In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and more specifically in
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
and
polyhedral combinatoricsPolyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher dimensional convex polytopes....
, the
Euler characteristic (or
Euler–Poincaré characteristic) is a topological invariant, a number that describes a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
's shape or structure regardless of the way it is bent. It is commonly denoted by
(
Greek letterThe Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
chiChi is the 22nd letter of the Greek alphabet, pronounced as in English.-Greek:-Ancient Greek:Its value in Ancient Greek was an aspirated velar stop .-Koine Greek:...
).
The Euler characteristic was originally defined for
polyhedraIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
and used to prove various theorems about them, including the classification of the
Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s.
Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from
homologyIn mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
and connects to many other invariants.
Polyhedra
The
Euler characteristic 
was classically defined for the surfaces of polyhedra, according to the formula
where
V,
E, and
F are respectively the numbers of
verticesIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
(corners),
edgeIn geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
s and
facesIn geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
in the given polyhedron. Any convex polyhedron's surface has Euler characteristic
This result is known as
Euler's polyhedron formula or
theorem. It corresponds to the Euler characteristic of the sphere (i.e. χ = 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below.
| Name |
Image |
Vertices
V |
Edges
E |
Faces
F |
Euler characteristic:
V − E + F |
TetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
 |
4 |
6 |
4 |
2 |
HexahedronA hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
or cube |
 |
8 |
12 |
6 |
2 |
OctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
|
 |
6 |
12 |
8 |
2 |
| Dodecahedron |
 |
20 |
30 |
12 |
2 |
IcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
 |
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 In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
|
 |
6 |
12 |
7 |
1 |
| Octahemioctahedron In geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U3. Its vertex figure is a crossed quadrilateral.It is one of nine hemipolyhedra with 4 hexagonal faces passing through the model center.- Related polyhedra :...
|
 |
12 |
24 |
12 |
0 |
| Cubohemioctahedron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces...
|
 |
12 |
24 |
10 |
−2 |
| Great icosahedron |
 |
12 |
30 |
20 |
2 |
A modified form of Euler's formula, using polyhedral density (
D) and
polygon density of the
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
s (
) and faces (
) was given by
Arthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, and holds both for convex polyhedra (where the correction factors are all 1), and the regular non-convex Kepler–Poinsot polyhedrons:
Further, projective polyhedra all have Euler characteristic 1, corresponding to the
real projective planeIn mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
, while toroidal polyhedra all have Euler characteristic 0, corresponding to the
torusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
.
Planar graphs
The Euler characteristic can be defined for connected
planar graphIn graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...
s by the same
formula as for polyhedral surfaces, where
F is the number of faces in the graph, including the exterior face.
The Euler characteristic of any planar 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, the same argument by induction on F shows that
. One of the few graph theory papers of Cauchy also proves this result.
Via
stereographic projectionThe stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
the plane maps to the two-dimensional 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.
Proof of Euler's formula
There are many proofs of Euler's formula. One was given by
CauchyBaron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
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, 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 theoremIn topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
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:
- 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 = 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
Nineteen Proofs of Euler's Formula by
David EppsteinDavid Arthur Eppstein is an American computer scientist and mathematician. He is professor of computer science at University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics.-Biography:Born in England of New Zealander...
. Multiple proofs, including their flaws and limitations, are used as examples in
Proofs and RefutationsProofs and Refutations is a book by the philosopher Imre Lakatos expounding his view ofthe progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron...
by
Imre LakatosImre Lakatos was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
.
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 complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es.) 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 spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, we can define the
nth
Betti numberIn algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
bn as the
rankIn mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
of the
n-th
singular homologyIn algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
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. 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
.
Properties
As a corollary of
Poincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
, the Euler characteristic of any
closedIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
odd-dimensional manifold is zero. This applies more generally to any
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
stratified spaceIn topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
all of whose strata are odd-dimensional. Furthermore, the Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.
Homotopy invariance
Since the homology is a topological invariant (in fact, a homotopy invariant — two topological spaces that are homotopy equivalent have
isomorphicIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
homology groups), so is the Euler characteristic.
For example, any convex polyhedron is homeomorphic to the three-dimensional
ballIn mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, which has Euler characteristic 2. This explains why convex polyhedra have Euler characteristic 2.
Inclusion-exclusion principle
If
M and
N are any two topological spaces, then the Euler characteristic of their
disjoint unionIn mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a 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. In some cases, the Euler characteristic obeys a version of the
inclusion-exclusion principleIn combinatorics, the inclusion–exclusion principle is an equation relating the sizes of two sets and their union...
:
This is true in the following cases:
- if M and N are an excisive couple. In particular, if the interiors
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of M and N inside the union still cover the union.
- if X is a locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
, and one uses Euler characteristics with compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
supportsIn mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
, no assumptions on M or N are needed.
- if X is a stratified space
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way...
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 complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
algebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
.
