4421:
2937:
4416:{\displaystyle {\begin{aligned}P_{SU(n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{5}\right)\cdots \left(1+x^{2n+1}\right)\\P_{SO(2n+1)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{Sp(n)}(x)&=\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-1}\right)\\P_{SO(2n)}(x)&=\left(1+x^{2n-1}\right)\left(1+x^{3}\right)\left(1+x^{7}\right)\cdots \left(1+x^{4n-5}\right)\\P_{G_{2}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\\P_{F_{4}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\\P_{E_{6}}(x)&=\left(1+x^{3}\right)\left(1+x^{9}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{17}\right)\left(1+x^{23}\right)\\P_{E_{7}}(x)&=\left(1+x^{3}\right)\left(1+x^{11}\right)\left(1+x^{15}\right)\left(1+x^{19}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\\P_{E_{8}}(x)&=\left(1+x^{3}\right)\left(1+x^{15}\right)\left(1+x^{23}\right)\left(1+x^{27}\right)\left(1+x^{35}\right)\left(1+x^{39}\right)\left(1+x^{47}\right)\left(1+x^{59}\right)\end{aligned}}}
4866:
5242:
5025:
5263:
5231:
5300:
5273:
5253:
1371:
1730:
1452:
647:
404:
921:
4655:
2206:
1200:
519:
1905:
2911:
302:
1618:
1104:
Given a topological space which has a finitely generated homology, the
Poincaré polynomial is defined as the generating function of its Betti numbers, via the polynomial where the coefficient of
2942:
2568:
767:-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so
1462:
with 0-simplices: a, b, c, and d, 1-simplices: E, F, G, H and I, and the only 2-simplex is J, which is the shaded region in the figure. There is one connected component in this figure (
208:
2027:
2759:
2707:
2365:
2821:
555:
122:
2646:
It is possible for spaces that are infinite-dimensional in an essential way to have an infinite sequence of non-zero Betti numbers. An example is the infinite-dimensional
2628:
948:
1099:
601:
446:
344:
250:
164:
1937:
4533:
2468:
2057:
1595:
1565:
1538:
1511:
1156:
1129:
2275:
5303:
4449:
2295:
2249:
349:
82:, which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if
4813:
1782:
in the homology groups, but they are very useful basic topological invariants. In the most intuitive terms, they allow one to count the number of
1366:{\displaystyle H_{k}(G)={\begin{cases}\mathbb {Z} ^{|C|}&k=0\\\mathbb {Z} ^{|E|+|C|-|V|}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}
2924:; thus the Poincaré series is expressible as a rational function if and only if the sequence of Betti numbers is a linear recursive sequence.
4793:
4766:
4729:
853:
55:), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite.
4549:
2085:
451:
1809:
4455:, the importance of the Betti numbers may arise from a different direction, namely that they predict the dimensions of vector spaces of
4937:
2829:
5334:
5291:
5286:
4906:
4887:
255:
4844:
2764:
More generally, any sequence that is periodic can be expressed as a sum of geometric series, generalizing the above. For example
4819:
5281:
4505:
gives a set of inequalities for alternating sums of Betti numbers in terms of a corresponding alternating sum of the number of
1725:{\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}
2068:
1380:
on the number of edges. A new edge either increments the number of 1-cycles or decrements the number of connected components.
4685:
5339:
5183:
4476:
2422:
988:
4506:
5324:
777:
The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions.
1059:
of its Betti numbers. For example, the Betti numbers of the torus are 1, 2, and 1; thus its
Poincaré polynomial is
5191:
2489:
5329:
4680:
2917:
169:
5262:
4990:
4456:
2647:
1973:
1757:) is a finite group - it does not have any infinite component. The finite component of the group is called the
1172:
815:
692:
The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional
5276:
4464:
4901:, Research Notes in Mathematics Series, vol. 395 (Second ed.), Boca Raton, FL: Chapman and Hall,
4865:
2719:
2660:
5211:
5206:
5132:
5009:
4997:
4970:
4930:
2395:
2303:
1377:
963:
2767:
5053:
4980:
1433:
5241:
524:
5201:
5153:
5127:
4975:
4690:
4468:
2635:
1940:
1437:
736:
85:
48:
1649:
1231:
5252:
5048:
4664:
4472:
2379:
2072:
1746:
of order 2. The 0-th Betti number is again 1. However, the 1-st Betti number is 0. This is because
1056:
995:
627:
558:
5246:
5196:
5117:
5107:
4985:
4965:
4809:
4725:
4460:
2921:
2593:
1459:
1101:. The same definition applies to any topological space which has a finitely generated homology.
951:
926:
693:
36:
20:
5216:
2631:
2212:
1062:
564:
409:
307:
213:
127:
5234:
5100:
5058:
4923:
4902:
4883:
4789:
4762:
4486:
There is an alternate reading, namely that the Betti numbers give the dimensions of spaces of
2399:
1913:
1425:
615:
40:
28:
4783:
4756:
5014:
4960:
4853:
2411:
1779:
1606:
1429:
631:
604:
44:
4511:
2446:
2035:
1573:
1543:
1516:
1489:
1134:
1107:
5073:
5068:
4668:
4495:
4480:
4452:
63:
2654:
2. In this case the
Poincaré function is not a polynomial but rather an infinite series
2254:
1570:
The Betti number sequence for this figure is 1, 1, 0, 0, ...; the
Poincaré polynomial is
5163:
5095:
4536:
4434:
2280:
2234:
1191:
839:
635:
70:
448:, etc. Note that only the ranks of infinite groups are considered, so for example if
5318:
5173:
5083:
5063:
4839:
4660:
4487:
2651:
819:
5266:
4663:
gave an explanation of these inequalities by using the Morse function to modify the
5078:
5024:
4540:
4502:
4491:
1743:
623:
619:
52:
4758:
Island
Networks: Communication, Kinship, and Classification Structures in Oceania
991:, in a very simple torsion-free case, shows that these definitions are the same.
