4010:
6153:
1194:
727:
755:
5936:
3362:
3381:
1210:
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
6174:
6142:
3246:
643:
699:
671:
3340:
125:
3290:
6211:
6184:
6164:
3569:
3559:
3510:
3505:
3500:
3312:
3424:
3446:
3402:
504:
532:
416:
476:
448:
3474:
3268:
1209:
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
4503:
4826:, 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
1257:
If there is a face with more than three sides, draw a diagonalâthat 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
1402:
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
3949:
1184:
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.
4983:
1314:
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
4343:
2336:
1205:
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.
3786:
3714:
2620:
2710:
108:, 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
2026:
1879:
1768:
2848:
2225:
3199:
2093:
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
3138:
3067:
2990:
3794:
1358:
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
5038:
934:
5117:). While every manifold has an integer Euler characteristic, an orbifold can have a fractional Euler characteristic. For example, the teardrop orbifold has Euler characteristic
5104:
2783:
2038:. 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
274:
2123:
630:
403:
5070:
4873:
2906:
1175:
4671:
4133:
4723:
1647:
1603:
1559:
1252:
321:
178:
2125:
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.
862:
826:
3642:
1397:
1354:
1297:
1075:
1018:
3987:
1512:
1477:
1443:
1110:
4165:
2056:
80:
2523:
2497:
6214:
5179:. 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
3540:
5209:, a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above. In this setting, the Euler characteristic of a finite
4888:
5301:
5707:
4498:{\displaystyle H_{k}(\mathrm {S} ^{n})={\begin{cases}\mathbb {Z} ~&k=0~~{\mathsf {or}}~~k=n\\\{0\}&{\mathsf {otherwise}}\ ,\end{cases}}}
3214:
The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface (that is, a description as a
2857:, the Euler characteristic of the cover can be computed from the above, with a correction factor for the ramification points, which yields the
5508:
5350:
4764:
5378:
5332:
5114:
1303:
that this operation increases the number of faces by one.) Continue adding edges in this manner until all of the faces are triangular.
5848:
2252:
2243:
6245:
6240:
6202:
6197:
5422:
3719:
3647:
2535:
5528:
6192:
4595:
all of whose strata have odd dimension. It also applies to closed odd-dimensional non-orientable manifolds, via the two-to-one
1306:
Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a
5669:
2647:
4592:
4554:. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1.
2403:
1950:
1803:
1692:
940:
6094:
5814:
5804:
2791:
2162:
5076:. This is an instance of the Euler characteristic of a chain complex, where the chain complex is a finite resolution of
4827:
3944:{\displaystyle V-E+F={\tfrac {1}{3}}\left(\ 5P+6H\ \right)-{\tfrac {1}{2}}\left(\ 5P+6H\ \right)+P+H={\tfrac {1}{6}}P~.}
748:
720:
3150:
2858:
2735:
6235:
5809:
3072:
3008:
2933:
976:
3594:
by stitching together pentagonal and hexagonal pieces, with three pieces meeting at each vertex (see for example the
6102:
2031:
This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index
5306:
5073:
4991:
4760:
6173:
5901:
4838:
3989:
That is, a soccer ball constructed in this way always has 12 pentagons. The number of hexagons can be any
2066:
The Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.
1929:
1181:
39:
6250:
6187:
4676:
The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus
4566:
4009:
2996:
2854:
2458:
869:
2242:, then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the
6122:
6117:
6043:
5920:
5908:
5881:
5841:
5440:
5206:
3239:
1658:
109:
2136:, which has Euler characteristic 2. This explains why convex polyhedra have Euler characteristic 2.
5964:
5891:
5172:
5079:
4796:
4681:
4565: circles. Its Euler characteristic is 0, by the product property. More generally, any compact
3393:
2750:
2384:
2377:
1202:
950:
55:
6152:
4785:
226:
2099:
6112:
6038:
5886:
5632:
4876:
4745:
2354:
1300:
946:
591:
364:
5051:
4854:
4830:
is the number of vertices minus the number of edges. (Olaf Post calls this a "well-known formula".)
4383:
2879:
6163:
5959:
5210:
4756:
4615:
4585:
3990:
3069:â note that this is a lifting and goes "the wrong way" â whose composition with the projection map
2358:
2079:
1404:
1127:
1036:
636:
324:
113:
101:
24:
4635:
4100:
1193:
6157:
6107:
6028:
6018:
5896:
5876:
5692:
5622:
5551:
4845:
4690:
4611:
4028:
3998:
2129:
2075:
1790:
1679:
1608:
1564:
1517:
1213:
954:
782:
692:
664:
294:
145:
35:
6127:
4799:) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on
939:
This version holds both for convex polyhedra (where the densities are all 1) and the non-convex
831:
795:
3609:
1361:
1318:
1261:
1042:
985:
754:
726:
6145:
6011:
5969:
5834:
5785:
5766:
5515:
5504:
5346:
5195:
5191:
4807:
4775:
4771:
4741:
4041:
3957:
2723:. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; see
2720:
2426:
2083:
1937:
1909:
1482:
1447:
1413:
217:
196:
59:
1080:
5925:
5871:
5684:
5543:
5338:
5168:
5045:
4543:
4138:
2995:
This includes product spaces and covering spaces as special cases, and can be proven by the
2156:
is the sum of their Euler characteristics, since homology is additive under disjoint union:
3380:
3361:
2041:
65:
5984:
5979:
5474:
5464:
William Fulton: Introduction to toric varieties, 1993, Princeton
University Press, p. 141.
5202:
4577:
4083:
4069:
4055:
2502:
2094:
441:
210:
203:
5769:
3417:
2476:
5636:
2086:
homology groups. It follows that the Euler characteristic is also a homotopy invariant.
1674:
The polyhedral surfaces discussed above are, in modern language, two-dimensional finite
1656:(2013). Multiple proofs, including their flaws and limitations, are used as examples in
1561:
Since each of the two above transformation steps preserved this quantity, we have shown
6074:
6006:
5788:
5730:
5726:
5370:
5366:
5291:
4849:
4841:
of the homology groups of the chain complex, assuming that all these ranks are finite.
4737:
4334:
3595:
3525:
2916:
2745:
2434:
2423:
2153:
1653:
280:
105:
97:
83:
19:
This article is about Euler characteristic number. For Euler characteristic class, see
5109:
Another generalization of the concept of Euler characteristic on manifolds comes from
6229:
6084:
5994:
5974:
5649:
5555:
5494:
5176:
4834:
4804:
4800:
4749:
4626:
4608:
4596:
4589:
4581:
4551:
2631:
2526:
2406:
all of whose strata are even-dimensional, the inclusionâexclusion principle holds if
2381:
789:
778:
6177:
2132:, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional
2128:
For another example, any convex polyhedron is homeomorphic to the three-dimensional
6069:
5989:
5935:
5700:
5696:
5436:
5404:
4823:
4759:, the Euler characteristic can also be found by integrating the curvature; see the
3439:
3374:
3355:
3003:
2090:
1917:
1663:
1307:
970:
642:
497:
96:
and used to prove various theorems about them, including the classification of the
86:
3339:
698:
670:
62:'s shape or structure regardless of the way it is bent. It is commonly denoted by
5498:
4978:{\displaystyle \chi ({\mathcal {F}})=\sum _{i}(-1)^{i}h^{i}(X,{\mathcal {F}})\ ,}
6167:
6079:
5583:
5296:
4733:
4532:
3591:
3245:
3141:
2716:
979:
525:
409:
139:
was classically defined for the surfaces of polyhedra, according to the formula
31:
20:
3568:
3558:
3509:
3504:
3499:
3311:
124:
6023:
5954:
5913:
5820:
5618:
5342:
5337:. Science Networks. Historical Studies. Vol. 59. BirkhÀuser. p. 71.
4779:
4524:
3289:
3215:
2870:
1675:
1031:
is 2. This is easily proved by induction on the number of faces determined by
469:
437:
93:
5547:
6048:
5793:
5774:
5613:
Post, Olaf (2009). "Spectral analysis of metric graphs and related spaces".
4570:
4014:
3994:
3445:
3423:
3401:
3283:
2442:
554:
The surfaces of nonconvex polyhedra can have various Euler characteristics:
100:. It was stated for Platonic solids in 1537 in an unpublished manuscript by
4782:, measured in full circles, is the Euler characteristic of the polyhedron.
4512:, and all other Betti numbers are 0. Its Euler characteristic is then
327:. An illustration of the formula on all Platonic polyhedra is given below.
