Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler-Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \(\chi\) (Greek lower-case letter chi).

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 It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. 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. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra.

The Euler characteristic was classically defined for the surface of a three-dimensional polyhedron. If the polyhedron has V vertices (corners), E edges, and F faces, then the Euler characteristic χ of its surface is \[\chi = V - E + F.\] Any three-dimensional convex polyhedron's surface has an Euler characteristic of \(\chi = 2\). This equation, stated by Euler in 1758, is known as Euler's polyhedron formula. It corresponds to the Euler characteristic of the sphere (i.e. \(\ \chi = 2\)), and applies identically to spherical polyhedra. An illustration of the formula for all Platonic solids is given below.

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 \(\ d_v\ ,\) and face density \(\ d_f\ :\)

\(\ d_v V - E + d_f F = 2 D ~.\)

This version holds both for convex polyhedra (where the densities are all 1) and the non-convex Kepler-Poinsot polyhedra.

Projective polyhedra all have Euler characteristic 1, like the real projective plane, while the surfaces of toroidal polyhedra all have Euler characteristic 0, like the torus.

The Euler characteristic can be defined for connected plane graphs by the same \(\ V - E + F\) formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face.

The Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of faces determined by G, starting with a tree as the base case. For trees, \(\ E = V - 1\) and \(\ F = 1 ~.\) If G has C components (disconnected graphs), the same argument by induction on F shows that \(\ V - E + F - C = 1 ~.\) One of the few graph theory papers of Cauchy also proves this result.

Via stereographic projection the plane maps to the 2-sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchy's proof of Euler's formula given below.

There are many proofs of Euler's formula. One was given by Cauchy in 1811, as follows. It applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks.

Remove one face of the polyhedral surface. By pulling the edges of the missing face away from each other, deform all the rest into a planar graph of points and curves, in such a way that the perimeter of the missing face is placed externally, surrounding the graph obtained, as illustrated by the first of the three graphs for the special case of the cube. (The assumption that the polyhedral surface is homeomorphic to the sphere at the beginning is what makes this possible.) After this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, but the number of faces has been reduced by 1. Therefore, proving Euler's formula for the polyhedron reduces to proving \(\ V - E + F = 1\) for this deformed, planar object.

If there is a face with more than three sides, draw a diagonal, that is, a curve through the face connecting two vertices that are not yet connected. Each new diagonal adds one edge and one face and does not change the number of vertices, so it does not change the quantity \(\ V - E + F ~.\) (The assumption that all faces are disks is needed here, to show via the Jordan curve theorem that this operation increases the number of faces by one.) Continue adding edges in this manner until all of the faces are triangular.

Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a simple cycle:

1. 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 \(\ V - E + F ~.\)
2. 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 \(\ V - E + F ~.\)

These transformations eventually reduce the planar graph to a single triangle. (Without the simple-cycle invariant, removing a triangle might disconnect the remaining triangles, invalidating the rest of the argument. A valid removal order is an elementary example of a shelling.)

At this point the lone triangle has \(\ V = 3\ ,\)\(\ E = 3\ ,\) and \(\ F = 1\ ,\) so that \(\ V - E + F = 1 ~.\) Since each of the two above transformation steps preserved this quantity, we have shown \(\ V - E + F = 1\) for the deformed, planar object thus demonstrating \(\ V - E + F = 2\) for the polyhedron. This proves the theorem.

For additional proofs, see Eppstein (2013). Multiple proofs, including their flaws and limitations, are used as examples in Proofs and Refutations by Lakatos (1976).

The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum

\(\chi = k_0 - k_1 + k_2 - k_3 + \cdots,\)

where k_n denotes the number of cells of dimension n in the complex.

Similarly, for a simplicial complex, the Euler characteristic equals the alternating sum

\(\chi = k_0 - k_1 + k_2 - k_3 + \cdots,\)

where k_n denotes the number of n-simplexes in the complex.

More generally still, for any topological space, we can define the nth Betti number b_n as the rank of the n-th singular homology group. The Euler characteristic can then be defined as the alternating sum

\(\chi = b_0 - b_1 + b_2 - b_3 + \cdots.\)

This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n₀. For simplicial complexes, this is not the same definition as in the previous paragraph but a homology computation shows that the two definitions will give the same value for \(\chi\).

For an oriented, compact smooth manifold without boundary, \(X\), one defines

\(\chi(X)= I(\Delta, \Delta)\)

The number of self oriented intersections of the diagonal of \(X\), \(\Delta\), inside \(X\times X.\)

Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It follows that the Euler characteristic is also a homotopy invariant.

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. Therefore, its Euler characteristic is 1. This case includes Euclidean space \(\mathbb{R}^n\) 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.

For another example, any convex polyhedron is homeomorphic to the three-dimensional ball, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional sphere, which has Euler characteristic 2. This explains why the surface of a convex polyhedron has Euler characteristic 2.

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:

\(\chi(M \sqcup N) = \chi(M) + \chi(N).\)

More generally, if M and N are subspaces of a larger space X, then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the inclusion, exclusion principle:

\(\chi(M \cup N) = \chi(M) + \chi(N) - \chi(M \cap N).\)

This is true in the following cases:

• if M and N are an excisive couple. In particular, if the interiors of M and N inside the union still cover the union.
• if X is a locally compact space, and one uses Euler characteristics with compact supports, no assumptions on M or N are needed.
• if X is a stratified space all of whose strata are even-dimensional, the inclusion, exclusion principle holds if M and N are unions of strata. This applies in particular if M and N are subvarieties of a complex algebraic variety.

In general, the inclusion, exclusion principle is false. 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.

If X is a finite CW-complex and A is a subcomplex, then

\(\chi(X) = \chi(A) + \chi(X/A) -1.\)

The formula could be more succinctly written for the reduced Euler characteristic, the alternating sum of ranks of reduced homology groups.

For two connected closed n-manifolds \(M, N\) one can obtain a new connected manifold \(M \mathbin{\#} N\) via the connected sum operation. The Euler characteristic is related by the formula

\(\chi(M \mathbin{\#} N) = \chi(M) + \chi(N) - \chi(S^n).\)

Also, the Euler characteristic of any product space M × N is

\(\chi(M \times N) = \chi(M) \cdot \chi(N).\)

Using this and the formula for quotient spaces, one obtains

\(\hat\chi(M \wedge N) = \hat\chi(M) \cdot \hat\chi(N),\)

where \(\hat\chi\) is the reduced Euler characteristic, and \(M\wedge N\) is the smash product.

These addition and multiplication properties are also enjoyed by cardinality of sets. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; see .

Similarly, for a k-sheeted covering space \(\tilde{M} \to M,\) one has

\(\chi(\tilde{M}) = k \cdot \chi(M).\)

More generally, for a ramified covering space, the Euler characteristic of the cover can be computed from the above, with a correction factor for the ramification points, which yields the Riemann-Hurwitz formula.

The product property holds much more generally, for fibrations with certain conditions.

If \(p\colon E \to B\) 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:

\(\chi(E) = \chi(F)\cdot \chi(B).\)

This includes product spaces and covering spaces as special cases, and can be proven by the Serre spectral sequence on homology of a fibration.

For fiber bundles, this can also be understood in terms of a transfer map \(\tau\colon H_*(B) \to H_*(E)\), note that this is a lifting and goes "the wrong way", whose composition with the projection map \(p_*\colon H_*(E) \to H_*(B)\) is multiplication by the Euler class of the fiber:

\(p_* \circ \tau = \chi(F) \cdot 1.\)