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Topological invariant in mathematics
topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that
Euler_characteristic
Concept in differential geometry
geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes
Euler characteristic of an orbifold
Euler_characteristic_of_an_orbifold
three cases: Euler–Lotka equation, a characteristic equation employed in mathematical demography Euler's pump and turbine equation Euler transform used
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Characteristic class of oriented, real vector bundles
in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted"
Euler_class
Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of
Local Euler characteristic formula
Local_Euler_characteristic_formula
Infinite series summing alternating 1 and -1 terms
infinite-dimensional sphere is contractible, its Euler characteristic is 1, and its 2-to-1 quotient should have an Euler characteristic of 1/2. This description of RP∞
Grandi's_series
Topological space that locally resembles Euclidean space
E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological
Manifold
Topological space
fibration becomes trivial after taking a finite cover of B. The orbifold Euler characteristic χ ( B ) {\displaystyle \chi (B)} of the orbifold B is given by χ
Seifert_fiber_space
Swiss mathematician (1707–1783)
no holes equals 2, a number now commonly known as the Euler characteristic. In physics, Euler reformulated Isaac Newton's laws of motion into new laws
Leonhard_Euler
Flat-sided three-dimensional shape
Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures
Polyhedron
Any of 4 regular star polyhedra
Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic Anthony Pugh (1976). Polyhedra: A Visual Approach. California: University
Kepler–Poinsot_polyhedron
{\displaystyle 2} before P ( ξ > a ) {\displaystyle P(\xi >a)} is in fact the Euler characteristic of the sphere (for the torus it vanishes). It is assumed that X {\displaystyle
Large deviations of Gaussian random functions
Large_deviations_of_Gaussian_random_functions
Tool in algebraic topology
cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem. In
Sheaf_cohomology
Notation for 2-dimensional spherical, euclidean and hyperbolic symmetry groups
orientable in the chiral case and non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each
Orbifold_notation
Association of cohomology classes to principal bundles
important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented
Characteristic_class
Mapping theorem in topology
reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic). In the particular
Lefschetz_fixed-point_theorem
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
is the Euler characteristic of M {\displaystyle M} . A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic
Poincaré–Hopf_theorem
Theorem in differential topology
all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero
Hairy_ball_theorem
Theorem in differential geometry
ds is the line element along the boundary of M. Here, χ(M) is the Euler characteristic of M. If the boundary ∂M is piecewise smooth, then we interpret the
Gauss–Bonnet_theorem
Concept in algebraic geometry
complex algebraic varieties. For a proper scheme X over a field k, the Euler characteristic of a coherent sheaf E on X is the integer χ ( X , E ) = ∑ j ( − 1
Coherent_sheaf_cohomology
Relates the integral of Gaussian curvature of surfaces to the Euler characteristic
the integral of Gaussian curvature of a non-compact surface to the Euler characteristic. It is akin to the Gauss–Bonnet theorem for a compact surface. A
Cohn-Vossen's_inequality
Term in mathematics
This implies that the middle homology group is determined by the Euler characteristic of the space. Hirzebruch gave a generating function computing the
Complete_intersection
Mathematical formula of two surfaces
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
Riemann–Hurwitz_formula
Mathematical conjectures attributed to Heinz Hopf
has positive Euler characteristic. A compact, (2d)-dimensional Riemannian manifold with negative sectional curvature has Euler characteristic of sign ( −
Hopf_conjecture
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the
Chern–Gauss–Bonnet_theorem
Number, approximately 3.14
{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma ),} where χ(Σ) is the Euler characteristic, which is an integer. An example is the surface area of a sphere
Pi
Roughly, the number of k-dimensional holes on a topological surface