5821:
An animated version of a proof of Euler's formula using spherical geometry
2724:
503:
6033:
6001:
5950:
5857:
5243:
5110:
4728:
For closed smooth manifolds, the Euler characteristic coincides with the
4302:
4276:
1299:(The assumption that all faces are disks is needed here, to show via the
1678:. (When only triangular faces are used, they are two-dimensional finite
531:
415:
4629:
decomposition of the surface; intuitively, the number of "handles") as
4250:
4224:
4018:
2152:
are any two topological spaces, then the Euler characteristic of their
1894:
475:
447:
3473:
1177:
One of the few graph theory papers of Cauchy also proves this result.
5688:
5214:
4198:
4172:
3305:
3261:
2450:
2133:
288:
3954:
Because the sphere has Euler characteristic 2, it follows that
1120:
components (disconnected graphs), the same argument by induction on
5627:
4793:
4622:
3327:
3267:
1192:
1024:
is the number of faces in the graph, including the exterior face.
958:
123:
2715:
These addition and multiplication properties are also enjoyed by
5653:
2331:{\displaystyle \chi (M\cup N)=\chi (M)+\chi (N)-\chi (M\cap N).}
5830:
4833:
More generally, one can define the Euler characteristic of any
2529:
operation. The Euler characteristic is related by the formula
5334:
A History of
Folding in Mathematics: Mathematizing the Margins
4744:
of a manifold. The Euler class, in turn, relates to all other
5732:
Euler's Gem: The polyhedron formula and the birth of topology
3002:
For fiber bundles, this can also be understood in terms of a
2923:
then the Euler characteristic with coefficients in the field
5088:
5057:
5019:
4961:
4900:
4860:
5826:
4491:
3781:{\displaystyle \ E={\tfrac {1}{2}}\left(\ 5P+6H\ \right)\ }
3709:{\displaystyle \ V={\tfrac {1}{3}}\left(\ 5P+6H\ \right)\ }
2615:{\displaystyle \chi (M\#N)=\chi (M)+\chi (N)-\chi (S^{n}).}
1201:
There are many proofs of Euler's formula. One was given by
2433:
In general, the inclusionâexclusion principle is false. A
5455:
Edwin
Spanier: Algebraic Topology, Springer 1966, p. 205.
2705:{\displaystyle \chi (M\times N)=\chi (M)\cdot \chi (N).}
4508:
hence has Betti number 1 in dimensions 0 and
5516:
Applications of the homology spectral sequence, p. 481
4580:
odd-dimensional manifold is also 0. The case for
3921:
3863:
3817:
3733:
3661:
2021:{\displaystyle \chi =b_{0}-b_{1}+b_{2}-b_{3}+\cdots .}
1874:{\displaystyle \chi =k_{0}-k_{1}+k_{2}-k_{3}+\cdots ,}
1763:{\displaystyle \chi =k_{0}-k_{1}+k_{2}-k_{3}+\cdots ,}
1027:
The Euler characteristic of any plane connected graph
5615:
Limits of graphs in group theory and computer science
5375:
5082:
5054:
4994:
4891:
4857:
4693:
4638:
4346:
4141:
4103:
4029:
4 dimensional analogues of the regular polyhedra
3960:
3797:
3722:
3650:
3612:
3528:
3153:
3075:
3011:
2936:
2882:
2794:
2753:
2650:
2538:
2505:
2479:
2414:
are unions of strata. This applies in particular if
2255:
2165:
2102:
2044:
1953:
1806:
1695:
1611:
1567:
1520:
1485:
1450:
1416:
1364:
1321:
1264:
1216:
1130:
1083:
1045:
988:
872:
834:
798:
781:
derived a modified form of Euler's formula using the
594:
367:
297:
229:
148:
68:
4684:
in a connected sum decomposition of the surface) as
2869:
The product property holds much more generally, for
2843:{\displaystyle \chi ({\tilde {M}})=k\cdot \chi (M).}
2074:
Homology is a topological invariant, and moreover a
287:. It corresponds to the Euler characteristic of the
92:
The Euler characteristic was originally defined for
6093:
6057:
5943:
5864:
2220:{\displaystyle \chi (M\sqcup N)=\chi (M)+\chi (N).}
1605:for the deformed, planar object thus demonstrating
5143:is a prime number corresponding to the cone angle
5098:
5064:
5032:
4977:
4867:
4717:
4665:
4497:
4159:
4127:
3981:
3943:
3780:
3708:
3636:
3534:
3193:
3132:
3061:
2984:
2900:
2842:
2777:
2704:
2614:
2517:
2491:
2330:
2219:
2117:
2050:
2020:
1873:
1762:
1641:
1597:
1553:
1506:
1471:
1437:
1391:
1348:
1291:
1246:
1169:
1104:
1069:
1012:
928:
856:
820:
624:
397:
315:
268:
172:
74:
5072:. In this case, the dimensions are all finite by
4770:A discrete analog of the GaussâBonnet theorem is
3194:{\displaystyle p_{*}\circ \tau =\chi (F)\cdot 1.}
4561: dimensional torus is the product space of
3133:{\displaystyle p_{*}\colon H_{*}(E)\to H_{*}(B)}
3062:{\displaystyle \tau \colon H_{*}(B)\to H_{*}(E)}
2985:{\displaystyle \chi (E)=\chi (F)\cdot \chi (B).}
2919:, and the fibration is orientable over a field
23:. For Euler number in 3-manifold topology, see
5568:Fowler, P.W. & Manolopoulos, D.E. (1995).
5282:for each connected component of the groupoid.
5201:This can be further generalized by defining a
4788:characterizes the Euler characteristic as the
4588:. This property applies more generally to any
3788: edges. The Euler characteristic is thus
1682:.) In general, for any finite CW-complex, the
1197:First steps of the proof in the case of a cube
5842:
4882:, one defines its Euler characteristic to be
1649:for the polyhedron. This proves the theorem.
1035:, starting with a tree as the base case. For
8:
5529:"Fibre bundles and the Euler characteristic"
4446:
4440:
3993:except 1. This result is applicable to
3230:
975:The Euler characteristic can be defined for
5273:, where we picked one representative group
5242:, and the Euler characteristic of a finite
5167:The concept of Euler characteristic of the
5033:{\displaystyle \ h^{i}(X,{\mathcal {F}})\ }
1944:can then be defined as the alternating sum
6210:
6183:
5849:
5835:
5827:
5373:[Elements of rubrics for solids].
4844:A version of Euler characteristic used in
4024:
2380:, and one uses Euler characteristics with
1020:formula as for polyhedral surfaces, where
957:all have Euler characteristic 0, like the
949:all have Euler characteristic 1, like the
5626:
5302:List of topics named after Leonhard Euler
5087:
5086:
5081:
5056:
5055:
5053:
5018:
5017:
5002:
4993:
4960:
4959:
4944:
4934:
4915:
4899:
4898:
4890:
4859:
4858:
4856:
4692:
4637:
4452:
4451:
4412:
4411:
4387:
4386:
4378:
4366:
4361:
4351:
4345:
4140:
4102:
3959:
3920:
3862:
3816:
3796:
3732:
3721:
3660:
3649:
3611:
3527:
3158:
3152:
3115:
3093:
3080:
3074:
3044:
3022:
3010:
2935:
2881:
2802:
2801:
2793:
2755:
2754:
2752:
2649:
2600:
2537:
2504:
2478:
2254:
2164:
2109:
2105:
2104:
2101:
2043:
2003:
1990:
1977:
1964:
1952:
1856:
1843:
1830:
1817:
1805:
1782:denotes the number of cells of dimension
1745:
1732:
1719:
1706:
1694:
1610:
1566:
1519:
1484:
1449:
1415:
1363:
1320:
1263:
1215:
1129:
1082:
1044:
987:
902:
880:
871:
842:
833:
806:
797:
593:
366:
296:
228:
147:
67:
5670:"The Euler characteristic of a category"
5600:
5391:
5205:Euler characteristic for certain finite
5175:is another generalization, important in
4814:that is "homogeneous of degree 0".
4008:
3220:
2499:one can obtain a new connected manifold
556:
329:
5405:"Twenty-one proofs of Euler's formula:
5323:
4013:Comparison of Euler characteristics of
2369:inside the union still cover the union.
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4416:
4413:
4333: dimensional sphere has singular
2630:Also, the Euler characteristic of any
1686:can be defined as the alternating sum
929:{\displaystyle \ d_{v}V-E+d_{f}F=2D~.}
4763:for the two-dimensional case and the
4607:The Euler characteristic of a closed
2473:For two connected closed n-manifolds
2341:This is true in the following cases:
7:
1410:At this point the lone triangle has
220:'s surface has Euler characteristic
16:Topological invariant in mathematics
5115:Euler characteristic of an orbifold
4573:, has Euler characteristic 0.
3218:) and using the above definitions.
