×
N is
These addition and multiplication properties are also enjoyed by cardinality of
sets. In this way, the Euler characterstic can be viewed as a generalisation of cardinality; see .
As a corollary of Poincaré duality, the Euler characteristic of any closed odd-dimensional manifold is zero.
Relations to other invariants
The Euler characteristic of a closed orientable
surface can be calculated from its genus
g as
The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus
k 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. The Euler class, in turn, relates to all other characteristic classes of
vector bundles.
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.
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 .
Hadwiger's theorem characterizes the Euler characteristic as the
unique 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".
Examples
Any contractible space has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore its Euler characteristic is 1. This case includes Euclidean space of any dimension, as well as the solid unit ball in any Euclidean space — the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.
The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface and using the above definitions.
Partially ordered sets
The concept of the Euler characteristic of a bounded finite
poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest elements, which let us call 0 and 1. The
Euler characteristic of such a poset is defined as μ, where μ is the
Möbius function in that poset's incidence algebra.
See also
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