5256:
5168:
4711:
2426:
1451:
399:{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }
5112:
5043:
5002:
4735:
2928:
4858:
4467:. The connection with the definition given above is via three basic results,
5137:
2713:
which, being a geometric series, can be expressed as the rational function
657:= 2 , which can be intuitively thought of as the number of circular "holes"
646:
5122:
5090:
5039:
4946:
4824:
4740:
916:{\displaystyle H_{k}=\ker \delta _{k}/\operatorname {Im} \delta _{k+1}}
786:
4650:{\displaystyle b_{i}(X)-b_{i-1}(X)+\cdots \leq N_{i}-N_{i-1}+\cdots .}
2201:{\displaystyle P_{X}(z)=b_{0}(X)+b_{1}(X)z+b_{2}(X)z^{2}+\cdots ,\,\!}
514:{\displaystyle H_{n}(X)\cong \mathbb {Z} ^{k}\oplus \mathbb {Z} /(2)}
983:) since the homology group in this case is a vector space over
1900:{\displaystyle \chi (K)=\sum _{i=0}^{\infty }(-1)^{i}b_{i}(K,F),\,}
16:
Roughly, the number of k-dimensional holes on a topological surface
2582:
2478:
728:
Thus, for example, a torus has one connected surface component so
645:
4919:
2906:{\displaystyle \left(a+bx+cx^{2}\right)/\left(1-x^{3}\right)\,}
2438:
The Betti number sequence for a circle is 1, 1, 0, 0, 0, ...;
297:{\displaystyle H_{n}(X)\cong \mathbb {Z} \oplus \mathbb {Z} }
4427:
Relationship with dimensions of spaces of differential forms
1450:
4915:
1718:
1359:
603:. These finite components of the homology groups are their
2650:, with sequence 1, 0, 1, 0, 1, ... that is periodic, with
962:
th Betti number. Equivalently, one can define it as the
746:= 2, and a single cavity enclosed within the surface so
2231:-dimensional manifold, there is symmetry interchanging
1398:|, which is simply the number of connected components.
724:
is the number of two-dimensional "voids" or "cavities".
4880:
Foundations of differentiable manifolds and Lie groups
4785:
4552:
4514:
4437:
2940:
2832:
2770:
2722:
2663:
2596:
2492:
2449:
2306:
2283:
2257:
2237:
2088:
2038:
1976:
1916:
1812:
1621:
1576:
1546:
1519:
1492:
1203:
1137:
1110:
1065:
929:
856:
715:
is the number of one-dimensional or "circular" holes;
567:
527:
454:
412:
352:
310:
258:
216:
172:
130:
88:
4899:
Elliptic
Operators, Topology, and Asymptotic Methods
5182:
5146:
5032:
4953:
818:(number of linearly independent generators) of the
4649:
4527:
4490:. This requires the use of some of the results of
4443:
4415:
2905:
2815:
2753:
2701:
2622:
2562:
2462:
2359:
2289:
2269:
2243:
2200:
2051:
2021:
1931:
1899:
1724:
1589:
1559:
1532:
1505:
1365:
1150:
1123:
1093:
942:
915:
735:= 1, two "circular" holes (one equatorial and one
595:
549:
513:
440:
398:
338:
296:
244:
202:
158:
116:
2197:
626:. Betti numbers are used today in fields such as
685:-dimensional cycle that is not a boundary of a (
2927:The Poincaré polynomials of the compact simple
2071:, for infinite-dimensional spaces), i.e., the
4931:
2563:{\displaystyle (1+x)^{3}=1+3x+3x^{2}+x^{3}\,}
39:. For the most reasonable finite-dimensional
8:
4820:"Delta complexes, Betti numbers and torsion"
4710:Barile, and Weisstein, Margherita and Eric.
1705:
1699:
1346:
1340:
1469:); one hole, which is the unshaded region (
5299:
5272:
4938:
4924:
4916:
4842:(1982), "Supersymmetry and Morse theory",
4761:. Cambridge University Press. p. 49.
2421:a prime number, is given in detail by the
203:{\displaystyle H_{n}(X)\cong \mathbb {Z} }
4857:
4788:. American Mathematical Soc. p. 20.
4714:. From MathWorld--A Wolfram Web Resource.
4626:
4613:
4579:
4557:
4551:
4519:
4513:
4436:
4398:
4372:
4346:
4320:
4294:
4268:
4242:
4216:
4177:
4172:
4153:
4127:
4101:
4075:
4049:
4023:
3997:
3958:
3953:
3934:
3908:
3882:
3856:
3830:
3804:
3765:
3760:
3741:
3715:
3689:
3663:
3624:
3619:
3600:
3574:
3535:
3530:
3502:
3473:
3447:
3412:
3360:
3332:
3303:
3277:
3228:
3200:
3171:
3145:
3087:
3059:
3030:
3004:
2949:
2941:
2939:
2902:
2891:
2871:
2860:
2831:
2769:
2739:
2723:
2721:
2687:
2674:
2662:
2619:
2613:
2595:
2559:
2553:
2540:
2509:
2491:
2459:
2448:
2333:
2311:
2305:
2282:
2256:
2236:
2196:
2181:
2162:
2137:
2115:
2093:
2087:
2043:
2037:
2022:{\displaystyle P_{X\times Y}=P_{X}P_{Y},}
2010:
2000:
1981:
1975:
1915:
1896:
1872:
1862:
1843:
1832:
1811:
1710:
1678:
1674:
1673:
1653:
1652:
1644:
1626:
1620:
1586:
1575:
1551:
1545:
1524:
1518:
1497:
1491:
1351:
1318:
1310:
1302:
1294:
1286:
1278:
1277:
1273:
1272:
1249:
1241:
1240:
1236:
1235:
1226:
1208:
1202:
1186:, and the set of connected components is
1142:
1136:
1115:
1109:
1085:
1064:
934:
928:
901:
886:
880:
861:
855:
572:
566:
533:
529:
528:
526:
497:
493:
492:
483:
479:
478:
459:
453:
417:
411:
392:
391:
384:
383:
376:
375:
357:
351:
315:
309:
290:
289:
282:
281:
263:
257:
221:
215:
196:
195:
177:
171:
135:
129:
93:
87:
1376:This may be proved straightforwardly by
665:th Betti number refers to the number of
4702:
2920:are exactly the sequences generated by
2402:, the Betti numbers are independent of
650:For a torus, the first Betti number is
614:The term "Betti numbers" was coined by
2477:The Betti number sequence for a three-
1383:Therefore, the "zero-th" Betti number
706:is the number of connected components;
2754:{\displaystyle {\frac {1}{1-x^{2}}}.}
2702:{\displaystyle 1+x^{2}+x^{4}+\dotsb }
1601:Betti numbers of the projective plane
1447:Betti numbers of a simplicial complex
1022:th Betti number with coefficients in
7:
2360:{\displaystyle b_{k}(X)=b_{n-k}(X),}
1194:, its homology groups are given by:
2816:{\displaystyle a,b,c,a,b,c,\dots ,}
1026:, as the vector space dimension of
622:. The modern formulation is due to
2410:-torsion and the Betti number for
1844:
1055:of a surface is defined to be the
14:
1476:); and no "voids" or "cavities" (
5298:
5271:
5261:
5251:
5240:
5230:
5229:
5023:
4864:
4845:Journal of Differential Geometry
2634:), so the Betti numbers are the
1178:in which the set of vertices is
550:{\displaystyle \mathbb {Z} /(2)}
4736:"History of algebraic topology"
1778:) do not take into account any
1765:. The (rational) Betti numbers
1443:All other Betti numbers are 0.