971:Planar graph § Euler's formula
52:Euler–PoincarĂ© characteristic
5099:{\displaystyle \ {\mathcal {F}}\ }
4362:
3606:hexagons are used, then there are
2778:{\displaystyle {\tilde {M}}\to M,}
2548:
2509:
2078:: Two topological spaces that are
1254:for this deformed, planar object.
14:
5586:& Stasheff, James D. (1974).
5074:Grothendieck's finiteness theorem
4837:to be the alternating sum of the
4027:Euler characteristics of the six
269:{\displaystyle \ \chi =V-E+F=2~.}
6209:
6182:
6172:
6162:
6151:
6141:
6140:
5934:
5536:Journal of Differential Geometry
5377:(in Latin): 109â140 – via
4765:generalized GaussâBonnet theorem
4576:The Euler characteristic of any
3567:
3557:
3508:
3503:
3498:
3472:
3444:
3422:
3400:
3379:
3360:
3338:
3310:
3288:
3266:
3244:
2927:satisfies the product property:
2238:are subspaces of a larger space
2118:{\displaystyle \mathbb {R} ^{n}}
753:
725:
697:
669:
641:
530:
502:
474:
446:
414:
195:are respectively the numbers of
5527:Gottlieb, Daniel Henry (1975),
625:{\displaystyle \ \chi =V-E+F\ }
398:{\displaystyle \ \chi =V-E+F\ }
128:Vertex, edge and face of a cube
5371:"Elementa doctrinae solidorum"
5065:{\displaystyle {\mathcal {F}}}
5024:
5008:
4966:
4950:
4931:
4921:
4905:
4895:
4868:{\displaystyle {\mathcal {F}}}
4372:
4357:
3182:
3176:
3127:
3121:
3108:
3105:
3099:
3056:
3050:
3037:
3034:
3028:
2976:
2970:
2961:
2955:
2946:
2940:
2901:{\displaystyle p\colon E\to B}
2892:
2834:
2828:
2813:
2807:
2798:
2766:
2760:
2696:
2690:
2681:
2675:
2666:
2654:
2606:
2593:
2584:
2578:
2569:
2563:
2554:
2542:
2322:
2310:
2301:
2295:
2286:
2280:
2271:
2259:
2211:
2205:
2196:
2190:
2181:
2169:
1908:More generally still, for any
323:), and applies identically to
1:
5735:. Princeton University Press.
5708:U. Illinois, Urbana-Champaign
5590:. Princeton University Press.
5445:. Cambridge Technology Press.
4603:Relations to other invariants
2244:inclusionâexclusion principle
2140:Inclusionâexclusion principle
1170:{\displaystyle \ V-E+F-C=1~.}
216:in the given polyhedron. Any
4666:{\displaystyle \chi =2-2g~.}
4128:{\displaystyle \chi =V-E\ +}
2999:on homology of a fibration.
2453:consisting of one point and
749:Great stellated dodecahedron
746:
721:Small stellated dodecahedron
718:
690:
662:
634:
523:
495:
467:
435:
407:
58:, a number that describes a
5810:Encyclopedia of Mathematics
5475:"Homology of connected sum"
4718:{\displaystyle \chi =2-k~.}
4614:can be calculated from its
4584:examples is a corollary of
1797:equals the alternating sum
1652:For additional proofs, see
1642:{\displaystyle \ V-E+F=2\ }
1598:{\displaystyle \ V-E+F=1\ }
1554:{\displaystyle \ V-E+F=1~.}
1247:{\displaystyle \ V-E+F=1\ }
316:{\displaystyle \ \chi =2\ }
173:{\displaystyle \chi =V-E+F}
34:, and more specifically in
6269:
6103:Banach fixed-point theorem
5706:on 2014-06-06 – via
5331:Friedman, Michael (2018).
4519:that is, either 0 if
3590:It is common to construct
2908:is a fibration with fiber
2733:
968:
857:{\displaystyle \ d_{f}\ :}
821:{\displaystyle \ d_{v}\ ,}
285:Euler's polyhedron formula
18:
6136:
5932:
5754:, Dover 2001, p. 40.
5752:From Geometry to Topology
5343:10.1007/978-3-319-72487-4
5307:List of uniform polyhedra
4021:of dimensions 1 to 4
3637:{\displaystyle \ F=P+H\ }
3140:is multiplication by the
2873:with certain conditions.
1392:{\displaystyle \ V-E+F~.}
1349:{\displaystyle \ V-E+F~.}
1292:{\displaystyle \ V-E+F~.}
1070:{\displaystyle \ E=V-1\ }
1013:{\displaystyle \ V-E+F\ }
279:This equation, stated by
6246:Polyhedral combinatorics
6241:Topological graph theory
5403:Eppstein, David (2013).
5040:is the dimension of the
4822:For every combinatorial
4774:theorem that the "total
4569:, including any compact
3982:{\displaystyle \ P=12~.}
2357:. In particular, if the
1904:Betti number alternative
1507:{\displaystyle \ F=1\ ,}
1472:{\displaystyle \ E=3\ ,}
1438:{\displaystyle \ V=3\ ,}
1189:Proof of Euler's formula
1182:stereographic projection
953:, while the surfaces of
941:KeplerâPoinsot polyhedra
40:polyhedral combinatorics
5803:Matveev, S.V. (2001) ,
4848:is as follows. For any
4597:orientable double cover
4567:parallelizable manifold
4546:is the quotient of the
4542: dimensional real
2997:Serre spectral sequence
2859:RiemannâHurwitz formula
2855:ramified covering space
2736:RiemannâHurwitz formula
1188:
1105:{\displaystyle \ F=1~.}
777:For regular polyhedra,
84:Greek lower-case letter
6158:Mathematics portal
6058:Metrics and properties
6044:Second-countable space
5805:"Euler characteristic"
5770:"Euler characteristic"
5668:Leinster, Tom (2008).
5588:Characteristic Classes
5570:An Atlas of Fullerenes
5548:10.4310/jdg/1214432674
5442:Proofs and Refutations
5100:
5066:
5034:
4979:
4869:
4767:for the general case.
4746:characteristic classes
4719:
4682:real projective planes
4667:
4499:
4161:
4160:{\displaystyle +\ F-C}
4129:
4022:
3983:
3945:
3782:
3710:
3638:
3536:
3195:
3134:
3063:
2986:
2902:
2853:More generally, for a
2844:
2779:
2706:
2616:
2519:
2493:
2422:are subvarieties of a
2332:
2221:
2119:
2052:
2022:
1893:denotes the number of
1875:
1764:
1670:Topological definition
1659:Proofs and Refutations
1643:
1599:
1555:
1508:
1473:
1439:
1393:
1350:
1293:
1248:
1198:
1171:
1106:
1071:
1014:
930:
858:
822:
626:
399:
317:
270:
174:
129:
112:and, more abstractly,
76:
5677:Documenta Mathematica
5101:
5067:
5035:
4980:
4870:
4797:scalar multiplication
4720:
4668:
4500:
4162:
4130:
4012:
3984:
3946:
3783:
3711:
3639:
3537:
3196:
3135:
3064:
2987:
2903:
2845:
2780:
2734:Further information:
2707:
2617:
2520:
2494:
2378:locally compact space
2333:
2222:
2120:
2053:
2051:{\displaystyle \chi }
2023:
1876:
1765:
1644:
1600:
1556:
1509:
1474:
1440:
1394:
1351:
1294:
1249:
1196:
1172:
1107:
1072:
1015:
951:real projective plane
931:
859:
823:
627:
587:Euler characteristic:
400:
360:Euler characteristic:
318:
283:in 1758, is known as
271:
175:
127:
77:
75:{\displaystyle \chi }
56:topological invariant
6113:Invariance of domain
6065:Euler characteristic
6039:Bundle (mathematics)
5789:"Polyhedral formula"
5650:Euler characteristic
5621:. pp. 109â140.
5495:Spanier, Edwin Henry
5171:of a bounded finite
5106:by acyclic sheaves.
