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In mathematics, 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 invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
and
polyhedral combinatorics Polyhedral 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. Research in polyhedral com ...
, 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 spa ...
, a number that describes a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter
chi Chi or CHI may refer to: Greek *Chi (letter), the Greek letter (uppercase Χ, lowercase χ); Chinese * ''Chi'' (length) (尺), a traditional unit of length, about ⅓ meter * Chi (mythology) (螭), a dragon *Chi (surname) (池, pinyin: ''chí' ...
). The Euler characteristic was originally defined for
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
and used to prove various theorems about them, including the classification of the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...
s. It was stated for Platonic solids in 1537 in an unpublished manuscript by
Francesco Maurolico Francesco Maurolico ( Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions ...
.
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
, 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 in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
.


Polyhedra

The Euler characteristic \chi was classically defined for the surfaces of polyhedra, according to the formula :\chi=V-E+F where ''V'', ''E'', and ''F'' are respectively the numbers of vertices (corners),
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
s and
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may af ...
in the given polyhedron. Any
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
's surface has Euler characteristic :V - E + F = 2. This equation, stated by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
in 1758, is known as Euler's polyhedron formula. It corresponds to the Euler characteristic of the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
(i.e. χ = 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 prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
derived a modified form of Euler's formula using the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematica ...
''D'',
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw l ...
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 is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, while the surfaces of toroidal polyhedra all have Euler characteristic 0, like the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
.


Plane graphs

The Euler characteristic can be defined for connected plane graphs by the same V - E + F formula as for polyhedral surfaces, where ''F'' is the number of faces in the graph, including the exterior face. The Euler characteristic of any plane connected graph G is 2. This is easily proved by induction on the number of faces determined by G, starting with a tree as the base case. For trees, E = V-1 and F = 1. If G has C components (disconnected graphs), the same argument by induction on F shows that V - E + F - C = 1. One of the few graph theory papers of Cauchy also proves this result. Via
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
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. 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 In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exteri ...
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 ''Twenty-one Proofs of Euler's Formula'' by
David Eppstein David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a Distinguished Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algori ...
. Multiple proofs, including their flaws and limitations, are used as examples in '' Proofs and Refutations'' by
Imre Lakatos Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) 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 ...
.


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 set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial s ...
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 set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial s ...
, 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, 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 Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
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''-d ...
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 have
isomorphic In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
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 th ...
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, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
\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 can sometimes be 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 fo ...
, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
, 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 union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
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 In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \c ...
: :\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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
supports, no assumptions on ''M'' or ''N'' are needed. *if ''X'' is a
stratified space In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset o ...
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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. M ...
. 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 "John Smith is not a lazy student" is a ...
is given by taking ''X'' to be the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, ''M'' a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...
consisting of one point and ''N'' the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
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 In 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. This construction plays a key role in the classifica ...
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-seem ...
''M'' × ''N'' is :\chi(M \times N) = \chi(M) \cdot \chi(N). These addition and multiplication properties are also enjoyed by
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of sets. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; se


Covering spaces

Similarly, for a ''k''-sheeted
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
\tilde \to M, one has :\chi(\tilde) = k \cdot \chi(M). More generally, for a
ramified covering space In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
, 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 In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramificat ...
.


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 In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homologica ...
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 of ...
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 Telstar is a football made by Adidas. The iconic 32-panel alternating black-and-white design of the ball, based on the work of Eigil Nielsen, has since become a global standard design used to portray a football in different media. Ball The ba ...
). 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 is thus :V - E + F = \frac - \frac + P + H = \frac. 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. In principle, the number of hexagons is unconstrained. This result is applicable to
fullerene A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
s and
Goldberg polyhedra In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three pr ...
.


Arbitrary dimensions

The ''n''-dimensional sphere has singular homology groups equal to :H_k(S^n) = \begin \mathbb Z & k=0, n \\ \ & \text \end hence has Betti number 1 in dimensions 0 and ''n'', and all other Betti numbers are 0. Its Euler characteristic is then 1 + (−1)''n'' — that is, either 0 or 2. The ''n''-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 ''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 In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equi ...
, including any compact
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
, has Euler characteristic 0. The Euler characteristic of any
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold ( compa ...
. 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
stratified space In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset o ...
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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
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 ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial no ...
''g'' (the number of tori in a
connected sum In 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. This construction plays a key role in the classifica ...
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 ''k'' (the number of
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
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 of ...
of its
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
evaluated on the
fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...
of a manifold. The Euler class, in turn, relates to all other
characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
es 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 every ...
s. For closed
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
s, the Euler characteristic can also be found by integrating the curvature; see the
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a tr ...
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 In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry). Hadwiger's theorem characterizes the Euler characteristic as the ''unique'' (
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
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 polyto ...
sets in R''n'' that is "homogeneous of degree 0".


Generalizations

For every combinatorial
cell complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
, 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 discr ...
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 included in the kernel of ...
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, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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 ''X'', 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 ''i''-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 to solving a geometric problem globally when ...
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 which 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 1 + 1/''p'', where ''p'' is a prime number corresponding to the cone angle 2' / ''p''. The concept of Euler characteristic of the
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise st ...
of a bounded finite
poset In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
is another generalization, important in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
. 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 The Möbius function 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 most of ...
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. Subalgebras called reduced incidence algebras give a natural construct ...
. This can be further generalized by defining a Q-valued Euler characteristic for certain finite
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) * ...
, a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above. In this setting, the Euler characteristic of a finite
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
or
monoid In abstract algebra, a branch of mathematics, 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 0. Monoids a ...
''G'' is 1/, ''G'', , 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 funct ...
is the sum of 1/, ''Gi'', , where we picked one representative group ''Gi'' for each connected component of the groupoid.Tom Leinster,
The Euler characteristic of a category
", ''Documenta Mathematica'', 13 (2008), pp. 21–49


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 of ...
*
List of topics named after Leonhard Euler 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
* List of uniform polyhedra


References


Notes


Bibliography

* Richeson, David S.; '' Euler's Gem: The Polyhedron Formula and the Birth of Topology''. Princeton University Press 2008.


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