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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a
topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
, a number that describes a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
'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 In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
and used to prove various theorems about them, including the classification of the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico.
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, 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 Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
.


Polyhedra

The Euler characteristic was classically defined for the surfaces of polyhedra, according to the formula : \chi = V - E + F where , , and are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any
convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
's surface has Euler characteristic :\ \chi = V - E + F = 2 ~. This equation, stated by
Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
in 1758, is known as Euler's polyhedron formula. It corresponds to the Euler characteristic of the
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(i.e. \ \chi = 2\ ), and applies identically to spherical polyhedra. An illustration of the formula on all Platonic polyhedra is given below. The surfaces of nonconvex polyhedra can have various Euler characteristics: For regular polyhedra,
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...
derived a modified form of Euler's formula using the
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
,
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
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 In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
, while the surfaces of
toroidal polyhedra In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topology (Mathematics), topological Genus (mathematics), genus () of 1 or greater. Notable examples include the Császár polyhedron, Császár a ...
all have Euler characteristic 0, like the
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
.


Plane graphs

The Euler characteristic can be defined for connected plane graphs by the same \ V - E + F\ formula as for polyhedral surfaces, where is the number of faces in the graph, including the exterior face. The Euler characteristic of any plane connected graph is 2. This is easily proved by induction on the number of faces determined by , starting with a tree as the base case. For
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only p ...
, \ E = V - 1\ and \ F = 1 ~. If has components (disconnected graphs), the same argument by induction on shows that \ V - E + F - C = 1 ~. One of the few graph theory papers of Cauchy also proves this result. Via
stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
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.


Proof of Euler's formula

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 In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be ...
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: #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 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).


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 complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
es.) 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 In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
, 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.


Betti number alternative

More generally still, for any
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, we can define the ''n''th
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
''b''''n'' as the rank of the ''n''-th
singular homology In algebraic topology, 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(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional ...
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 ''n''0. 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.


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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
have
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
homology groups. It follows that the Euler characteristic is also a homotopy invariant. For example, any
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...
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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\mathbb^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 A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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 In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
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 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. More precisely, it is a topological space in which ...
, and one uses Euler characteristics with
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
. In general, the inclusion–exclusion principle is false. A
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
is given by taking ''X'' to be the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
, ''M'' a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
consisting of one point and ''N'' the complement of ''M''.


Connected sum

For two connected closed n-manifolds M, N one can obtain a new connected manifold M \# N via the connected sum operation. The Euler characteristic is related by the formula : \chi(M \# N) = \chi(M) + \chi(N) - \chi(S^n).


Product property

Also, the Euler characteristic of any
product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
''M'' × ''N'' is :\chi(M \times N) = \chi(M) \cdot \chi(N). These addition and multiplication properties are also enjoyed by
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; se


Covering spaces

Similarly, for a ''k''-sheeted
covering space In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
\tilde \to M, one has :\chi(\tilde) = 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 p\colon E \to B is a fibration with fiber ''F,'' with the base ''B''
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
, 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 In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
of the fiber: :p_* \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.


Soccer ball

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 pentagons and hexagons are used, then there are \ F = P + H\  faces, \ V = \tfrac\left(\ 5 P + 6 H\ \right)\  vertices, and \ E = \tfrac \left(\ 5P + 6H\ \right)\  edges. The Euler characteristic is thus : V - E + F = \tfrac \left(\ 5 P + 6 H\ \right) - \tfrac \left(\ 5 P + 6 H\ \right) + P + H = \tfrac P ~. Because the sphere has Euler characteristic 2, it follows that \ P = 12 ~. That is, a soccer ball constructed in this way always has 12 pentagons. The number of hexagons can be any
nonnegative integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
except 1. This result is applicable to fullerenes and Goldberg polyhedra.


Arbitrary dimensions

The  dimensional sphere has singular
homology group In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
s equal to :H_k(\mathrm^n) = \begin \mathbb ~& k = 0 ~~ \mathsf ~~ k = n \\ \ & \mathsf\ , \end hence has Betti number 1 in dimensions 0 and , and all other Betti numbers are 0. Its Euler characteristic is then that is, either 0 if is odd, or 2 if is even. The  dimensional real
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
is the quotient of the  sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The  dimensional torus is the product space of  circles. Its Euler characteristic is 0, by the product property. More generally, any compact
parallelizable manifold In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, ...
, including any compact
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, 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 In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...
. This property applies more generally to any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
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 A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
can be calculated from its
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
(the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as :\chi = 2 - 2g ~. The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus (the number of
real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
s in a connected sum decomposition of the surface) as :\chi = 2 - k ~. For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the
Euler class In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
of its
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
evaluated on the fundamental class of a manifold. The Euler class, in turn, relates to all other characteristic classes of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s. For closed
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
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 A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
for the general case. A discrete analog of the Gauss–Bonnet theorem is Descartes' theorem that the "total defect" of a
polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
, measured in full circles, is the Euler characteristic of the polyhedron. Hadwiger's theorem characterizes the Euler characteristic as the ''unique'' (
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on finite unions of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
sets in 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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
is the number of vertices minus the number of edges. More generally, one can define the Euler characteristic of any
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
to be the alternating sum of the ranks of the homology groups of the chain complex, assuming that all these ranks are finite. A version of Euler characteristic used in
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
is as follows. For any
coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
\mathcal on a proper scheme , one defines its Euler characteristic to be : \chi ( \mathcal)= \sum_i (-1)^i h^i(X,\mathcal)\ , where \ h^i(X, \mathcal)\ is the dimension of the -th
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...
group of \mathcal. 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 \ \mathcal\ by acyclic sheaves. Another generalization of the concept of Euler characteristic on manifolds comes from
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...
s (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 where is a prime number corresponding to the cone angle . The concept of Euler characteristic of the reduced homology of a bounded finite
poset In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
is another generalization, important in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
. 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 , where is the
Möbius function The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...
in that poset's
incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
. 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 In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
is , and the Euler characteristic of a finite
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...
is the sum of , where we picked one representative group for each connected component of the groupoid.


See also

* Euler calculus *
Euler class In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
* List of topics named after Leonhard Euler * List of uniform polyhedra


References


Notes


Bibliography

*


Further reading

*Flegg, H. Graham; ''From Geometry to Topology'', Dover 2001, p. 40.


External links

* * *
An animated version of a proof of Euler's formula using spherical geometry
{{Topology Algebraic topology Topological graph theory Polyhedral combinatorics Articles containing proofs Leonhard Euler