5080:
5052:
4992:
4889:
4855:
4761:GaussâBonnet theorem
4757:Riemannian manifolds
4691:
4636:
4550: sphere by the
4344:
4139:
4101:
4005:Arbitrary dimensions
3958:
3795:
3720:
3716: vertices, and
3648:
3610:
3526:
3151:
3073:
3009:
2934:
2880:
2792:
2751:
2648:
2536:
2518:{\displaystyle M\#N}
2503:
2477:
2387:, no assumptions on
2253:
2163:
2100:
2042:
1951:
1942:Euler characteristic
1912:, we can define the
1804:
1795:Euler characteristic
1693:
1684:Euler characteristic
1680:simplicial complexes
1609:
1565:
1518:
1483:
1448:
1414:
1362:
1319:
1301:Jordan curve theorem
1262:
1214:
1128:
1081:
1043:
986:
947:Projective polyhedra
870:
832:
796:
592:
365:
295:
227:
146:
134:Euler characteristic
66:
44:Euler characteristic
6123:Tychonoff's theorem
6118:Poincaré conjecture
5872:General (point-set)
5750:Flegg, H. Graham;
5637:2007arXiv0712.1507P
4032:
3991:nonnegative integer
2492:{\displaystyle M,N}
2437:is given by taking
2230:More generally, if
2080:homotopy equivalent
2070:Homotopy invariance
637:Tetrahemihexahedron
325:spherical polyhedra
114:homological algebra
102:Francesco Maurolico
25:Seifert fiber space
6236:Algebraic topology
6108:De Rham cohomology
6029:Polyhedral complex
6019:Simplicial complex
5786:Weisstein, Eric W.
5767:Weisstein, Eric W.
5500:Algebraic Topology
5413:(acad. pers. wbs.)
5096:
5062:
5030:
4975:
4920:
4865:
4846:algebraic geometry
4786:Hadwiger's theorem
4715:
4663:
4495:
4490:
4157:
4125:
4025:
4023:
3999:Goldberg polyhedra
3979:
3941:
3930:
3872:
3826:
3778:
3742:
3706:
3670:
3634:
3532:
3191:
3130:
3059:
2982:
2898:
2865:Fibration property
2840:
2775:
2702:
2612:
2515:
2489:
2328:
2217:
2115:
2076:homotopy invariant
2048:
2018:
1871:
1791:simplicial complex
1760:
1639:
1595:
1551:
1504:
1469:
1435:
1389:
1346:
1289:
1244:
1199:
1167:
1102:
1067:
1010:
955:toroidal polyhedra
926:
854:
818:
693:Cubohemioctahedron
665:Octahemioctahedron
622:
395:
313:
266:
170:
130:
72:
36:algebraic topology
6223:
6222:
6012:fundamental group
5510:978-0-387-94426-5
5352:978-3-319-72486-7
5196:incidence algebra
5095:
5085:
5029:
4997:
4971:
4911:
4742:fundamental class
4740:evaluated on the
4711:
4659:
4515:χ = 1 + (â1)
4484:
4426:
4423:
4410:
4407:
4393:
4327:
4326:
4147:
4121:
4031:
3975:
3963:
3937:
3929:
3899:
3881:
3871:
3853:
3835:
3825:
3777:
3769:
3751:
3741:
3725:
3705:
3697:
3679:
3669:
3653:
3633:
3615:
3583:
3582:
3535:{\displaystyle n}
3494:of three spheres)
2810:
2763:
2740:Similarly, for a
2427:algebraic variety
2089:For example, any
1938:singular homology
1910:topological space
1789:Similarly, for a
1638:
1614:
1594:
1570:
1547:
1523:
1500:
1488:
1465:
1453:
1431:
1419:
1385:
1367:
1342:
1324:
1285:
1267:
1243:
1219:
1163:
1133:
1098:
1086:
1066:
1048:
1009:
991:
922:
875:
850:
837:
828:and face density
814:
801:
775:
774:
621:
597:
552:
551:
394:
370:
312:
300:
262:
232:
218:convex polyhedron
60:topological space
6258:
6213:
6212:
6186:
6185:
6176:
6166:
6156:
6155:
6144:
6143:
5938:
5851:
5844:
5837:
5828:
5817:
5799:
5798:
5780:
5779:
5736:
5712:
5711:
5705:
5699:. Archived from
5674:
5665:
5659:
5647:
5641:
5640:
5630:
5617:. Lausanne, CH:
5610:
5604:
5598:
5592:
5591:
5580:
5574:
5573:
5565:
5559:
5558:
5533:
5524:
5518:
5513:
5491:
5485:
5484:
5482:
5481:
5471:
5465:
5462:
5456:
5453:
5447:
5446:
5433:
5427:
5426:
5420:
5418:
5410:
5409:
5400:
5394:
5389:
5383:
5382:
5363:
5357:
5356:
5328:
5280:
5276:
5272:
5270:
5269:
5267:
5265:
5261:
5255:
5252:
5241:
5239:
5238:
5236:
5235:
5229:
5226:
5219:
5194:in that poset's
5189:
5185:
5169:reduced homology
5163:
5161:
5160:
5159:
5155:
5152:
5151:
5142:
5138:
5136:
5134:
5133:
5132:
5128:
5125:
5105:
5103:
5102:
5097:
5093:
5092:
5091:
5083:
5071:
5069:
5068:
5063:
5061:
5060:
5046:sheaf cohomology
5043:
5039:
5037:
5036:
5031:
5027:
5023:
5022:
5007:
5006:
4995:
4984:
4982:
4981:
4976:
4969:
4965:
4964:
4949:
4948:
4939:
4938:
4919:
4904:
4903:
4881:
4874:
4872:
4871:
4866:
4864:
4863:
4813:
4724:
4722:
4721:
4716:
4709:
4679:
4672:
4670:
4669:
4664:
4657:
4620:
4593:stratified space
4586:Poincaré duality
4564:
4560:
4549:
4544:projective space
4541:
4530:
4522:
4518:
4516:
4511:
4504:
4502:
4501:
4496:
4494:
4493:
4482:
4481:
4480:
4424:
4421:
4420:
4419:
4408:
4405:
4391:
4390:
4371:
4370:
4365:
4356:
4355:
4332:
4166:
4164:
4163:
4158:
4145:
4134:
4132:
4131:
4126:
4119:
4091:
4086:
4077:
4072:
4063:
4058:
4049:
4044:
4038:4 polytope
4033:
4026:
3988:
3986:
3985:
3980:
3973:
3961:
3950:
3948:
3947:
3942:
3935:
3931:
3922:
3904:
3900:
3897:
3879:
3873:
3864:
3858:
3854:
3851:
3833:
3827:
3818:
3787:
3785:
3784:
3779:
3775:
3774:
3770:
3767:
3749:
3743:
3734:
3723:
3715:
3713:
3712:
3707:
3703:
3702:
3698:
3695:
3677:
3671:
3662:
3651:
3643:
3641:
3640:
3635:
3631:
3613:
3605:
3601:
3579:
3571:
3561:
3544:(not connected)
3541:
3539:
3538:
3533:
3512:
3507:
3502:
3489:(not connected)
3476:
3463:(not connected)
3454:
3448:
3432:
3426:
3410:
3404:
3383:
3364:
3348:
3342:
3320:
3314:
3298:
3292:
3276:
3270:
3254:
3248:
3234:
3221:
3200:
3198:
3197:
3192:
3163:
3162:
3139:
3137:
3136:
3131:
3120:
3119:
3098:
3097:
3085:
3084:
3068:
3066:
3065:
3060:
3049:
3048:
3027:
3026:
2991:
2989:
2988:
2983:
2907:
2905:
2904:
2899:
2849:
2847:
2846:
2841:
2812:
2811:
2803:
2784:
2782:
2781:
2776:
2765:
2764:
2756:
2711:
2709:
2708:
2703:
2626:Product property
2621:
2619:
2618:
2613:
2605:
2604:
2524:
2522:
2521:
2516:
2498:
2496:
2495:
2490:
2404:stratified space
2337:
2335:
2334:
2329:
2226:
2224:
2223:
2218:
2124:
2122:
2121:
2116:
2114:
2113:
2108:
2057:
2055:
2054:
2049:
2027:
2025:
2024:
2019:
2008:
2007:
1995:
1994:
1982:
1981:
1969:
1968:
1900:in the complex.
1880:
1878:
1877:
1872:
1861:
1860:
1848:
1847:
1835:
1834:
1822:
1821:
1786:in the complex.
