In mathematics , and more specifically in algebraic topology and polyhedral combinatorics , the EULER CHARACTERISTIC (or EULER NUMBER, or EULER–POINCARé CHARACTERISTIC) is a topological invariant , a number that describes a topological space 's shape or structure regardless of the way it is bent. It is commonly denoted by {displaystyle chi } (Greek lower-case letter chi ). The
CONTENTS * 1 Polyhedra * 1.1 Plane graphs * 1.2 Proof of Euler\'s formula * 2 Topological definition * 3 Properties * 3.1 Homotopy invariance
* 3.2
* 4 Examples * 4.1 Surfaces
* 4.2
* 5 Relations to other invariants * 6 Generalizations * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 Further reading * 10 External links POLYHEDRA The EULER CHARACTERISTIC {displaystyle chi } was classically defined for the surfaces of polyhedra, according to the formula = V E + F {displaystyle chi =V-E+F} where V, E, and F are respectively the numbers of vertices (corners),
edges and faces in the given polyhedron. Any convex polyhedron 's
surface has
This equation is known as EULER\'S POLYHEDRON FORMULA. It
corresponds to the
NAME IMAGE Vertices V Edges E Faces F Euler characteristic: V − E + F 4 6 4 2
8 12 6 2 6 12 8 2 20 30 12 2 12 30 20 2 The surfaces of nonconvex polyhedra can have various Euler characteristics; NAME IMAGE Vertices V Edges E Faces F Euler characteristic: V − E + F
6 12 7 1
12 24 12 0
12 24 10 −2
12 30 20 2 For regular polyhedra,
This version holds both for convex polyhedra (where the densities are all 1) and the non-convex Kepler-Poinsot polyhedra .
PLANE GRAPHS See also: Planar graph § Euler\'s formula The
The
Via stereographic projection the plane maps to the two-dimensional
sphere, such that a connected graph maps to a polygonal decomposition
of the sphere, which has
PROOF OF EULER\'S FORMULA First steps of the proof in the case of a cube There are many proofs of Euler's formula. One was given by Cauchy in 1811, as follows. It applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks. Remove one face of the polyhedral surface. By pulling the edges of the missing face away from each other, deform all the rest into a planar graph of points and curves, as illustrated by the first of the three graphs for the special case of the cube. (The assumption that the polyhedral surface is homeomorphic to the sphere at the beginning is what makes this possible.) After this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, but the number of faces has been reduced by 1. Therefore, proving Euler's formula for the polyhedron reduces to proving V − E + F =1 for this deformed, planar object. If there is a face with more than three sides, draw a diagonal—that
is, a curve through the face connecting two vertices that aren't
connected yet. This adds one edge and one face and does not change the
number of vertices, so it does not change the quantity V − E + F.
(The assumption that all faces are disks is needed here, to show via
the
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 Proofs of Euler's Formula by David
Eppstein . Multiple proofs, including their flaws and limitations,
are used as examples in
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 complexes .) In general, for any finite CW-complex, the EULER CHARACTERISTIC can be defined as the alternating sum = k 0 k 1 + k 2 k 3 + , {displaystyle chi =k_{0}-k_{1}+k_{2}-k_{3}+cdots ,} where kn denotes the number of cells of dimension n in the complex. Similarly, for a simplicial complex, the EULER CHARACTERISTIC equals the alternating sum = k 0 k 1 + k 2 k 3 + , {displaystyle chi =k_{0}-k_{1}+k_{2}-k_{3}+cdots ,} where kn denotes the number of n-simplexes in the complex. More generally still, for any topological space , we can define the
nth
This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n0. For simplicial complexes, this is not the same definition as in the previous paragraph but a homology computation shows that the two definitions will give the same value for {displaystyle chi } . PROPERTIES The
HOMOTOPY INVARIANCE Homology is a topological invariant, and moreover a homotopy invariant : Two topological spaces that are homotopy equivalent have isomorphic homology groups. It follows that the Euler characteristic is also a homotopy invariant. For example, any contractible space (that is, one homotopy equivalent