In general, the inclusion-exclusion principle is false. A
counterexampleIn logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
is given by taking
X to be the
real lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
,
M a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
consisting of one point and
N the
complementIn set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of
M.
Product property
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 characteristic can be viewed as a generalisation of cardinality; see
http://math.ucr.edu/home/baez/counting/.
Covering spaces
Similarly, for an
k-sheeted covering space
one has
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
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:
This includes product spaces and covering spaces as special cases,
and can be proven by the
Serre spectral sequenceIn mathematics, the Serre spectral sequence is an important tool in algebraic topology...
on homology of a fibration.
For fiber bundles, this can also be understood in terms of a transfer map
– note that this is a lifting and goes "the wrong way" – whose composition with the projection map
is multiplication by the
Euler classIn mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
of the fiber:
Relations to other invariants
The Euler characteristic of a closed
orientableIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
surfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
can be calculated from its
genusIn mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g (the number of
toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
in a
connected sumIn mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...
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 planeIn mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
s 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 classIn mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
of its
tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
evaluated on the
fundamental classIn mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
of a manifold. The Euler class, in turn, relates to all other
characteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es of
vector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s.
For closed
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s, 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'René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
theorem that the "total defect" of a
polyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
, measured in full circles, is the Euler characteristic of the polyhedron; see
defect (geometry)In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...
.
Hadwiger's theoremIn integral geometry , Hadwiger's theorem characterises the valuations on convex bodies in Rn. It was proved by Hugo Hadwiger.-Valuations:...
characterizes the Euler characteristic as the
unique (
up toIn mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
scalar multiplicationIn mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on finite unions of
compactIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
convex sets in
Rn that is "homogeneous of degree 0".
Examples
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.
| Name |
Image |
Euler characteristic |
| Interval In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
|
 |
1 |
CircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
|
 |
0 |
| Disk |
 |
1 |
SphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
|
 |
2 |
TorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
(Product of two circles) |
 |
0 |
| Double torus In mathematics, a genus-2 surface is a surface formed by the connected sum of two tori. That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified , forming a double torus.This is the simplest case of the connected sum of n tori...
|
 |
−2 |
| Triple torus Triple torus or three-torus can refer to one of the two following concepts, both related to a torus.-Three-dimensional torus:The three-dimensional torus, or triple torus, is defined as the Cartesian product of three circles,...
|
 |
−4 |
Real projective planeIn mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
|
 |
1 |
Möbius stripThe Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
|
 |
0 |
| Klein bottle In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
|
 |
0 |
Two spheres (not connected)
(Disjoint union of two spheres) |
|
2 + 2 = 4 |
Three spheres (not connected)
(Disjoint union of three spheres) |
|
2 + 2 + 2 = 6 |
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 spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
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
n-dimensional sphere has Betti number 1 in dimensions 0 and
n, and all other Betti numbers 0. Hence its Euler characteristic is
— that is, either 0 or 2.
The
n-dimensional real
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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.
The soccer ball example
How many pentagons and hexagons does it take to make a soccer ball? Assume we use
hexagons and
pentagons; then we have
faces. Every pentagon (hexagon) has 5 vertices (6 vertices), and each one is shared between 3 faces, hence we have
vertices. Similarly, every pentagon (hexagon) has 5 edges (6 edges), and each one is shared between 2 faces, hence we have
edges. The Euler characteristic is thus
. Since the sphere has Euler characteristic 2, it follows that
. The result is that we always need 12 pentagons on a football/soccer ball; the number of hexagons is in principle unconstrained (but for a real football/soccer ball one obviously chooses a number that makes the ball as spherical as possible). This result is also applicable to fullerenes.
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
graphIn mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
is the number of vertices minus the number of edges.
More generally, one can define the Euler characteristic of any
chain complexIn mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
to be the alternating sum of the
ranksIn mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
of the homology groups of the chain complex.
A version used in
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
is as follows. For any
sheafIn mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
on a projective
schemeIn mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
X, one defines its Euler characteristic
where
is the dimension of the
i-th
sheaf cohomologyIn mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
group of
.
Another generalization of the concept of Euler characteristic on manifolds comes from
orbifoldIn the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
s. 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 2
π /
p.
The concept of Euler characteristic of a bounded finite
posetIn mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
is another generalization, important in
combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. 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 functionThe classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
in that poset's
incidence algebraIn order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered setand commutative ring with unity.-Definition:...
.
This can be further generalized by defining a
Q-valued Euler characteristic for certain finite
categoriesIn mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above. In this setting, the Euler characteristic of a finite
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
or
monoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
G is 1/|
G|, and the Euler characteristic of a finite
groupoidIn mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
is the sum of 1/|
Gi|, where we picked one representative group
Gi for each connected component of the groupoid.
See also
- Euler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
- List of uniform polyhedra
- List of topics named after Leonhard Euler
- Euler calculus
Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the euler characteristic as a finitely-additive measure...
Further reading
- Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
- H. Graham Flegg: From Geometry to Topology. Dover 2001, p. 40
External links