1769:
1767:
1766:
1761:
1750:
1749:
1737:
1736:
1724:
1723:
1711:
1710:
1648:
1646:
1645:
1640:
1636:
1612:
1604:
1602:
1601:
1596:
1592:
1568:
1560:
1558:
1557:
1552:
1545:
1521:
1513:
1511:
1510:
1505:
1498:
1486:
1478:
1476:
1475:
1470:
1463:
1451:
1444:
1442:
1441:
1436:
1429:
1417:
1398:
1396:
1395:
1390:
1383:
1365:
1355:
1353:
1352:
1347:
1340:
1322:
1298:
1296:
1295:
1290:
1283:
1265:
1253:
1251:
1250:
1245:
1241:
1217:
1176:
1174:
1173:
1168:
1161:
1131:
1123:
1119:
1115:
1111:
1109:
1108:
1103:
1096:
1084:
1076:
1074:
1073:
1068:
1064:
1046:
1034:
1030:
1023:
1019:
1017:
1016:
1011:
1007:
989:
935:
933:
932:
927:
920:
907:
906:
885:
884:
873:
863:
861:
860:
855:
848:
847:
846:
835:
827:
825:
824:
819:
812:
811:
810:
799:
787:
757:
729:
701:
673:
645:
631:
629:
628:
623:
619:
595:
584:
577:
570:
557:
534:
506:
478:
450:
418:
404:
402:
401:
396:
392:
368:
357:
350:
343:
330:
322:
320:
319:
314:
310:
298:
275:
273:
272:
267:
260:
230:
194:
190:
186:
179:
177:
176:
171:
138:
81:
79:
78:
73:
6268:
6267:
6261:
6260:
6259:
6257:
6256:
6255:
6226:
6225:
6224:
6219:
6150:
6132:
6128:Urysohn's lemma
6089:
6053:
5939:
5930:
5902:low-dimensional
5860:
5855:
5802:
5784:
5783:
5765:
5764:
5761:
5747:
5745:Further reading
5742:
5740:
5725:
5721:
5716:
5715:
5703:
5672:
5667:
5666:
5662:
5648:
5644:
5612:
5611:
5607:
5601:Richeson (2008)
5599:
5595:
5582:
5581:
5577:
5567:
5566:
5562:
5531:
5526:
5525:
5521:
5511:
5493:
5492:
5488:
5479:
5477:
5473:
5472:
5468:
5463:
5459:
5454:
5450:
5435:
5434:
5430:
5416:
5414:
5407:
5406:
5402:
5401:
5397:
5392:Richeson (2008)
5390:
5386:
5381:, Stockton, CA.
5365:
5364:
5360:
5353:
5330:
5329:
5325:
5320:
5315:
5288:
5281:
5278:
5274:
5266:
5263:
5259:
5258:
5256:
5253:
5250:
5249:
5247:
5233:
5232:
5230:
5227:
5224:
5223:
5221:
5217:
5203:rational valued
5192:Möbius function
5187:
5180:
5157:
5156:
5153:
5149:
5147:
5146:
5144:
5140:
5130:
5129:
5126:
5123:
5122:
5120:
5118:
5078:
5077:
5050:
5049:
5041:
4998:
4990:
4989:
4940:
4930:
4887:
4886:
4879:
4853:
4852:
4820:
4818:Generalizations
4811:
4689:
4688:
4680:(the number of
4677:
4634:
4633:
4621:(the number of
4618:
4605:
4562:
4558:
4547:
4539:
4528:
4527:, or 2 if
4520:
4514:
4513:
4509:
4489:
4488:
4449:
4437:
4436:
4394:
4379:
4360:
4347:
4342:
4341:
4335:homology groups
4330:
4323:
4297:
4271:
4245:
4219:
4193:
4167:
4137:
4136:
4135:
4099:
4098:
4094:
4089:
4088:
4084:
4080:
4075:
4074:
4070:
4066:
4061:
4060:
4056:
4052:
4047:
4046:
4042:
4037:
4007:
3956:
3955:
3878:
3874:
3832:
3828:
3793:
3792:
3748:
3744:
3718:
3717:
3676:
3672:
3646:
3645:
3608:
3607:
3603:
3599:
3588:
3574:
3565:
3548:
3547:(Disjoint union
3545:
3543:
3524:
3523:
3493:
3492:(Disjoint union
3490:
3488:
3468:of two spheres)
3467:
3466:(Disjoint union
3464:
3462:
3452:
3430:
3408:
3395:
3394:Real projective
3346:
3333:
3330:
3318:
3296:
3274:
3252:
3212:
3207:
3154:
3149:
3148:
3111:
3089:
3076:
3071:
3070:
3040:
3018:
3007:
3006:
2932:
2931:
2878:
2877:
2867:
2790:
2789:
2749:
2748:
2738:
2732:
2730:Covering spaces
2646:
2645:
2628:
2596:
2534:
2533:
2501:
2500:
2475:
2474:
2471:
2355:excisive couple
2251:
2250:
2161:
2160:
2142:
2103:
2098:
2097:
2095:Euclidean space
2072:
2064:
2040:
2039:
2037:
1999:
1986:
1973:
1960:
1949:
1948:
1927:
1906:
1892:
1852:
1839:
1826:
1813:
1802:
1801:
1781:
1741:
1728:
1715:
1702:
1691:
1690:
1672:
1607:
1606:
1563:
1562:
1516:
1515:
1481:
1480:
1446:
1445:
1412:
1411:
1360:
1359:
1317:
1316:
1260:
1259:
1212:
1211:
1191:
1126:
1125:
1121:
1117:
1113:
1079:
1078:
1041:
1040:
1032:
1028:
1021:
984:
983:
973:
967:
898:
876:
868:
867:
838:
830:
829:
802:
794:
793:
785:
590:
589:
588:
582:
581:
575:
574:
568:
567:
363:
362:
361:
355:
354:
348:
347:
341:
340:
293:
292:
225:
224:
192:
188:
184:
144:
143:
136:
122:
98:Platonic solids
64:
63:
28:
17:
12:
11:
5:
6266:
6265:
6262:
6254:
6253:
6251:Leonhard Euler
6248:
6243:
6238:
6228:
6227:
6221:
6220:
6218:
6217:
6207:
6206:
6205:
6200:
6195:
6180:
6170:
6160:
6148:
6137:
6134:
6133:
6131:
6130:
6125:
6120:
6115:
6110:
6105:
6099:
6097:
6091:
6090:
6088:
6087:
6082:
6077:
6075:Winding number
6072:
6067:
6061:
6059:
6055:
6054:
6052:
6051:
6046:
6041:
6036:
6031:
6026:
6021:
6016:
6015:
6014:
6009:
6007:homotopy group
5999:
5998:
5997:
5992:
5987:
5982:
5977:
5967:
5962:
5957:
5947:
5945:
5941:
5940:
5933:
5931:
5929:
5928:
5923:
5918:
5917:
5916:
5906:
5905:
5904:
5894:
5889:
5884:
5879:
5874:
5868:
5866:
5862:
5861:
5856:
5854:
5853:
5846:
5839:
5831:
5825:
5824:
5818:
5800:
5781:
5760:
5759:External links
5757:
5756:
5755:
5746:
5743:
5738:
5737:
5727:Richeson, D.S.