950:s are the boundary maps of the
117:{\displaystyle H_{n}(X)\cong 0}
4818:Wildberger, Norman J. (2012).
4597:
4591:
4569:
4563:
4191:
4185:
3972:
3966:
3779:
3773:
3638:
3632:
3549:
3543:
3387:
3381:
3376:
3367:
3252:
3246:
3241:
3235:
3120:
3114:
3109:
3094:
2979:
2973:
2968:
2956:
2610:
2597:
2506:
2493:
2481:is 1, 3, 3, 1, 0, 0, 0, ... .
2351:
2345:
2323:
2317:
2174:
2168:
2149:
2143:
2127:
2121:
2105:
2099:
1926:
1920:
1890:
1878:
1859:
1849:
1822:
1816:
1638:
1632:
1319:
1311:
1303:
1295:
1287:
1279:
1250:
1242:
1220:
1214:
1190:. As explained in the page on
584:
578:
544:
538:
508:
502:
471:
465:
429:
423:
369:
363:
327:
321:
275:
269:
233:
227:
189:
183:
147:
141:
105:
99:
1:
4878:Warner, Frank Wilson (1983),
4782:Peter Robert Kotiuga (2010).
4477:universal coefficient theorem
4431:In geometric situations when
2823:has the generating function
2423:universal coefficient theorem
2398:. If the homology groups are
989:universal coefficient theorem
673:on a topological surface. A "
31:based on the connectivity of
4475:(when those apply), and the
2390:The dependence on the field
1486:This means that the rank of
62:Betti number represents the
2623:{\displaystyle (1+x)^{n}\,}
2588:the Poincaré polynomial is
2484:the Poincaré polynomial is
2441:the Poincaré polynomial is
1605:The homology groups of the
943:{\displaystyle \delta _{k}}
5356:
5192:Banach fixed-point theorem
2918:linear recursive sequences
1432:before Betti's paper. See
1094:{\displaystyle 1+2x+x^{2}}
756:Another interpretation of
607:, and they are denoted by
596:{\displaystyle b_{n}(X)=k}
441:{\displaystyle b_{n}(X)=3}
339:{\displaystyle b_{n}(X)=2}
245:{\displaystyle b_{n}(X)=1}
159:{\displaystyle b_{n}(X)=0}
5225:
5021:
4681:Topological data analysis
4457:closed differential forms
2429:, but in a simple case).
1786:of different dimensions.
1424:|. It is also called the
763:is the maximum number of
5335:Topological graph theory
4465:exact differential forms
2648:complex projective space
2075:of the Betti numbers of
1932:{\displaystyle \chi (K)}
1799:For a finite CW-complex
1167:Betti numbers of a graph
994:More generally, given a
689:+1)-dimensional object.
642:Geometric interpretation
27:are used to distinguish
2069:Hilbert–Poincaré series
2067:, (more generally, the
1401:The first Betti number
5247:Mathematics portal
5147:Metrics and properties
5133:Second-countable space
4882:, New York: Springer,
4859:10.4310/jdg/1214437492
4734:Albin, Pierre (2019).
4651:
4529:
4445:
4417:
2907:
2817:
2755:
2703:
2624:
2564:
2464:
2386:Different coefficients
2361:
2291:
2271:
2245:
2202:
2053:
2023:
1933:
1901:
1848:
1726:
1591:
1561:
1534:
1507:
1455:
1436:for an application to
1428:—a term introduced by
1378:mathematical induction
1367:
1182:, the set of edges is
1152:
1125:
1095:
964:vector space dimension
944:
917:
658:
597:
551:
515:
442:
400:
340:
298:
246:
204:
160:
118:
4652:
4530:
4528:{\displaystyle N_{i}}
4446:
4418:
2908:
2818:
2756:
2704:
2636:binomial coefficients
2625:
2565:
2465:
2463:{\displaystyle 1+x\,}
2362:
2292:
2272:
2246:
2203:
2054:
2052:{\displaystyle P_{X}}
2024:
1934:
1902:
1828:
1727:
1592:
1590:{\displaystyle 1+x\,}
1562:
1560:{\displaystyle H_{2}}
1540:is 1 and the rank of
1535:
1533:{\displaystyle H_{1}}
1508:
1506:{\displaystyle H_{0}}
1454:
1434:cyclomatic complexity
1368:
1153:
1151:{\displaystyle b_{n}}
1126:
1124:{\displaystyle x^{n}}
1096:
945:
918:
850:th homology group is
649:
598:
552:
516:
443:
401:
341:
299:
247:
205:
161:
119:
5340:Generating functions
5202:Invariance of domain
5154:Euler characteristic
5128:Bundle (mathematics)
4691:Euler characteristic
4550:
4512:
4435:
2938:
2830:
2768:
2720:
2661:
2594:
2490:
2447:
2412:characteristic
2406:. The connection of
2394:is only through its
2370:under conditions (a
2304:
2281:
2255:
2235:
2086:
2036:
1974:
1941:Euler characteristic
1914:
1810:
1795:Euler characteristic
1619:
1574:
1544:
1517:
1490:
1438:software engineering
1201:
1135:
1108:
1063:
927:
854:
694:simplicial complexes
609:torsion coefficients
565:
525:
452:
410:
350:
308:
256:
214:
170:
128:
86:
49:simplicial complexes
37:simplicial complexes
5212:Tychonoff's theorem
5207:Poincaré conjecture
4961:General (point-set)
4686:Torsion coefficient
4665:exterior derivative
2916:and more generally
2270:{\displaystyle n-k}
2073:generating function
2061:Poincaré polynomial
1959:For any two spaces
1947:and any field
1759:torsion coefficient
1057:generating function
1053:Poincaré polynomial
1047:Poincaré polynomial
785:For a non-negative
628:simplicial homology
559:finite cyclic group
5325:Algebraic topology
5197:De Rham cohomology
5118:Polyhedral complex
5108:Simplicial complex
4897:Roe, John (1998),
4647:
4525:
4441:
4413:
4411:
2922:rational functions
2903:
2813:
2751:
2699:
2620:
2577:Similarly, for an
2560:
2460:
2357:
2287:
2267:
2241:
2198:
2049:
2019:
1929:
1897:
1722:
1717:
1587:
1557:
1530:
1513:is 1, the rank of
1503:
1460:simplicial complex
1456:
1363:
1358:
1148:
1121:
1091:
952:simplicial complex
940:
913:
814:is defined as the
659:
593:
547:
511:
438:
396:
336:
294:
242:
200:
156:
114:
29:topological spaces
21:algebraic topology
5312:
5311:
5101:fundamental group
4795:978-0-8218-8381-5
4768:978-0-521-55232-5
4755:Per Hage (1996).
