Euler characteristic

Topological definition

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

${\displaystyle \chi =k_0-k_1+k_2-k_3+\cdots ,}$

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

${\displaystyle \chi =k_0-k_1+k_2-k_3+\cdots ,}$

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

Betti number alternative

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

${\displaystyle \chi =b_0-b_1+b_2-b_3+\cdots .}$

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 ${\displaystyle \chi }$.

Differentiable point of view

For an oriented, compact smooth manifold without boundary, ${\displaystyle X}$, one defines

${\displaystyle \chi (X)=I(\mathrm{\Delta },\mathrm{\Delta })}$

The number of self oriented intersections of the diagonal of ${\displaystyle X}$, ${\displaystyle \mathrm{\Delta }}$, inside ${\displaystyle X\times X.}$

Properties

The Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.

Homotopy invariance

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 ${\displaystyle {\mathbb{R}}^n}$ 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.

Inclusion–exclusion principle

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:

${\displaystyle \chi (M\bigsqcup N)=\chi (M)+\chi (N).}$

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:

${\displaystyle \chi (M\cup N)=\chi (M)+\chi (N)-\chi (M\cap N).}$

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.

Quotient space

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

${\displaystyle \chi (X)=\chi (A)+\chi (X/A)-1.}$

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

Connected sum

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

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

Product property

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

${\displaystyle \chi (M\times N)=\chi (M)\cdot \chi (N).}$

Using this and the formula for quotient spaces, one obtains

${\displaystyle \hat{\chi }(M\wedge N)=\hat{\chi }(M)\cdot \hat{\chi }(N),}$

where ${\displaystyle \hat{\chi }}$ is the reduced Euler characteristic, and ${\displaystyle M\wedge N}$ 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 .

Covering spaces

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

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

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.

Fibration property

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

If ${\displaystyle p:E\to B}$ 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:

${\displaystyle \chi (E)=\chi (F)\cdot \chi (B).}$

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 ${\displaystyle \tau :H_{\ast }(B)\to H_{\ast }(E)}$ – note that this is a lifting and goes "the wrong way" – whose composition with the projection map ${\displaystyle p_{\ast }:H_{\ast }(E)\to H_{\ast }(B)}$ is multiplication by the Euler class of the fiber:

${\displaystyle p_{\ast }\circ \tau =\chi (F)\cdot 1.}$

Examples

Surfaces

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.

Football

It is common to construct footballs by stitching together regular pentagons and hexagons, with three faces meeting at each vertex. If P pentagons and H hexagons are used, then there are ${\displaystyle F=P+H}$ faces, ${\displaystyle V=\frac{1}{3}\left(5P+6H\right)}$ vertices, and ${\displaystyle E=\frac{1}{2}\left(5P+6H\right)}$ edges. The Euler characteristic is thus

${\displaystyle V-E+F=\frac{1}{3}\left(5P+6H\right)-\frac{1}{2}\left(5P+6H\right)+P+H=\frac{1}{6}P.}$

Because the sphere has Euler characteristic 2, it follows that ${\displaystyle P=12}$. That is, a football constructed in this way always has 12 pentagons. The number of hexagons can be any nonnegative integer except 1. This result is applicable to fullerenes and Goldberg polyhedra.

Specifically, footballs are commonly made using 12 pentagonal and 20 hexagonal panels. The result is a truncated icosahedron spherical geometry. An example is the Adidas Telstar.

Arbitrary dimensions

The n-dimensional sphere has singular homology groups equal to

${\displaystyle H_k({\mathrm{S}}^n)=\{\begin{array}{ll}\mathbb{Z},& \text{if }k=0\text{ or }k=n,\\ \{0\}& \text{otherwise,}\end{array}}$

hence has Betti number 1 in dimensions 0 and n, and all other Betti numbers are 0. Its Euler characteristic is then χ = 1 + (−1) ⁿ ; that is, either 0 if n is odd, or 2 if n is even.

The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1.

The n-dimensional torus is the product space of n circles. Its Euler characteristic is 0 by the product property. More generally, any compact parallelizable manifold, including any compact Lie group, has Euler characteristic 0.

The Euler characteristic of any closed odd-dimensional manifold is also 0. The case for orientable examples is a corollary of Poincaré duality. 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.

Relations to other invariants

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

${\displaystyle \chi =2-2g.}$

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

${\displaystyle \chi =2-k.}$

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.

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".

Generalizations

For every combinatorial cell complex, one defines the Euler characteristic as the number of 0-cells, minus the number of 1-cells, plus the number of 2-cells, etc., if this alternating sum is finite. 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.

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 and only finitely many are non-zero.

A version of Euler characteristic used in algebraic geometry is as follows. For any coherent sheaf ${\displaystyle \mathcal{F}}$ on a proper scheme X, one defines its Euler characteristic to be

${\displaystyle \chi (\mathcal{F})=\sum _i(-1{)}^i{h}^i(X,\mathcal{F}),}$

where ${\displaystyle h^i(X,\mathcal{F})}$ is the dimension of the i-th sheaf cohomology group of ${\displaystyle \mathcal{F}}$. In this case, the dimensions are all finite by Grothendieck's finiteness theorem. This is an instance of the Euler characteristic of a chain complex, where the chain complex is a finite resolution of ${\displaystyle \mathcal{F}}$ by acyclic sheaves.

Another generalization of the concept of Euler characteristic on manifolds comes from orbifolds (see Euler characteristic of an orbifold). While every manifold has an integer Euler characteristic, an orbifold can have a fractional Euler characteristic. 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. A poset is "bounded" if it has smallest and largest elements; call them 0 and 1. The Euler characteristic of such a poset is defined as the integer μ(0,1), where μ is the Möbius function in that poset's incidence algebra.

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/ |G_i |⁠, where we picked one representative group G_i for each connected component of the groupoid.