5722:
5720:
5717:
5714:
5713:
5689:10.4171/dm/240
5660:
5642:
5605:
5593:
5575:
5560:
5519:
5509:
5486:
5466:
5457:
5448:
5428:
5408:V â E + F = 2
5395:
5384:
5358:
5351:
5322:
5321:
5319:
5316:
5314:
5311:
5310:
5309:
5304:
5299:
5294:
5292:Euler calculus
5287:
5284:
5277:
5262:
5246:is the sum of
5090:
5059:
5026:
5021:
5016:
5013:
5010:
5005:
5001:
4986:
4985:
4974:
4968:
4963:
4958:
4955:
4952:
4947:
4943:
4937:
4933:
4929:
4926:
4923:
4918:
4914:
4910:
4907:
4902:
4897:
4894:
4862:
4850:coherent sheaf
4819:
4816:
4750:vector bundles
4738:tangent bundle
4726:
4725:
4714:
4708:
4705:
4702:
4699:
4696:
4674:
4673:
4662:
4656:
4653:
4650:
4647:
4644:
4641:
4604:
4601:
4506:
4505:
4492:
4487:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4450:
4448:
4445:
4442:
4439:
4438:
4435:
4432:
4429:
4418:
4415:
4404:
4401:
4398:
4395:
4389:
4385:
4384:
4382:
4377:
4374:
4369:
4364:
4359:
4354:
4350:
4325:
4324:
4319:
4317:
4314:
4311:
4308:
4305:
4299:
4298:
4293:
4291:
4288:
4285:
4282:
4279:
4273:
4272:
4267:
4265:
4262:
4259:
4256:
4253:
4247:
4246:
4241:
4239:
4236:
4233:
4230:
4227:
4221:
4220:
4215:
4213:
4210:
4207:
4204:
4201:
4195:
4194:
4189:
4187:
4184:
4181:
4178:
4175:
4169:
4168:
4156:
4153:
4150:
4144:
4124:
4118:
4115:
4112:
4109:
4106:
4097:
4095:
4092:
4081:
4078:
4067:
4064:
4053:
4050:
4039:
4006:
4003:
3978:
3972:
3969:
3966:
3952:
3951:
3940:
3934:
3928:
3925:
3919:
3916:
3913:
3910:
3907:
3903:
3896:
3893:
3890:
3887:
3884:
3877:
3870:
3867:
3861:
3857:
3850:
3847:
3844:
3841:
3838:
3831:
3824:
3821:
3815:
3812:
3809:
3806:
3803:
3800:
3773:
3766:
3763:
3760:
3757:
3754:
3747:
3740:
3737:
3731:
3728:
3701:
3694:
3691:
3688:
3685:
3682:
3675:
3668:
3665:
3659:
3656:
3644: faces,
3630:
3627:
3624:
3621:
3618:
3602:pentagons and
3596:Adidas Telstar
3587:
3584:
3581:
3580:
3575:2 + ... + 2 =
3572:
3563:
3555:
3531:
3520:
3519:
3513:
3496:
3484:
3483:
3477:
3470:
3458:
3457:
3449:
3442:
3436:
3435:
3427:
3420:
3414:
3413:
3405:
3398:
3390:
3389:
3384:
3377:
3371:
3370:
3365:
3358:
3352:
3351:
3343:
3336:
3324:
3323:
3315:
3308:
3302:
3301:
3293:
3286:
3280:
3279:
3271:
3264:
3258:
3257:
3249:
3242:
3236:
3235:
3228:
3225:
3211:
3208:
3206:
3203:
3202:
3201:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3161:
3157:
3144:of the fiber:
3129:
3126:
3123:
3118:
3114:
3110:
3107:
3104:
3101:
3096:
3092:
3088:
3083:
3079:
3058:
3055:
3052:
3047:
3043:
3039:
3036:
3033:
3030:
3025:
3021:
3017:
3014:
2993:
2992:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2917:path-connected
2912:with the base
2897:
2894:
2891:
2888:
2885:
2866:
2863:
2851:
2850:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2809:
2806:
2800:
2797:
2774:
2771:
2768:
2762:
2759:
2746:covering space
2731:
2728:
2713:
2712:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2627:
2624:
2623:
2622:
2611:
2608:
2603:
2599:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2514:
2511:
2508:
2488:
2485:
2482:
2470:
2467:
2435:counterexample
2431:
2430:
2396:
2370:
2339:
2338:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2228:
2227:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2154:disjoint union
2141:
2138:
2112:
2107:
2071:
2068:
2063:
2060:
2047:
2035:
2029:
2028:
2017:
2014:
2011:
2006:
2002:
1998:
1993:
1989:
1985:
1980:
1976:
1972:
1967:
1963:
1959:
1956:
1923:
1905:
1902:
1888:
1882:
1881:
1870:
1867:
1864:
1859:
1855:
1851:
1846:
1842:
1838:
1833:
1829:
1825:
1820:
1816:
1812:
1809:
1777:
1771:
1770:
1759:
1756:
1753:
1748:
1744:
1740:
1735:
1731:
1727:
1722:
1718:
1714:
1709:
1705:
1701:
1698:
1671:
1668:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1550:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1503:
1497:
1494:
1491:
1468:
1462:
1459:
1456:
1434:
1428:
1425:
1422:
1400:
1399:
1388:
1382:
1379:
1376:
1373:
1370:
1356:
1345:
1339:
1336:
1333:
1330:
1327:
1288:
1282:
1279:
1276:
1273:
1270:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1190:
1187:
1166:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1101:
1095:
1092:
1089:
1063:
1060:
1057:
1054:
1051:
1006:
1003:
1000:
997:
994:
966:
963:
937:
936:
925:
919:
916:
913:
910:
905:
901:
897:
894:
891:
888:
883:
879:
853:
845:
841:
817:
809:
805:
773:
772:
767:
764:
761:
758:
751:
745:
744:
739:
736:
733:
730:
723:
717:
716:
711:
708:
705:
702:
695:
689:
688:
683:
680:
677:
674:
667:
661:
660:
655:
652:
649:
646:
639:
633:
632:
618:
615:
612:
609:
606:
603:
600:
585:
578:
571:
564:
561:
550:
549:
544:
541:
538:
535:
528:
522:
521:
516:
513:
510:
507:
500:
494:
493:
488:
485:
482:
479:
472:
466:
465:
460:
457:
454:
451:
444:
434:
433:
428:
425:
422:
419:
412:
406:
405:
391:
388:
385:
382:
379:
376:
373:
358:
351:
344:
337:
334:
309:
306:
303:
277:
276:
265:
259:
256:
253:
250:
247:
244:
241:
238:
235:
181:
180:
169:
166:
163:
160:
157:
154:
151:
121:
118:
106:Leonhard Euler
71:
15:
13:
10:
9:
6:
4:
3:
2:
6264:
6263:
6252:
6249:
6247:
6244:
6242:
6239:
6237:
6234:
6233:
6231:
6216:
6208:
6204:
6201:
6199:
6196:
6194:
6191:
6190:
6189:
6181:
6179:
6175:
6171:
6169:
6165:
6161:
6159:
6154:
6149:
6147:
6139:
6138:
6135:
6129:
6126:
6124:
6121:
6119:
6116:
6114:
6111:
6109:
6106:
6104:
6101:
6100:
6098:
6096:
6092:
6086:
6085:Orientability
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6066:
6063:
6062:
6060:
6056:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6013:
6010:
6008:
6005:
6004:
6003:
6000:
5996:
5993:
5991:
5988:
5986:
5983:
5981:
5978:
5976:
5973:
5972:
5971:
5968:
5966:
5963:
5961:
5958:
5956:
5952:
5949:
5948:
5946:
5942:
5937:
5927:
5924:
5922:
5921:Set-theoretic
5919:
5915:
5912:
5911:
5910:
5907:
5903:
5900:
5899:
5898:
5895:
5893:
5890:
5888:
5885:
5883:
5882:Combinatorial
5880:
5878:
5875:
5873:
5870:
5869:
5867:
5863:
5859:
5852:
5847:
5845:
5840:
5838:
5833:
5832:
5829:
5822:
5819:
5816:
5812:
5811:
5806:
5801:
5796:
5795:
5790:
5787:
5782:
5777:
5776:
5771:
5768:
5763:
5762:
5758:
5753:
5749:
5748:
5744:
5741:
5734:
5733:
5728:
5724:
5723:
5718:
5709:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5671:
5664:
5661:
5658:
5656:
5651:
5646:
5643:
5638:
5634:
5629:
5624:
5620:
5616:
5609:
5606:
5603:, p. 261
5602:
5597:
5594:
5589:
5585:
5579:
5576:
5572:. p. 32.