4501:In this setting,
4469:de Rham's theorem
4444:{\displaystyle X}
2746:
2290:{\displaystyle k}
2244:{\displaystyle k}
1955:Cartesian product
1713:
1426:cyclomatic number
1354:
1173:topological graph
954:and the rank of H
781:Formal definition
605:torsion subgroups
561:of order 2, then
45:compact manifolds
5347:
5330:Graph invariants
5302:
5301:
5275:
5274:
5265:
5255:
5245:
5244:
5233:
5232:
5027:
4940:
4933:
4926:
4917:
4911:
4892:
4870:
4869:
4868:
4862:
4861:
4836:
4830:
4829:
4806:
4800:
4799:
4779:
4773:
4772:
4752:
4746:
4745:
4722:
4716:
4715:
4707:
4656:
4654:
4653:
4648:
4637:
4636:
4618:
4617:
4590:
4589:
4562:
4561:
4534:
4532:
4531:
4526:
4524:
4523:
4473:Poincaré duality
4450:
4448:
4447:
4442:
4422:
4420:
4419:
4414:
4412:
4408:
4404:
4403:
4402:
4382:
4378:
4377:
4376:
4356:
4352:
4351:
4350:
4330:
4326:
4325:
4324:
4304:
4300:
4299:
4298:
4278:
4274:
4273:
4272:
4252:
4248:
4247:
4246:
4226:
4222:
4221:
4220:
4184:
4183:
4182:
4181:
4163:
4159:
4158:
4157:
4137:
4133:
4132:
4131:
4111:
4107:
4106:
4105:
4085:
4081:
4080:
4079:
4059:
4055:
4054:
4053:
4033:
4029:
4028:
4027:
4007:
4003:
4002:
4001:
3965:
3964:
3963:
3962:
3944:
3940:
3939:
3938:
3918:
3914:
3913:
3912:
3892:
3888:
3887:
3886:
3866:
3862:
3861:
3860:
3840:
3836:
3835:
3834:
3814:
3810:
3809:
3808:
3772:
3771:
3770:
3769:
3751:
3747:
3746:
3745:
3725:
3721:
3720:
3719:
3699:
3695:
3694:
3693:
3673:
3669:
3668:
3667:
3631:
3630:
3629:
3628:
3610:
3606:
3605:
3604:
3584:
3580:
3579:
3578:
3542:
3541:
3540:
3539:
3521:
3517:
3516:
3515:
3483:
3479:
3478:
3477:
3457:
3453:
3452:
3451:
3431:
3427:
3426:
3425:
3380:
3379:
3351:
3347:
3346:
3345:
3313:
3309:
3308:
3307:
3287:
3283:
3282:
3281:
3245:
3244:
3219:
3215:
3214:
3213:
3181:
3177:
3176:
3175:
3155:
3151:
3150:
3149:
3113:
3112:
3078:
3074:
3073:
3072:
3040:
3036:
3035:
3034:
3014:
3010:
3009:
3008:
2972:
2971:
2912:
2910:
2909:
2904:
2901:
2897:
2896:
2895:
2875:
2870:
2866:
2865:
2864:
2822:
2820:
2819:
2814:
2760:
2758:
2757:
2752:
2747:
2745:
2744:
2743:
2724:
2708:
2706:
2705:
2700:
2692:
2691:
2679:
2678:
2629:
2627:
2626:
2621:
2618:
2617:
2569:
2567:
2566:
2561:
2558:
2557:
2545:
2544:
2514:
2513:
2469:
2467:
2466:
2461:
2380:Poincaré duality
2366:
2364:
2363:
2358:
2344:
2343:
2316:
2315:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2250:
2248:
2247:
2242:
2207:
2205:
2204:
2199:
2186:
2185:
2167:
2166:
2142:
2141:
2120:
2119:
2098:
2097:
2058:
2056:
2055:
2050:
2048:
2047:
2028:
2026:
2025:
2020:
2015:
2014:
2005:
2004:
1992:
1991:
1938:
1936:
1935:
1930:
1906:
1904:
1903:
1898:
1877:
1876:
1867:
1866:
1847:
1842:
1731:
1729:
1728:
1723:
1721:
1720:
1714:
1711:
1683:
1682:
1677:
1656:
1631:
1630:
1607:projective plane
1596:
1594:
1593:
1588:
1566:
1564:
1563:
1558:
1556:
1555:
1539:
1537:
1536:
1531:
1529:
1528:
1512:
1510:
1509:
1504:
1502:
1501:
1430:Gustav Kirchhoff
1372:
1370:
1369:
1364:
1362:
1361:
1355:
1352:
1324:
1323:
1322:
1314:
1306:
1298:
1290:
1282:
1276:
1255:
1254:
1253:
1245:
1239:
1213:
1212:
1157:
1155:
1154:
1149:
1147:
1146:
1130:
1128:
1127:
1122:
1120:
1119:
1100:
1098:
1097:
1092:
1090:
1089:
949:
947:
946:
941:
939:
938:
922:
920:
919:
914:
912:
911:
890:
885:
884:
866:
865:
797:th Betti number
661:Informally, the
632:computer science
602:
600:
599:
594:
577:
576:
556:
554:
553:
548:
537:
532:
520:
518:
517:
512:
501:
496:
488:
487:
482:
464:
463:
447:
445:
444:
439:
422:
421:
405:
403:
402:
397:
395:
387:
379:
362:
361:
345:
343:
342:
337:
320:
319:
303:
301:
300:
295:
293:
285:
268:
267:
251:
249:
248:
243:
226:
225:
209:
207:
206:
201:
199:
182:
181:
165:
163:
162:
157:
140:
139:
123:
121:
120:
115:
98:
97:
5355:
5354:
5350:
5349:
5348:
5346:
5345:
5344:
5315:
5314:
5313:
5308:
5239:
5221:
5217:Urysohn's lemma
5178:
5142:
5028:
5019:
4991:low-dimensional
4949:
4944:
4909:
4896:
4890:
4877:
4874:
4873:
4863:
4838:
4837:
4833:
4817:
4814:Wayback Machine
4807:
4803:
4796:
4781:
4780:
4776:
4769:
4754:
4753:
4749:
4733:
4730:Wayback Machine
4723:
4719:
4709:
4708:
4704:
4699:
4677:
4669:de Rham complex
4622:
4609:
4575:
4553:
4548:
4547:
4515:
4510:
4509:
4507:critical points
4496:Hodge Laplacian
4481:homology theory
4453:closed manifold
4433:
4432:
4429:
4410:
4409:
4394:
4387:
4383:
4368:
4361:
4357:
4342:
4335:
4331:
4316:
4309:
4305:
4290:
4283:
4279:
4264:
4257:
4253:
4238:
4231:
4227:
4212:
4205:
4201:
4194:
4173:
4168:
4165:
4164:
4149:
4142:
4138:
4123:
4116:
4112:
4097:
4090:
4086:
4071:
4064:
4060:
4045:
4038:
4034:
4019:
4012:
4008:
3993:
3986:
3982:
3975:
3954:
3949:
3946:
3945:
3930:
3923:
3919:
3904:
3897:
3893:
3878:
3871:
3867:
3852:
3845:
3841:
3826:
3819:
3815:
3800:
3793:
3789:
3782:
3761:
3756:
3753:
3752:
3737:
3730:
3726:
3711:
3704:
3700:
3685:
3678:
3674:
3659:
3652:
3648:
3641:
3620:
3615:
3612:
3611:
3596:
3589:
3585:
3570:
3563:
3559:
3552:
3531:
3526:
3523:
3522:
3498:
3491:
3487:
3469:
3462:
3458:
3443:
3436:
3432:
3408:
3401:
3397:
3390:
3356:
3353:
3352:
3328:
3321:
3317:
3299:
3292:
3288:
3273:
3266:
3262:
3255:
3224:
3221:
3220:
3196:
3189:
3185:
3167:
3160:
3156:
3141:
3134:
3130:
3123:
3083:
3080:
3079:
3055:
3048:
3044:
3026:
3019:
3015:
3000:
2993:
2989:
2982:
2945:
2936:
2935:
2887:
2880:
2876:
2856:
2837:
2833:
2828:
2827:
2766:
2765:
2735:
2728:
2718:
2717:
2683:
2670:
2659:
2658:
2632:KĂĽnneth theorem
2609:
2592:
2591:
2549:
2536:
2505:
2488:
2487:
2445:
2444:
2435:
2388:
2378:manifold); see
2329:
2307:
2302:
2301:
2279:
2278:
2253:
2252:
2233:
2232:
2221:
2213:KĂĽnneth theorem
2177:
2158:
2133:
2111:
2089:
2084:
2083:
2039:
2034:
2033:
2006:
1996:
1977:
1972:
1971:
1957:
1912:
1911:
1868:
1858:
1808:
1807:
1797:
1792:
1773:
1752:
1741:
1716:
1715:
1708:
1696:
1695:
1684:
1672:
1669:
1668:
1657:
1645:
1622:
1617:
1616:
1603:
1572:
1571:
1547:
1542:
1541:
1520:
1515:
1514:
1493:
1488:
1487:
1482:
1475:
1468:
1449:
1407:
1389:
1357:
1356:
1349:
1337:
1336:
1325:
1271:
1268:
1267:
1256:
1234:
1227:
1204:
1199:
1198:
1169:
1164:
1138:
1133:
1132:
1111:
1106:
1105:
1081:
1061:
1060:
1049:
1034:
1009:
1001:one can define
974:
957:
930:
925:
924:
897:
876:
857:
852:
851:
829:
810:) of the space
805:
783:
773:
762:
752:
745:
734:
723:
714:
705:
656:
644:
568:
563:
562:
523:
522:
477:
455:
450:
449:
413:
408:
407:
353:
348:
347:
311:
306:
305:
259:
254:
253:
217:
212:
211:
173:
168:
167:
131:
126:
125:
89:
84:
83:
81:
17:
12:
11:
5:
5353:
5351:
5343:
5342:
5337:
5332:
5327:
5317:
5316:
5310:
5309:
5307:
5306:
5296:
5295:
5294:
5289:
5284:
5269:
5259:
5249:
5237:
5226:
5223:
5222:
5220:
5219:
5214:
5209:
5204:
5199:
5194:
5188:
5186:
5180:
5179:
5177:
5176:
5171:
5166:
5164:Winding number
5161:
5156:
5150:
5148:
5144:
5143:
5141:
5140:
5135:
5130:
5125:
5120:
5115:
5110:
5105:
5104:
5103:
5098:
5096:homotopy group
5088:
5087:
5086:
5081:
5076:
5071:
5066:
5056:
5051:
5046:
5036:
5034:
5030:
5029:
5022:
5020:
5018:
5017:
5012:
5007:
5006:
5005:
4995:
4994:
4993:
4983:
4978:
4973:
4968:
4963:
4957:
4955:
4951:
4950:
4945:
4943:
4942:
4935:
4928:
4920:
4914:
4913:
4907:
4894:
4888:
4872:
4871:
4852:(4): 661–692,
4840:Witten, Edward
4831:
4801:
4794:
4774:
4767:
4747:
4717:
4712:"Betti number"
4701:
4700:
4698:
4695:
4694:
4693:
4688:
4683:
4676:
4673:
4658:
4657:
4646:
4643:
4640:
4635:
4632:
4629:
4625:
4621:
4616:
4612:
4608:
4605:
4602:
4599:
4596:
4593:
4588:
4585:
4582:
4578:
4574:
4571:
4568:
4565:
4560:
4556:
4537:Morse function
4522:
4518:
4488:harmonic forms
4440:
4428:
4425:
4424:
4423:
4407:
4401:
4397:
4393:
4390:
4386:
4381:
4375:
4371:
4367:
4364:
4360:
4355:
4349:
4345:
4341:
4338:
4334:
4329:
4323:
4319:
4315:
4312:
4308:
4303:
4297:
4293:
4289:
4286:
4282:
4277:
4271:
4267:
4263:
4260:
4256:
4251:
4245:
4241:
4237:
4234:
4230:
4225:
4219:
4215:
4211:
4208:
4204:
4200:
4197:
4195:
4193:
4190:
4187:
4180:
4176:
4171:
4167:
4166:
4162:
4156:
4152:
4148:
4145:
4141:
4136:
4130:
4126:
4122:
4119:
4115:
4110:
4104:
4100:
4096:
4093:
4089:
4084:
4078:
4074:
4070:
4067:
4063:
4058:
4052:
4048:
4044:
4041:
4037:
4032:
4026:
4022:
4018:
4015:
4011:
4006:
4000:
3996:
3992:
3989:
3985:
3981:
3978:
3976:
3974:
3971:
3968:
3961:
3957:
3952:
3948:
3947:
3943:
3937:
3933:
3929:
3926:
3922:
3917:
3911:
3907:
3903:
3900:
3896:
3891:
3885:
3881:
3877:
3874:
3870:
3865:
3859:
3855:
3851:
3848:
3844:
3839:
3833:
3829:
3825:
3822:
3818:
3813:
3807:
3803:
3799:
3796:
3792:
3788:
3785:
3783:
3781:
3778:
3775:
3768:
3764:
3759:
3755:
3754:
3750:
3744:
3740:
3736:
3733:
3729:
3724:
3718:
3714:
3710:
3707:
3703:
3698:
3692:
3688:
3684:
3681:
3677:
3672:
3666:
3662:
3658:
3655:
3651:
3647:
3644:
3642:
3640:
3637:
3634:
3627:
3623:
3618:
3614:
3613:
3609:
3603:
3599:
3595:
3592:
3588:
3583:
3577:
3573:
3569:
3566:
3562:
3558:
3555:
3553:
3551:
3548:
3545:
3538:
3534:
3529:
3525:
3524:
3520:
3514:
3511:
3508:
3505:
3501:
3497:
3494:
3490:
3486:
3482:
3476:
3472:
3468:
3465:
3461:
3456:
3450:
3446:
3442:
3439:
3435:
3430:
3424:
3421:
3418:
3415:
3411:
3407:
3404:
3400:
3396:
3393:
3391:
3389:
3386:
3383:
3378:
3375:
3372:
3369:
3366:
3363:
3359:
3355:
3354:
3350:
3344:
3341:
3338:
3335:
3331:
3327:
3324:
3320:
3316:
3312:
3306:
3302:
3298:
3295:
3291:
3286:
3280:
3276:
3272:
