In mathematics, and more specifically in algebraic topology and
polyhedral combinatorics, the
Euler characteristic
χ displaystyle chi (Greek lowercase letter chi).
The
Euler characteristic
Contents 1 Polyhedra 1.1 Plane graphs 1.2 Proof of Euler's formula 2 Topological definition 3 Properties 3.1 Homotopy invariance 3.2 Inclusion–exclusion principle 3.3 Connected sum 3.4 Product property 3.5 Covering spaces 3.6 Fibration property 4 Examples 4.1 Surfaces 4.2 Soccer ball 4.3 Arbitrary dimensions 5 Relations to other invariants 6 Generalizations 7 See also 8 References 8.1 Notes 8.2 Bibliography 9 Further reading 10 External links Polyhedra[edit]
The
Euler characteristic
χ displaystyle chi was classically defined for the surfaces of polyhedra, according to the formula χ = V − E + F displaystyle chi =VE+F where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic V − E + F = 2. displaystyle VE+F=2. This equation is known as Euler's polyhedron formula.[1] It
corresponds to the
Euler characteristic
Name Image Vertices V Edges E Faces F Euler characteristic: V − E + F Tetrahedron 4 6 4 2
Hexahedron
8 12 6 2 Octahedron 6 12 8 2 Dodecahedron 20 30 12 2 Icosahedron 12 30 20 2 The surfaces of nonconvex polyhedra can have various Euler characteristics; Name Image Vertices V Edges E Faces F Euler characteristic: V − E + F Tetrahemihexahedron 6 12 7 1 Octahemioctahedron 12 24 12 0 Cubohemioctahedron 12 24 10 −2 Great icosahedron 12 30 20 2 For regular polyhedra,
Arthur Cayley
d f displaystyle d_ f : d v V − E + d f F = 2 D . displaystyle d_ v VE+d_ f F=2D. This version holds both for convex polyhedra (where the densities are
all 1) and the nonconvex KeplerPoinsot polyhedra.
Projective polyhedra
V − E + F displaystyle VE+F formula as for polyhedral surfaces, where F is the number of faces in
the graph, including the exterior face.
The
Euler characteristic
E = V − 1 displaystyle E=V1 and F = 1 displaystyle F=1 . If G has C components (disconnected graphs), the same argument by induction on F shows that V − E + F − C = 1 displaystyle VE+FC=1 . One of the few graph theory papers of Cauchy also proves this
result.
Via stereographic projection the plane maps to the twodimensional
sphere, such that a connected graph maps to a polygonal decomposition
of the sphere, which has
Euler characteristic
First steps of the proof in the case of a cube 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 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.
(The assumption that all faces are disks is needed here, to show via
the
Jordan curve theorem
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. These transformations eventually reduce the planar graph to a single
triangle. (Without the simplecycle 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 Twenty Proofs of Euler's Formula by David
Eppstein.[2] Multiple proofs, including their flaws and limitations,
are used as examples in
Proofs and Refutations by Imre Lakatos.[3]
Topological definition[edit]
The polyhedral surfaces discussed above are, in modern language,
twodimensional finite CWcomplexes. (When only triangular faces are
used, they are twodimensional finite simplicial complexes.) In
general, for any finite CWcomplex, the
Euler characteristic
χ = k 0 − k 1 + k 2 − k 3 + ⋯ , displaystyle chi =k_ 0 k_ 1 +k_ 2 k_ 3 +cdots , where kn denotes the number of cells of dimension n in the complex.
Similarly, for a simplicial complex, the
Euler characteristic
χ = k 0 − k 1 + k 2 − k 3 + ⋯ , displaystyle chi =k_ 0 k_ 1 +k_ 2 k_ 3 +cdots , where kn denotes the number of nsimplexes in the complex.
More generally still, for any topological space, we can define the nth
Betti number
χ = b 0 − b 1 + b 2 − b 3 + ⋯ . displaystyle chi =b_ 0 b_ 1 +b_ 2 b_ 3 +cdots . This quantity is welldefined 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 χ displaystyle chi .