}(-1)^{i}b_{i}(K,F),\,} where χ ( K ) {\displaystyle \chi (K)} denotes Euler characteristic of K and any field F. For any two spaces X and Y we have P X × Y
Betti_number
Relation between genus, degree, and dimension of function spaces over surfaces
The left hand side thus equals the Euler characteristic of the divisor D. When D = 0, we find the Euler characteristic for the structure sheaf is 1 − g
Riemann–Roch_theorem
states that the Euler characteristic of a compact affine manifold is zero. If any compact manifold is odd-dimensional, its Euler characteristic vanishes by
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
Classification of a two-dimensional repetitive pattern
orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group. The orbifold Euler characteristic is 2
Wallpaper_group
2008 mathematics book
to be equivalent to Euler's formula). It surveys the life of Euler, his discovery in the early 1750s that the Euler characteristic V − E + F {\displaystyle
Euler's_Gem
Theory in algebraic topology
j ( X j , X j − 1 ) {\displaystyle {H_{j}}(X_{j},X_{j-1})} . The Euler characteristic of X {\displaystyle X} is then defined by χ ( X ) = ∑ j = 0 n ( −
Cellular_homology
^{n}C)y^{n}u^{i-n}={\frac {(1+y)^{2g}}{(1-uy)(1-u^{-1}y)}}} and their Euler characteristics e(ΣnC) are given by the generating function ∑ n = 0 ∞ e ( Σ n C
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
gives the total curvature as 2 π {\displaystyle 2\pi } times the Euler characteristic χ = 2 {\displaystyle \chi =2} , so for a convex polyhedron the sum
Angular_defect
The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field.
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
Representation of mathematical space
quantities arising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows one to assign such quantities to topological
Triangulation_(topology)
Research topic in computational geometry
holes). So in this case, the Euler characteristic is -1. To bring this into the discrete world, the Euler characteristic of a mesh is computed in terms
Geometry_processing
Associative algebra used in combinatorics
(not to be confused with the 0 and 1 of the ring of scalars). The Euler characteristic of a bounded finite poset is μ(0,1). The reason for this terminology
Incidence_algebra
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf cohomology, namely the alternating sum χ ( X , E )
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Number of "holes" of a surface
handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ {\displaystyle \chi } , via the relationship χ = 2 − 2 g {\displaystyle
Genus_(mathematics)
Two-dimensional manifold
family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of
Surface_(topology)
Type of geometric object
by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher
Uniform_9-polytope
Polyhedron with regular congruent polygons as faces
also contain tetrahedral symmetry. The five Platonic solids have an Euler characteristic of 2. This simply reflects that the surface is a topological 2-sphere
Regular_polyhedron
Invariant of mathematical knots
turns out to be an invariant of L {\displaystyle L} , and its graded Euler characteristic is the Jones polynomial of L {\displaystyle L} . This definition
Khovanov_homology
Mathematical theorem
{1}{2}}D.(D-K)\,} where χ {\displaystyle \chi } is the holomorphic Euler characteristic, the dot . {\displaystyle .} is the intersection number, and K {\displaystyle
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Planar maps require at most four colors
positive genus, the maximum number p of colors needed depends on the Euler characteristic χ of the surface. Except for the Klein bottle, the formula is as
Four_color_theorem
Connects homology and cohomology groups for oriented closed manifolds
odd-dimensional manifold M has Euler characteristic zero, which in turn gives that any manifold that bounds has even Euler characteristic. Poincaré duality is closely
Poincaré_duality
Property that is not changed by mathematical transformations
presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number
Invariant_(mathematics)
Algebraic structure associated with a topological space
software. Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemann's definition of genus
Homology_(mathematics)
Twenty-second letter of the Greek alphabet
target model In algebraic topology, Chi is used to represent the Euler characteristic of a surface. The chromatic number of a graph in graph theory In
Chi_(letter)
Computes the Euler characteristic of an orbifold