5571:
5564:
5561:
5557:
5553:
5549:
5545:
5541:
5537:
5530:
5523:
5520:
5517:
5512:
5506:
5502:
5501:
5496:
5490:
5487:
5476:
5470:
5467:
5461:
5458:
5452:
5449:
5444:
5443:
5438:
5432:
5429:
5424:
5412:
5399:
5396:
5393:
5388:
5385:
5380:
5376:
5372:
5368:
5362:
5359:
5354:
5348:
5344:
5340:
5336:
5335:
5327:
5324:
5317:
5312:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5289:
5285:
5283:
5245:
5216:
5212:
5208:
5204:
5199:
5197:
5193:
5183:
5178:
5177:combinatorics
5174:
5170:
5165:
5116:
5112:
5107:
5075:
5047:
5014:
5011:
5003:
4999:
4972:
4956:
4953:
4945:
4941:
4935:
4927:
4924:
4916:
4912:
4908:
4892:
4885:
4884:
4883:
4878:
4851:
4847:
4842:
4840:
4836:
4835:chain complex
4831:
4829:
4825:
4817:
4815:
4809:
4806:
4802:
4801:finite unions
4798:
4795:
4791:
4787:
4783:
4781:
4777:
4773:
4768:
4766:
4762:
4758:
4753:
4751:
4747:
4743:
4739:
4735:
4731:
4712:
4706:
4703:
4700:
4697:
4694:
4687:
4686:
4685:
4683:
4660:
4654:
4651:
4648:
4645:
4642:
4639:
4632:
4631:
4630:
4628:
4627:connected sum
4624:
4617:
4613:
4610:
4602:
4600:
4598:
4594:
4591:
4587:
4583:
4579:
4574:
4572:
4568:
4555:
4553:
4552:antipodal map
4545:
4536:
4534:
4526:
4485:
4443:
4433:
4430:
4427:
4402:
4399:
4396:
4380:
4375:
4367:
4352:
4348:
4340:
4339:
4338:
4336:
4322:
4318:
4315:
4312:
4309:
4306:
4304:
4303:600 cell
4301:
4300:
4296:
4292:
4289:
4286:
4283:
4280:
4278:
4277:120 cell
4275:
4274:
4270:
4266:
4263:
4260:
4257:
4254:
4252:
4249:
4248:
4244:
4240:
4237:
4234:
4231:
4228:
4226:
4223:
4222:
4218:
4214:
4211:
4208:
4205:
4202:
4200:
4197:
4196:
4192:
4188:
4185:
4182:
4179:
4176:
4174:
4171:
4170:
4154:
4151:
4148:
4142:
4122:
4116:
4113:
4110:
4107:
4104:
4096:
4087:
4082:
4073:
4068:
4059:
4054:
4045:
4040:
4035:
4034:
4030:
4020:
4016:
4011:
4004:
4002:
4000:
3996:
3992:
3976:
3970:
3967:
3964:
3938:
3932:
3926:
3923:
3917:
3914:
3911:
3908:
3905:
3901:
3894:
3891:
3888:
3885:
3882:
3875:
3868:
3865:
3859:
3855:
3848:
3845:
3842:
3839:
3836:
3829:
3822:
3819:
3813:
3810:
3807:
3804:
3801:
3798:
3791:
3790:
3789:
3771:
3764:
3761:
3758:
3755:
3752:
3745:
3738:
3735:
3729:
3726:
3699:
3692:
3689:
3686:
3683:
3680:
3673:
3666:
3663:
3657:
3654:
3628:
3625:
3622:
3619:
3616:
3597:
3593:
3585:
3578:
3573:
3570:
3566:
3560:
3556:
3554:
3550:
3529:
3522:
3521:
3518:
3514:
3511:
3506:
3501:
3497:
3495:
3487:Three spheres
3486:
3485:
3482:
3478:
3475:
3471:
3469:
3460:
3459:
3456:
3450:
3447:
3443:
3441:
3438:
3437:
3434:
3428:
3425:
3421:
3419:
3416:
3415:
3412:
3406:
3403:
3399:
3397:
3392:
3391:
3388:
3385:
3382:
3378:
3376:
3373:
3372:
3369:
3366:
3363:
3359:
3357:
3354:
3353:
3350:
3344:
3341:
3337:
3335:
3329:
3326:
3325:
3322:
3316:
3313:
3309:
3307:
3304:
3303:
3300:
3294:
3291:
3287:
3285:
3282:
3281:
3278:
3272:
3269:
3265:
3263:
3260:
3259:
3256:
3250:
3247:
3243:
3241:
3238:
3237:
3233:
3229:
3226:
3223:
3222:
3219:
3217:
3209:
3204:
3188:
3185:
3179:
3173:
3170:
3167:
3164:
3159:
3155:
3147:
3146:
3145:
3143:
3124:
3116:
3112:
3102:
3094:
3090:
3086:
3081:
3077:
3053:
3045:
3041:
3031:
3023:
3019:
3015:
3012:
3005:
3000:
2998:
2979:
2973:
2967:
2964:
2958:
2952:
2949:
2943:
2937:
2930:
2929:
2928:
2926:
2922:
2918:
2915:
2911:
2895:
2889:
2886:
2883:
2874:
2872:
2864:
2862:
2860:
2856:
2837:
2831:
2825:
2822:
2819:
2816:
2804:
2795:
2788:
2787:
2786:
2772:
2769:
2757:
2747:
2743:
2737:
2729:
2727:
2725:
2722:
2718:
2699:
2693:
2687:
2684:
2678:
2672:
2669:
2663:
2660:
2657:
2651:
2644:
2643:
2642:
2640:
2636:
2633:
2632:product space
2625:
2609:
2601:
2597:
2590:
2587:
2581:
2575:
2572:
2566:
2560:
2557:
2551:
2545:
2539:
2532:
2531:
2530:
2528:
2527:connected sum
2512:
2506:
2486:
2483:
2480:
2469:Connected sum
2468:
2466:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2436:
2428:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2394:
2390:
2386:
2383:
2379:
2375:
2371:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2343:
2342:
2325:
2319:
2316:
2313:
2307:
2304:
2298:
2292:
2289:
2283:
2277:
2274:
2268:
2265:
2262:
2256:
2249:
2248:
2247:
2245:
2241:
2237:
2233:
2214:
2208:
2202:
2199:
2193:
2187:
2184:
2178:
2175:
2172:
2166:
2159:
2158:
2157:
2155:
2151:
2147:
2139:
2137:
2135:
2131:
2126:
2110:
2096:
2092:
2087:
2085:
2081:
2077:
2069:
2067:
2061:
2059:
2045:
2034:
2015:
2012:
2009:
2004:
2000:
1996:
1991:
1987:
1983:
1978:
1974:
1970:
1965:
1961:
1957:
1954:
1947:
1946:
1945:
1943:
1939:
1935:
1931:
1926:
1922:
1919:
1915:
1911:
1903:
1901:
1899:
1897:
1891:
1887:
1868:
1865:
1862:
1857:
1853:
1849:
1844:
1840:
1836:
1831:
1827:
1823:
1818:
1814:
1810:
1807:
1800:
1799:
1798:
1796:
1792:
1787:
1785:
1780:
1776:
1757:
1754:
1751:
1746:
1742:
1738:
1733:
1729:
1725:
1720:
1716:
1712:
1707:
1703:
1699:
1696:
1689:
1688:
1687:
1685:
1681:
1677:
1669:
1667:
1665:
1661:
1660:
1655:
1650:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1548:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1501:
1495:
1492:
1489:
1466:
1460:
1457:
1454:
1432:
1426:
1423:
1420:
1408:
1406:
1386:
1380:
1377:
1374:
1371:
1368:
1357:
1343:
1337:
1334:
1331:
1328:
1325:
1313:
1312:
1311:
1309:
1304:
1302:
1286:
1280:
1277:
1274:
1271:
1268:
1255:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1207:
1204:
1195:
1186:
1183:
1178:
1164:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1099:
1093:
1090:
1087:
1061:
1058:
1055:
1052:
1049:
1038:
1025:
1004:
1001:
998:
995:
992:
981:
978:
972:
964:
962:
960:
956:
952:
948:
944:
942:
923:
917:
914:
911:
908:
903:
899:
895:
892:
889:
886:
881:
877:
866:
865:
864:
851:
843:
839:
815:
807:
803:
791:
790:vertex figure
784:
780:
779:Arthur Cayley
771:
768:
765:
762:
759:
756:
752:
750:
747:
743:
740:
737:
734:
731:
728:
724:
722:
719:
715:
712:
709:
706:
703:
700:
696:
694:
691:
687:
684:
681:
678:
675:
672:
668:
666:
663:
659:
656:
653:
650:
647:
644:
640:
638:
635:
616:
613:
610:
607:
604:
601:
598:
586:
579:
572:
565:
562:
559:
558:
555:
548:
545:
542:
539:
536:
533:
529:
527:
524:
520:
517:
514:
511:
508:
505:
501:
499:
496:
492:
489:
486:
483:
480:
477:
473:
471:
468:
464:
461:
458:
455:
452:
449:
445:
443:
439:
436:
432:
429:
426:
423:
420:
417:
413:
411:
408:
389:
386:
383:
380:
377:
374:
371:
359:
352:
345:
338:
335:
332:
331:
328:
326:
307:
304:
301:
290:
286:
282:
263:
257:
254:
251:
248:
245:
242:
239:
236:
233:
223:
222:
221:
219:
215:
213:
208:
206:
201:
199:
167:
164:
161:
158:
155:
152:
149:
142:
141:
140:
135:
126:
119:
117:
115:
111:
107:
103:
99:
95:
90:
88:
85:
69:
61:
57:
53:
49:
45:
41:
37:
33:
26:
22:
6215:Publications
6080:Chern number
6070:Betti number
6064:
5953: /
5944:Key concepts
5892:Differential
5808:
5792:
5773:
5751:
5739:
5731:
5719:Bibliography
5701:the original
5680:
5676:
5663:
5654:
5645:
5614:
5608:
5596:
5587:
5584:Milnor, J.W.