Properties[edit]
The
Euler characteristic
R n displaystyle mathbb R ^ n of any dimension, as well as the solid unit ball in any Euclidean space — the onedimensional interval, the twodimensional disk, the threedimensional ball, etc. For another example, any convex polyhedron is homeomorphic to the threedimensional ball, so its surface is homeomorphic (hence homotopy equivalent) to the twodimensional sphere, which has Euler characteristic 2. This explains why convex polyhedra have Euler characteristic 2. Inclusion–exclusion principle[edit] 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: χ ( M ⊔ N ) = χ ( M ) + χ ( N ) . displaystyle chi (Msqcup 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: χ ( M ∪ N ) = χ ( M ) + χ ( N ) − χ ( M ∩ N ) . displaystyle chi (Mcup N)=chi (M)+chi (N)chi (Mcap 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.[4] 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 evendimensional, 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.[5] 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. Connected sum[edit] For two connected closed nmanifolds M , N displaystyle M,N one can obtain a new connected manifold M # N displaystyle M#N via the connected sum operation. The
Euler characteristic
χ ( M # N ) = χ ( M ) + χ ( N ) − χ ( S n ) . displaystyle chi (M#N)=chi (M)+chi (N)chi (S^ n ). Product property[edit]
Also, the
Euler characteristic
χ ( M × N ) = χ ( M ) ⋅ χ ( N ) . displaystyle chi (Mtimes N)=chi (M)cdot chi (N). These addition and multiplication properties are also enjoyed by
cardinality of sets. In this way, the
Euler characteristic
M ~ → M , displaystyle tilde M to M, one has χ ( M ~ ) = k ⋅ χ ( M ) . displaystyle chi ( tilde M )=kcdot 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[edit] The product property holds much more generally, for fibrations with certain conditions. If p : E → B displaystyle pcolon Eto B is a fibration with fiber F, with the base B pathconnected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:[7] χ ( E ) = χ ( F ) ⋅ χ ( B ) . 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 τ : H ∗ ( B ) → H ∗ ( E ) displaystyle 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 ∗ : H ∗ ( E ) → H ∗ ( B ) displaystyle p_ * colon H_ * (E)to H_ * (B) is multiplication by the Euler class of the fiber:[8] p ∗ ∘ τ = χ ( F ) ⋅ 1. displaystyle p_ * circ tau =chi (F)cdot 1. Examples[edit]
Surfaces[edit]
The
Euler characteristic
Name Image Euler characteristic Interval 1 Circle 0 Disk 1 Sphere 2 Torus (Product of two circles) 0 Double torus −2 Triple torus −4 Real projective plane 1 Möbius strip 0 Klein bottle 0 Two spheres (not connected)
(
Disjoint union
2 + 2 = 4 Three spheres (not connected)
(
Disjoint union
2 + 2 + 2 = 6 Soccer ball[edit]
It is common to construct soccer balls by stitching together
pentagonal and hexagonal pieces, with three pieces meeting at each
vertex (see for example the Adidas Telstar). If P pentagons and H
hexagons are used, then there are F = P + H faces, V = (5 P + 6 H) / 3
vertices, and E = (5 P + 6 H) / 2 edges. The
Euler characteristic
V − E + F = 5 P + 6 H 3 − 5 P + 6 H 2 + P + H = P 6 . displaystyle VE+F= frac 5P+6H 3  frac 5P+6H 2 +P+H= frac P 6 . Because the sphere has
Euler characteristic
H k ( S n ) = Z k = 0 , n 0 otherwise, displaystyle H_ k (S^ n )= begin cases mathbb Z &k=0,n\ 0 & text otherwise, end cases hence has
Betti number
χ = 2 − 2 g . displaystyle chi =22g. The
Euler characteristic
χ = 2 − k . displaystyle chi =2k. For closed smooth manifolds, the
Euler characteristic
F displaystyle mathcal F on a proper scheme X, one defines its
Euler characteristic
χ ( F ) = ∑ i ( − 1 ) i h i ( X , F ) , displaystyle chi ( mathcal F )=sum _ i (1)^ i h^ i (X, mathcal F ), where h i ( X , F ) displaystyle h^ i (X, mathcal F ) is the dimension of the ith sheaf cohomology group of F displaystyle mathcal F . In this case, the dimensions are all finite by Grothendieck's
finiteness theorem. This is an instance of the
Euler characteristic
F displaystyle mathcal F by acyclic sheaves.
Another generalization of the concept of
Euler characteristic
Euler calculus Euler class List of topics named after Leonhard Euler List of uniform polyhedra References[edit] Notes[edit] ^ Richeson 2008
^ Eppstein, David. "Twenty Proofs of Euler's Formula: VE+F=2".
Retrieved 3 June 2013.
^ Imre Lakatos: Proofs and Refutations, Cambridge Technology Press,
1976
^ Edwin Spanier: Algebraic Topology, Springer 1966, p. 205.
^ William Fulton: Introduction to toric varieties, 1993, Princeton
University Press, p. 141.
^ "Homology of connected sum". Retrieved 20160713.
^ Spanier, Edwin Henry (1982), Algebraic Topology, Springer,
ISBN 9780387944265 , Applications of the homology
spectral sequence, p. 481
^ Gottlieb, Daniel Henry (1975), "Fibre bundles and the Euler
characteristic" (PDF), Journal of Differential Geometry, 10 (1):
39–48
^ Milnor, John W. and Stasheff, James D.: Characteristic Classes,
Princeton University Press, 1974
^ Richeson 2008, p. 261
^ Olaf Post calls this a "wellknown formula": Post, Olaf (2009),
"Spectral analysis of metric graphs and related spaces", Limits of
graphs in group theory and computer science, Lausanne, Switzerland:
EPFL Press, pp. 109–140, arXiv:0712.1507 .
^ nLab, "Euler characteristic"
^ Tom Leinster, "The
Euler characteristic
Bibliography[edit] Richeson, David S.; Euler's Gem: The
Polyhedron
Further reading[edit] Flegg, H. Graham; From Geometry to Topology, Dover 2001, p. 40. External links[edit] Weisstein, Eric W. "Euler characteristic". MathWorld.
Weisstein, Eric W. "Polyhedral formula". MathWorld.
Matveev, S.V. (2001) [1994], "Euler characteristic", in Hazewinkel,
Michiel, Encyclopedia of Mathematics, Springer Science+Business Media
B.V. / Kluwer Academic Publishers, ISBN 9781556080104
Euler Characteristic of the Barycentric Subdivision of an nSimplex.
In math.stackexchange.
Euler characteristic