Kawasaki, is the Riemann–Roch formula for orbifolds. It can compute the Euler characteristic of an orbifold. Kawasaki's original proof made a use of the equivariant
Kawasaki's Riemann–Roch formula
Kawasaki's_Riemann–Roch_formula
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
Euler characteristic and the zeros of vector fields. For example, the circle S1 has Betti numbers 1, 1, 0, 0, 0, and therefore Euler characteristic 0
Diagonal
recently definable functions by integrating with respect to the Euler characteristic as a finitely-additive measure. In the presence of a metric, it can
Euler_calculus
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Plane figure, bounded by circle
groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is
Disk_(mathematics)
Type of geometrical object
by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher
Uniform_10-polytope
Three-holed sphere
compact surface of genus zero with three boundary components. The Euler characteristic of a pair of pants is equal to −1. The only other orientable surface
Pair_of_pants_(mathematics)
Geometric object with flat sides
respectively in two and three dimensions. Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development
Polytope
Mathematics of smooth surfaces
value of the Gaussian curvature is completely determined by the Euler characteristic of the surface together with its surface area. Any regular surface
Differential geometry of surfaces
Differential_geometry_of_surfaces
Concept in algebraic topology
the Euler characteristic of the total space is given by: χ ( E ) = χ ( B ) χ ( F ) . {\displaystyle \chi (E)=\chi (B)\chi (F).} Here the Euler characteristics
Fibration
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
hence positive Euler characteristic (equal to 2). The second gives all flat 2-manifolds, i.e. the tori, which have Euler characteristic 0. The third case
Uniformization_theorem
Generalized manifold
{\displaystyle O} is Z 2 {\displaystyle \mathbb {Z} _{2}} and its orbifold Euler characteristic is 1. Like a manifold, an orbifold is specified by local conditions;
Orbifold
Exceptional functor
property for the Euler characteristic of a fiber bundle. Gottlieb, Daniel Henry (1975), "Fibre bundles and the Euler characteristic" (PDF), Journal of
Shriek_map
Type of smooth complex surface of kodaira dimension 0
{O}}_{X})=h^{0}(X,K_{X})=1.} As a result, the arithmetic genus (or holomorphic Euler characteristic) of X is: χ ( X , O X ) := ∑ i ( − 1 ) i h i ( X , O X ) = 1 − 0
K3_surface
Study of space and shapes locally given by a convergent power series
its origin) and algebraic curves. For an orientable surface S the Euler characteristic χ(S) is 2 − 2 g {\displaystyle 2-2g\,} where g is the genus (the
Geometric_function_theory
Euler showed that V-E+F= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic
History of manifolds and varieties
History_of_manifolds_and_varieties
Toroidal polyhedron with 7 faces
edge with each other face, it follows by some manipulation of the Euler characteristic that h = ( f − 4 ) ( f − 3 ) 12 . {\displaystyle h={\frac {(f-4)(f-3)}{12}}
Szilassi_polyhedron
Topics referred to by the same term
eigenvector of a matrix Characteristic word, a subclass of Sturmian word Euler characteristic, a topological invariant Method of characteristics, a technique for
Characteristic
Family of tetrahedra on an integer lattice
counting lattice points from finer lattices and incorporating the Euler characteristic of the polyhedron. All vertices of a Reeve tetrahedron are integer
Reeve_tetrahedra
Point where two or more curves, lines, or edges meet
its zero-dimensional faces. Any convex polyhedron's surface has Euler characteristic V − E + F = 2 , {\displaystyle V-E+F=2,} where V is the number of
Vertex_(geometry)
Branch of topology
family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of
Low-dimensional_topology
Natural number
25^{2}-23^{2}} . Skilling's figure, a degenerate uniform polyhedron, has Euler characteristic χ = − 96. {\displaystyle \chi =-96.} Every integer greater than 96
96_(number)
Self-intersecting polyhedron with 12 faces
yielding an Euler characteristic of 20 − 36 + 12 = −4. Escher's solid instead has 48 triangular faces, 72 edges, and 26 vertices, yielding an Euler characteristic
First stellation of the rhombic dodecahedron
First_stellation_of_the_rhombic_dodecahedron
Surface of revolution with a hole in the middle
surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1−g). The torus is an example of a toroid
Toroid