3269:
3265:
3261:
3258:
3256:
3254:
3251:
3248:
3243:
3240:
3237:
3234:
3231:
3227:
3223:
3222:
3218:
3212:
3209:
3206:
3203:
3199:
3195:
3192:
3188:
3184:
3180:
3174:
3170:
3166:
3163:
3159:
3154:
3148:
3144:
3140:
3137:
3133:
3129:
3126:
3124:
3122:
3119:
3116:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3086:
3082:
3081:
3077:
3071:
3068:
3065:
3062:
3058:
3054:
3051:
3047:
3043:
3039:
3033:
3029:
3025:
3022:
3018:
3013:
3007:
3003:
2999:
2996:
2992:
2988:
2985:
2983:
2981:
2978:
2975:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2948:
2944:
2943:
2914:
2913:
2900:
2894:
2890:
2886:
2883:
2879:
2874:
2869:
2863:
2859:
2855:
2852:
2849:
2846:
2843:
2840:
2836:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2762:
2761:
2750:
2742:
2738:
2734:
2731:
2727:
2711:
2710:
2698:
2695:
2690:
2686:
2682:
2677:
2673:
2669:
2666:
2644:
2643:
2642:
2641:
2640:
2639:
2616:
2612:
2608:
2605:
2602:
2599:
2575:
2574:
2573:
2572:
2571:
2556:
2552:
2548:
2543:
2539:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2512:
2508:
2504:
2501:
2498:
2495:
2475:
2474:
2473:
2472:
2471:
2458:
2455:
2452:
2434:
2431:
2396:characteristic
2387:
2384:
2368:
2367:
2356:
2353:
2350:
2347:
2342:
2339:
2336:
2332:
2328:
2325:
2322:
2319:
2314:
2310:
2286:
2266:
2263:
2260:
2240:
2220:
2217:
2209:
2208:
2195:
2192:
2189:
2184:
2180:
2176:
2173:
2170:
2165:
2161:
2157:
2154:
2151:
2148:
2145:
2140:
2136:
2132:
2129:
2126:
2123:
2118:
2114:
2110:
2107:
2104:
2101:
2096:
2092:
2046:
2042:
2030:
2029:
2018:
2013:
2009:
2003:
1999:
1995:
1990:
1987:
1984:
1980:
1956:
1953:
1928:
1925:
1922:
1919:
1908:
1907:
1895:
1892:
1889:
1886:
1883:
1880:
1875:
1871:
1865:
1861:
1857:
1854:
1851:
1846:
1841:
1838:
1835:
1831:
1827:
1824:
1821:
1818:
1815:
1796:
1793:
1791:
1788:
1769:
1750:
1739:
1733:
1732:
1719:
1709:
1707:
1704:
1701:
1698:
1697:
1694:
1691:
1688:
1685:
1681:
1676:
1671:
1670:
1667:
1664:
1661:
1658:
1655:
1651:
1650:
1648:
1643:
1640:
1637:
1634:
1629:
1625:
1602:
1599:
1585:
1582:
1579:
1554:
1550:
1527:
1523:
1500:
1496:
1480:
1473:
1466:
1448:
1445:
1405:
1387:
1374:
1373:
1360:
1350:
1348:
1345:
1342:
1339:
1338:
1335:
1332:
1329:
1326:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1275:
1270:
1269:
1266:
1263:
1260:
1257:
1252:
1248:
1244:
1238:
1233:
1232:
1230:
1225:
1222:
1219:
1216:
1211:
1207:
1192:graph homology
1168:
1165:
1163:
1160:
1145:
1141:
1118:
1114:
1088:
1084:
1080:
1077:
1074:
1071:
1068:
1048:
1045:
1030:
1005:
970:
955:
937:
933:
910:
907:
904:
900:
896:
893:
889:
883:
879:
875:
872:
869:
864:
860:
840:homology group
825:
801:
782:
779:
771:
760:
750:
743:
732:
726:
725:
721:
716:
712:
707:
703:
654:
643:
640:
636:digital images
616:Henri Poincaré
592:
589:
586:
583:
580:
575:
571:
546:
543:
540:
536:
531:
510:
507:
504:
500:
495:
491:
486:
481:
476:
473:
470:
467:
462:
458:
437:
434:
431:
428:
425:
420:
416:
394:
390:
386:
382:
378:
374:
371:
368:
365:
360:
356:
335:
332:
329:
326:
323:
318:
314:
292:
288:
284:
280:
277:
274:
271:
266:
262:
241:
238:
235:
232:
229:
224:
220:
198:
194:
191:
188:
185:
180:
176:
155:
152:
149:
146:
143:
138:
134:
113:
110:
107:
104:
101:
96:
92:
77:
71:homology group
15:
13:
10:
9:
6:
4:
3:
2:
5352:
5341:
5338:
5336:
5333:
5331:
5328:
5326:
5323:
5322:
5320:
5305:
5297:
5293:
5290:
5288:
5285:
5283:
5280:
5279:
5278:
5270:
5268:
5264:
5260:
5258:
5254:
5250:
5248:
5243:
5238:
5236:
5228:
5227:
5224:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5189:
5187:
5185:
5181:
5175:
5174:Orientability
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5151:
5149:
5145:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5119:
5116:
5114:
5111:
5109:
5106:
5102:
5099:
5097:
5094:
5093:
5092:
5089:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5067:
5065:
5062:
5061:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5041:
5038:
5037:
5035:
5031:
5026:
5016:
5013:
5011:
5010:Set-theoretic
5008:
5004:
5001:
5000:
4999:
4996:
4992:
4989:
4988:
4987:
4984:
4982:
4979:
4977:
4974:
4972:
4971:Combinatorial
4969:
4967:
4964:
4962:
4959:
4958:
4956:
4952:
4948:
4941:
4936:
4934:
4929:
4927:
4922:
4921:
4918:
4910:
4908:0-582-32502-1
4904:
4900:
4895:
4891:
4889:0-387-90894-3
4885:
4881:
4876:
4875:
4867:
4860:
4855:
4851:
4847:
4846:
4841:
4835:
4832:
4827:
4826:
4821:
4815:
4811:
4805:
4802:
4797:
4791:
4787:
4786:
4778:
4775:
4770:
4764:
4760:
4759:
4751:
4748:
4743:
4742:
4737:
4731:
4727:
4721:
4718:
4713:
4706:
4703:
4696:
4692:
4689:
4687:
4684:
4682:
4679:
4678:
4674:
4672:
4670:
4666:
4662:
4661:Edward Witten
4644:
4641:
4638:
4633:
4630:
4627:
4623:
4619:
4614:
4610:
4606:
4603:
4600:
4594:
4586:
4583:
4580:
4576:
4572:
4566:
4558:
4554:
4546:
4545:
4544:
4542:
4538:
4520:
4516:
4508:
4504:
4499:
4497:
4493:
4489:
4484:
4482:
4478:
4474:
4470:
4466:
4463:
4462:
4458:
4454:
4438:
4426:
4405:
4399:
4395:
4391:
4388:
4384:
4379:
4373:
4369:
4365:
4362:
4358:
4353:
4347:
4343:
4339:
4336:
4332:
4327:
4321:
4317:
4313:
4310:
4306:
4301:
4295:
4291:
4287:
4284:
4280:
4275:
4269:
4265:
4261:
4258:
4254:
4249:
4243:
4239:
4235:
4232:
4228:
4223:
4217:
4213:
4209:
4206:
4202:
4198:
4196:
4188:
4178:
4174:
4169:
4160:
4154:
4150:
4146:
4143:
4139:
4134:
4128:
4124:
4120:
4117:
4113:
4108:
4102:
4098:
4094:
4091:
4087:
4082:
4076:
4072:
4068:
4065:
4061:
4056:
4050:
4046:
4042:
4039:
4035:
4030:
4024:
4020:
4016:
4013:
4009:
4004:
3998:
3994:
3990:
3987:
3983:
3979:
3977:
3969:
3959:
3955:
3950:
3941:
3935:
3931:
3927:
3924:
3920:
3915:
3909:
3905:
3901:
3898:
3894:
3889:
3883:
3879:
3875:
3872:
3868:
3863:
3857:
3853:
3849:
3846:
3842:
3837:
3831:
3827:
3823:
3820:
3816:
3811:
3805:
3801:
3797:
3794:
3790:
3786:
3784:
3776:
3766:
3762:
3757:
3748:
3742:
3738:
3734:
3731:
3727:
3722:
3716:
3712:
3708:
3705:
3701:
3696:
3690:
3686:
3682:
3679:
3675:
3670:
3664:
3660:
3656:
3653:
3649:
3645:
3643:
3635:
3625:
3621:
3616:
3607:
3601:
3597:
3593:
3590:
3586:
3581:
3575:
3571:
3567:
3564:
3560:
3556:
3554:
3546:
3536:
3532:
3527:
3518:
3512:
3509:
3506:
3503:
3499:
3495:
3492:
3488:
3484:
3480:
3474:
3470:
3466:
3463:
3459:
3454:
3448:
3444:
3440:
3437:
3433:
3428:
3422:
3419:
3416:
3413:
3409:
3405:
3402:
3398:
3394:
3392:
3384:
3373:
3370:
3364:
3361:
3357:
3348:
3342:
3339:
3336:
3333:
3329:
3325:
3322:
3318:
3314:
3310:
3304:
3300:
3296:
3293:
3289:
3284:
3278:
3274:
3270:
3267:
3263:
3259:
3257:
3249:
3238:
3232:
3229:
3225:
3216:
3210:
3207:
3204:
3201:
3197:
3193:
3190:
3186:
3182:
3178:
3172:
3168:
3164:
3161:
3157:
3152:
3146:
3142:
3138:
3135:
3131:
3127:
3125:
3117:
3106:
3103:
3100:
3097:
3091:
3088:
3084:
3075:
3069:
3066:
3063:
3060:
3056:
3052:
3049:
3045:
3041:
3037:
3031:
3027:
3023:
3020:
3016:
3011:
3005:
3001:
2997:
2994:
2990:
2986:
2984:
2976:
2965:
2962:
2959:
2953:
2950:
2946:
2934:
2933:
2932:
2930:
2925:
2923:
2919:
2898:
2892:
2888:
2884:
2881:
2877:
2872:
2867:
2861:
2857:
2853:
2850:
2847:
2844:
2841:
2838:
2834:
2826:
2825:
2824:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2748:
2740:
2736:
2732:
2729:
2725:
2716:
2715:
2714:
2696:
2693:
2688:
2684:
2680:
2675:
2671:
2667:
2664:
2657:
2656:
2655:
2653:
2652:period length
2649:
2637:
2633:
2614:
2606:
2603:
2600:
2590:
2589:
2587:
2586:
2584:
2580:
2576:
2554:
2550:
2546:
2541:
2537:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2510:
2502:
2499:
2496:
2486:
2485:
2483:
2482:
2480:
2476:
2456:
2453:
2450:
2443:
2442:
2440:
2439:
2437:
2436:
2433:More examples
2432:
2430:
2428:
2424:
2420:
2416:
2415:
2409:
2405:
2401:
2397:
2393:
2385:
2383:
2381:
2377:
2373:
2354:
2348:
2340:
2337:
2334:
2330:
2326:
2320:
2312:
2308:
2300:
2299:
2298:
2284:
2264:
2261:
2258:
2238:
2230:
2226:
2218:
2216:
2214:
2193:
2190:
2187:
2182:
2178:
2171:
2163:
2159:
2155:
2152:
2146:
2138:
2134:
2130:
2124:
2116:
2112:
2108:
2102:
2094:
2090:
2082:
2081:
2080:
2078:
2074:
2070:
2066:
2062:
2044:
2040:
2016:
2011:
2007:
2001:
1997:
1993:
1988:
1985:
1982:
1978:
1970:
1969:
1968:
1966:
1962:
1954:
1952:
1950:
1946:
1942:
1923:
1917:
1893:
1887:
1884:
1881:
1873:
1869:
1863:
1855:
1852:
1839:
1836:
1833:
1829:
1825:
1819:
1813:
1806:
1805:
1804:
1802:
1794:
1789:
1787:
1785:
1781:
1777:
1772:
1768:
1764:
1760:
1756:
1749:
1745:
1738:
1702:
1692:
1689:
1686:
1679:
1665:
1662:
1659:
1646:
1641:
1635:
1627:
1623:
1615:
1614:
1613:
1611:
1608:
1600:
1598:
1583:
1580:
1577:
1568:
1552:
1548:
1525:
1521:
1498:
1494:
1484:
1479:
1472:
1465:
1461:
1453:
1446:
1444:
1441:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1404:
1399:
1397:
1393:
1386:
1381:
1379:
1343:
1333:
1330:
1327:
1315:
1307:
1299:
1291:
1283:
1264:
1261:
1258:
1246:
1228:
1223:
1217:
1209:
1205:
1197:
1196:
1195:
1193:
1189:
1185:
1181:
1177:
1174:
1166:
1161:
1159:
1143:
1139:
1116:
1112:
1102:
1086:
1082:
1078:
1075:
1072:
1069:
1066:
1058:
1054:
1046:
1044:
1042:
1038:
1033:
1029:
1025:
1021:
1017:
1013:
1008:
1004:
1000:
997:
992:
990:
986:
982:
978:
973:
969:
965:
961:
953:
935:
931:
908:
905:
902:
898:
894:
891:
887:
881:
877:
873:
870:
867:
862:
858:
849:
845:
841:
837:
833:
828:
824:
821:
820:abelian group
817:
813:
809:
804:
800:
796:
792:
788:
780:
778:
775:
770:
766:
759:
754:
749:
742:
738:
731:
720:
717:
711:
708:
702:
699:
698:
697:
695:
690:
688:
684:
680:
677:-dimensional
676:
672:
669:-dimensional
668:
664:
653:
648:
641:
639:
637:
633:
629:
625:
621:
617:
612:
610:
606:
590:
587:
581:
573:
569:
560:
541:
534:
505:
498:
489:
484:
474:
468:
460:
456:
435:
432:
426:
418:
414:
388:
380:
372:
366:
358:
354:
333:
330:
324:
316:
312:
286:
278:
272:
264:
260:
239:
236:
230:
222:
218:
192:
186:
178:
174:
153:
150:
144:
136:
132:
111:
108:
102:
94:
90:
80:
76:
72:
69:
65:
61:
56:
54:
50:
46:
42:
38:
35:-dimensional
34:
30:
26:
25:Betti numbers
22:
5304:Publications
5169:Chern number
5159:Betti number
5158:
5042: /
5033:Key concepts
4981:Differential
4898:
4879:
4849:
4843:
4834:
4823:
4810:Ghostarchive
4808:Archived at
4804:
4784:
4777:
4757:
4750:
4739:
4726:Ghostarchive
4724:Archived at
4720:
4705:
4659:
4503:Morse theory
4500:
4492:Hodge theory
4485:
4459:
4430:
2926:
2915:
2763:
2712:
2645:
2578:
2427:Tor functors
2418:
2413:
2407:
2403:
2400:torsion-free
2391:
2389:
2375:
2371:
2369:
2228:
2224:
2222:
2210:
2076:
2064:
2060:
2059:denotes the
2031:
1964:
1960:
1958:
1948:
1944:
1909:
1800:
1798:
1783:
1775:
1770:
1766:
1762:
1758:
1754:
1747:
1744:cyclic group
1736:
1734:
1609:
1604:
1569:
1485:
1477:
1470:
1463:
1457:
1442:
1421:
1417:
1413:
1409:
1402:
1400:
1395:
1391:
1384:
1382:
1375:
1187:
1183:
1179:
1175:
1170:
1103:
1052:
1050:
1040:
1036:
1031:
1027:
1023:
1019:
1015:
1011:
1006:
1002:
998:
993:
984:
980:
976:
971:
967:
959:
847:
843:
835:
831:
826:
822:
811:
807:
802:
798:
794:
790:
784:
776:
768:
764:
757:
755:
747:
740:
729:
727:
718:
709:
700:
691:
686:
682:
678:
674:
670:
666:
662:
660:
651:
624:Emmy Noether
620:Enrico Betti
613:
608:
78:
74:
67:
59:
57:
53:CW complexes
32:
24:
18:
5267:Wikiversity
5184:Key results
4539:of a given
1458:Consider a
1171:Consider a
5319:Categories
5113:CW complex
5054:Continuity
5044:Closed set
5003:cohomology
4697:References
2929:Lie groups
2425:(based on
2277:, for any
1790:Properties
1412:) equals |
1394:) equals |
737:meridional
73:, denoted
5292:geometric
5287:algebraic
5138:Cobordism
5074:Hausdorff
5069:connected
4986:Geometric
4976:Continuum
4966:Algebraic
4642:⋯
4631:−
4620:−
4607:≤
4604:⋯
4584:−
4573:−
3510:−
3485:⋯
3420:−
3340:−
3315:⋯
3208:−
3183:⋯
3042:⋯
2885:−
2808:…
2733:−
2697:⋯
2338:−
2262:−
2191:⋯
1986:×
1918:χ
1853:−
1845:∞
1830:∑
1814:χ
1712:otherwise
1353:otherwise
1308:−
932:δ
899:δ
895:
878:δ
874:
490:⊕
475:≅
389:⊕
381:⊕
373:≅
287:⊕
279:≅
193:≅
109:≅
47:, finite
43:(such as
5257:Wikibook
5235:Category
5123:Manifold
5091:Homotopy
5049:Interior
5040:Open set
4998:Homology
4947:Topology
4812:and the
4728:and the
4675:See also
2630:(by the
2376:oriented
2219:Symmetry
1967:we have
1939:denotes
1803:we have
1162:Examples
842:of
521:, where
5282:general
5084:uniform
5064:compact
5015:Digital
4825:YouTube
4741:YouTube
4667:in the
4494:on the
1780:torsion
1742:is the
1039:,
1018:), the
1014:,
979:;
958:is the
923:, the
834:), the
787:integer
681:" is a
557:is the
66:of the
5277:Topics
5079:metric
4954:Fields
4905:
4886:
4792:
4765:
4461:modulo
2417:, for
2372:closed
2032:where
1910:where
1735:Here,
1567:is 0.
987:. The
846:. The
793:, the
789:
618:after
41:spaces
23:, the
5059:Space
4541:index
4535:of a
4451:is a
2931:are:
2583:torus
2479:torus
2211:see
1784:holes
1612:are:
1420:| - |
1416:| + |
996:field
774:= 2.
753:= 1.
739:) so
671:holes
406:then
346:, if
304:then
252:, if
210:then
166:, if
124:then
4903:ISBN
4884:ISBN
4790:ISBN
4763:ISBN
4471:and
2374:and
2251:and
1963:and
1051:The
816:rank
679:hole
634:and
64:rank
58:The
4854:doi
4479:of
2227:is
2223:If
2063:of
1943:of
1761:of
1483:).
1131:is
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966:of
871:ker
838:th
51:or
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5321::
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4822:.
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2805:,
2802:c
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2103:z
2100:(
2095:X
2091:P
2077:X
2065:X
2045:X
2041:P
2017:,
2012:Y
2008:P
2002:X
1998:P
1994:=
1989:Y
1983:X
1979:P
1965:Y
1961:X
1949:F
1945:K
1927:)
1924:K
1921:(
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1703:0
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1390:(
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1347:}
1344:0
1341:{
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1328:k
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830:(
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806:(
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