to a point) has trivial homology, meaning that the 0th
For another example, any convex polyhedron is homeomorphic to the three-dimensional ball , so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional sphere , 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 is the sum of their Euler characteristics, since homology is additive under disjoint union: ( M N ) = ( M ) + ( N ) . {displaystyle chi (Msqcup N)=chi (M)+chi (N).} More generally, if M and N are subspaces of a larger space X, then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the inclusion–exclusion principle : ( M N ) = ( M ) + ( N ) ( M N ) . {displaystyle chi (Mcup N)=chi (M)+chi (N)-chi (Mcap N).} This is true in the following cases: * if M and N are an excisive couple . In particular, if the interiors of M and N inside the union still cover the union. * if X is a locally compact space , and one uses Euler characteristics with compact supports , no assumptions on M or N are needed. * if X is a stratified space all of whose strata are 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 . In general, the inclusion–exclusion principle is false. A counterexample is given by taking X to be the real line , M a subset consisting of one point and N the complement of M. CONNECTED SUM For two connected closed n-manifolds M , N {displaystyle M,N}
one can obtain a new connected manifold M N {displaystyle M#N}
via the connected sum operation. The
PRODUCT PROPERTY Also, the
These addition and multiplication properties are also enjoyed by
cardinality of sets . In this way, the
COVERING SPACES For more details on this topic, see
Similarly, for an k-sheeted covering space M M , {displaystyle {tilde {M}}to M,} one has ( M ) = k ( M ) . {displaystyle chi ({tilde {M}})=kcdot chi (M).} More generally, for a ramified covering space , the Euler
characteristic of the cover can be computed from the above, with a
correction factor for the ramification points, which yields the
FIBRATION PROPERTY The product property holds much more generally, for fibrations with certain conditions. If p E B {displaystyle pcolon Eto B} is a fibration with
fiber F, with the base B path-connected , and the fibration is
orientable over a field K, then the
This includes product spaces and covering spaces as special cases,
and can be proven by the
For fiber bundles, this can also be understood in terms of a transfer
map H ( B ) H ( E ) {displaystyle tau colon
H_{*}(B)to H_{*}(E)} – note that this is a lifting and goes "the
wrong way" – whose composition with the projection map p
H ( E ) H ( B ) {displaystyle p_{*}colon H_{*}(E)to
H_{*}(B)} is multiplication by the
EXAMPLES SURFACES The
NAME IMAGE EULER CHARACTERISTIC Interval 1 0 Disk 1 2
−2 −4 1 0 0 Two spheres (not connected)
(
Three spheres (not connected)
(
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
Because the sphere has
ARBITRARY DIMENSIONS The n-dimensional sphere has singular homology groups equal to H k ( S n ) = { Z k = 0 , n { 0 } otherwise, {displaystyle H_{k}(S^{n})={begin{cases}mathbb {Z} &k=0,n\{0} width:29.41ex; height:6.176ex;" alt="{displaystyle H_{k}(S^{n})={begin{cases}mathbb {Z} &k=0,n\{0} intuitively, the number of "handles") as = 2 2 g . {displaystyle chi =2-2g.} The
For closed smooth manifolds, the
For closed Riemannian manifolds , the
A discrete analog of the
Hadwiger\'s theorem characterizes the
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
More generally, one can define the
A version of
where h i ( X , F ) {displaystyle h^{i}(X,{mathcal
{F}})} is the dimension of the i-th sheaf cohomology group of
F {displaystyle {mathcal {F}}} . In this case, the dimensions are
all finite by Grothendieck\'s finiteness theorem . This is an instance
of the
Another generalization of the concept of
The concept of
This can be further generalized by defining a Q-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
SEE ALSO *
REFERENCES NOTES * ^ Richeson 2008
* ^ Eppstein, David. "Twenty Proofs of Euler\'s Formula: V-E+F=2".
Retrieved 3 June 2013.
* ^
BIBLIOGRAPHY * Richeson, David S.; Euler's Gem: The
FURTHER READING * Flegg, H. Graham; From Geometry to Topology, Dover 2001, p. 40. EXTERNAL LINKS * Weisstein, Eric Wolfgang . "Euler characteristic". |