No-go theorem concerning chirality of regularized fermions
the Euler characteristic of that manifold. In this case, the vector field lives on the Brillouin zone which is topologically a 4-torus which has Euler characteristic
Nielsen–Ninomiya_theorem
integral of the Gaussian curvature may be computed in terms of the Euler characteristic χ ( S ) {\displaystyle \chi (S)} of the surface, so ∫ S K d A = 2
Willmore_energy
Non-orientable mathematical surface
structure with one 0-cell P, two 1-cells C1, C2 and one 2-cell D. Its Euler characteristic is therefore 1 − 2 + 1 = 0. The boundary homomorphism is given by
Klein_bottle
Mathematical theory
Riemann–Hurwitz formula holds, in particular, the Euler characteristic of X is at most the Euler characteristic of Y times the degree. Now suppose that some
Ahlfors_theory
Serre defined the intersection multiplicity of R/P and R/Q by the Euler characteristic-like formula: χ ( R / P , R / Q ) := ∑ i = 0 ∞ ( − 1 ) i ℓ R ( Tor
Serre's multiplicity conjectures
Serre's_multiplicity_conjectures
Doughnut-shaped surface of revolution
is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•( T n {\displaystyle
Torus
Subject area in mathematics
Riemann surface, the Euler characteristic of a line bundle equals the difference in dimensions mentioned previously, the Euler characteristic of the trivial
Algebraic_K-theory
Rational function of the form (az + b)/(cz + d)
transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: χ ( C ^ ) = 2. {\displaystyle \chi ({\hat
Möbius_transformation
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem. The
Kodaira_vanishing_theorem
Concept in geometry
computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties (in the mixed characteristic case). Let Y be a pure-dimensional
Localized_Chern_class
Four-dimensional geometric object with flat sides
by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher
4-polytope
Four-dimensional analogues of the regular polyhedra in three dimensions
the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2). That
Regular_4-polytope
Branch of mathematics
17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably
Topology
Polyhedral compound
with degree 8, and 20 with degree 12), and 540 edges, yielding an Euler characteristic of 182 − 540 + 360 = 2. Its convex hull is a regular dodecahedron
Compound_of_five_cubes
Smooth closed surface with g holes
Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. The genus (sometimes called the demigenus or Euler
Genus_g_surface
non-regular hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8. It is an isohedron, meaning that all of its faces are symmetric
Small_triambic_icosahedron
{\displaystyle \chi } is the Euler characteristic. The crosscap number of the unknot is zero, as the Euler characteristic of the disk is one. The crosscap
Crosscap_number
Type of polyhedron
interior, defining the removed central rhombic dodecahedron. With Euler characteristic χ = f + v - e = -20, its genus, g = (2-χ)/2 is 11. Rhombicuboctahedron
Expanded_cuboctahedron
Cartan–Hadamard manifolds? Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. Chern's conjecture for hypersurfaces
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Symbols for constants, special functions
distribution) the chromatic number of a graph in graph theory the Euler characteristic in algebraic topology electronegativity in the periodic table the
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Theorem on graph coloring on surfaces
x\right\rfloor } is the floor function. Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable
Heawood_conjecture
Function of a knot that takes the same value for equivalent knots
well-known invariants. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective
Knot_invariant
Covalent chemical bond
Natoms + Nrings − 1 This rule is a special-case application of the Euler characteristic of the graph which represents the molecule. A molecule with no rings
Sigma_bond
summing of homology 3-spheres. The Casson invariant is a sort of Euler characteristic for Floer homology. For any integer n λ ( M + 1 n + 1 ⋅ K ) − λ (
Casson_invariant
Plane tiling corresponding to a polyhedron
decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast
Projective_polyhedron
Uniform star polyhedron with 204 faces
has 4 square faces passing through the center of the model. The Euler characteristic of the abstract polyhedron is −96. If the pairs of coinciding edges
Great disnub dirhombidodecahedron
Great_disnub_dirhombidodecahedron
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