5578:
5569:
5563:
5542:(1): 39â48,
5539:
5535:
5522:
5503:, Springer,
5499:
5489:
5478:. Retrieved
5469:
5460:
5451:
5441:
5431:
5421:– via
5415:. Retrieved
5398:
5387:
5374:
5361:
5333:
5326:
5200:
5181:
5166:
5108:
4987:
4875:on a proper
4843:
4832:
4824:cell complex
4821:
4789:
4784:
4769:
4754:
4732:, i.e., the
4730:Euler number
4729:
4727:
4675:
4606:
4575:
4556:
4537:
4507:
4328:
4320:
4294:
4268:
4251:24 cell
4242:
4225:16 cell
4216:
4190:
3953:
3592:soccer balls
3589:
3576:
3562:
3552:
3546:
3516:
3515:2 + 2 + 2 =
3491:
3480:
3465:
3451:
3440:Klein bottle
3429:
3418:Möbius strip
3407:
3386:
3375:Triple torus
3367:
3356:Double torus
3345:
3334:two circles)
3331:
3317:
3295:
3273:
3251:
3231:
3213:
3004:transfer map
3001:
2994:
2924:
2920:
2913:
2909:
2875:
2868:
2852:
2741:
2739:
2714:
2638:
2634:
2629:
2472:
2462:
2454:
2446:
2438:
2432:
2419:
2415:
2411:
2407:
2399:
2392:
2388:
2373:
2366:
2362:
2350:
2346:
2340:
2239:
2235:
2231:
2229:
2149:
2145:
2143:
2127:
2091:contractible
2088:
2073:
2065:
2032:
2030:
1941:
1933:
1924:
1920:
1918:Betti number
1913:
1907:
1895:
1889:
1885:
1883:
1794:
1788:
1783:
1778:
1774:
1772:
1683:
1676:CW-complexes
1673:
1657:
1651:
1409:
1401:
1308:simple cycle
1305:
1256:
1208:
1200:
1179:
1026:
982:by the same
980:plane graphs
974:
965:Plane graphs
945:
938:
776:
769:
741:
713:
685:
657:
553:
546:
518:
498:Dodecahedron
490:
462:
430:
284:
278:
211:
204:
197:
182:
133:
131:
91:
51:
48:Euler number
47:
43:
29:
6178:Wikiversity
6095:Key results
5437:Lakatos, I.
5297:Euler class
4755:For closed
4734:Euler class
4199:8 cell
4173:5 cell
3586:Soccer ball
3461:Two spheres
3332:(Product of
3142:Euler class
2717:cardinality
2395:are needed.
1940:group. The
1124:shows that
526:Icosahedron
410:Tetrahedron
202:(corners),
32:mathematics
21:Euler class
6230:Categories
6024:CW complex
5965:Continuity
5955:Closed set
5914:cohomology
5619:EPFL Press
5480:2016-07-13
5379:U. Pacific
5313:References
5207:categories
4780:polyhedron
4772:Descartes'
4609:orientable
4582:orientable
4015:hypercubes
3995:fullerenes
3216:CW-complex
2871:fibrations
2459:complement
2441:to be the
2084:isomorphic
2062:Properties
1898:-simplexes
969:See also:
470:Octahedron
438:Hexahedron
6203:geometric
6198:algebraic
6049:Cobordism
5985:Hausdorff
5980:connected
5897:Geometric
5887:Continuum
5877:Algebraic
5815:EMS Press
5794:MathWorld
5775:MathWorld
5683:: 21â49.
5628:0712.1507
5556:118905134
5423:UC Irvine
5367:Euler, L.
5111:orbifolds
5048:group of
4925:−
4913:∑
4893:χ
4704:−
4695:χ
4649:−
4640:χ
4571:Lie group
4337:equal to
4152:−
4114:−
4105:χ
4019:simplices
3860:−
3802:−
3186:⋅
3174:χ
3168:τ
3165:∘
3160:∗
3117:∗
3109:→
3095:∗
3087::
3082:∗
3046:∗
3038:→
3024:∗
3016::
3013:τ
2968:χ
2965:⋅
2953:χ
2938:χ
2893:→
2887::
2826:χ
2823:⋅
2808:~
2796:χ
2767:→
2761:~
2744:-sheeted
2688:χ
2685:⋅
2673:χ
2661:×
2652:χ
2591:χ
2588:−
2576:χ
2561:χ
2549:#
2540:χ
2510:#
2443:real line
2359:interiors
2317:∩
2308:χ
2305:−
2293:χ
2278:χ
2266:∪
2257:χ
2203:χ
2188:χ
2176:⊔
2167:χ
2046:χ
2013:⋯
1997:−
1971:−
1955:χ
1866:⋯
1850:−
1824:−
1808:χ
1755:⋯
1739:−
1713:−
1697:χ
1619:−
1575:−
1528:−
1372:−
1329:−
1272:−
1224:−
1150:−
1138:−
1059:−
996:−
977:connected
890:−
608:−
599:χ
381:−
372:χ
302:χ
243:−
234:χ
159:−
150:χ
120:Polyhedra
94:polyhedra
70:χ
6168:Wikibook
6146:Category
6034:Manifold
6002:Homotopy
5960:Interior
5951:Open set
5909:Homology
5858:Topology
5729:(2008).
5497:(1982),
5439:(1976).
5369:(1758).
5286:See also
5244:groupoid
5186:, where
4810:sets in
3553:spheres)
3479:2 + 2 =
3387:−4
3368:−2
3240:Interval
3210:Surfaces
3205:Examples
2785:one has
2525:via the
2385:supports
1666:(1976).
1654:Eppstein
1514:so that
1405:shelling
792:density
714:−2
566:Vertices
339:Vertices
110:homology
6193:general
5995:uniform
5975:compact
5926:Digital
5697:1046313
5652:at the
5633:Bibcode
5271:
5248:
5240:
5237:|
5222:
5190:is the
5162:
5145:
5135:
5121:
4805:compact
4778:" of a
4736:of its
4612:surface
4590:compact
4517: ;
4036:Regular
3542:spheres
2637:×
2424:complex
2382:compact
2353:are an
1932:of the
1928:as the
1664:Lakatos
783:density
200:ertices
54:) is a
6188:Topics
5990:metric
5865:Fields
5695:
5554:
5507:
5417:27 May
5349:
5268:|
5257:|
5231:|
5215:monoid
5139:where
5094:
5084:
5028:
4996:
4988:where
4970:
4877:scheme
4808:convex
4790:unique
4776:defect
4710:
4658:
4578:closed
4483:
4425:
4422:
4409:
4406:
4392:
4146:
4120:
3974:
3962:
3936:
3898:
3880:
3852:
3834:
3776:
3768:
3750:
3724:
3704:
3696:
3678:
3652:
3632:
3614:
3598:). If
3564:. . .
3306:Sphere
3262:Circle
3227:Image
2451:subset
2134:sphere
1884:where
1793:, the
1773:where
1637:
1613:
1593:
1569:
1546:
1522:
1499:
1487:
1464:
1452:
1430:
1418:
1384:
1366:
1341:
1323:
1284:
1266:
1242:
1218:
1203:Cauchy
1162:
1132:
1097:
1085:
1065:
1047:
1008:
990:
921:
874:
849:
836:
813:
800:
620:
596:
393:
369:
311:
299:
291:(i.e.
289:sphere
261:
231:
191:, and
183:where
42:, the
5970:Space
5704:(PDF)
5693:S2CID
5673:(PDF)
5623:arXiv
5552:S2CID
5532:(PDF)
5318:Notes
5211:group
5184:(0,1)
5173:poset
5113:(see
4839:ranks
4828:graph
4794:up to
4625:in a
4616:genus
3396:plane
3328:Torus
3224:Name
2402:is a
2376:is a
2082:have
1037:trees
959:torus
580:Faces
573:Edges
563:Image
353:Faces
346:Edges
336:Image
281:Euler
50:, or
5505:ISBN
5419:2022
5347:ISBN
5119:1 +
5044:-th
4623:tori
4557:The
4538:The
4533:even
4329:The
4313:1200
4284:1200
4017:and
3997:and
3284:Disk
2721:sets
2457:the
2418:and
2410:and
2365:and
2349:and
2234:and
2148:and
2130:ball
1936:-th
1930:rank
1479:and
1180:Via
1116:has
1077:and
560:Name
442:cube
333:Name
214:aces
209:and
207:dges
132:The
46:(or
38:and
5685:doi
5657:Lab
5544:doi
5339:doi
5220:is
5213:or
4803:of
4748:of
4531:is
4525:odd
4523:is
4316:600
4310:720
4307:120
4290:120
4287:720
4281:600
3549:of
2876:If
2719:of
2641:is
2461:of
2398:if
2391:or
2372:if
2361:of
2345:if
2144:If
1916:th
1662:by
1407:.)
1112:If
440:or
89:).
87:chi
30:In
6232::
5813:,
5807:,
5791:.
5772:.
5691:.
5681:13
5679:.
5675:.
5631:.
5550:,
5540:10
5538:,
5534:,
5514:,
5345:.
5198:.
5164:.
4812:â
4752:.
4599:.
4535:.
4264:24
4261:96
4258:96
4255:24
4238:16
4235:32
4232:24
4209:24
4206:32
4203:16
4183:10
4180:10
4001:.
3971:12
3577:2n
3189:1.
2921:K,
2910:F,
2861:.
2726:.
2465:.
2449:a
2445:,
2246::
2058:.
1310::
1039:,
961:.
943:.
788:,
766:12
763:30
760:20
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