1125:
1420:
986:
802:
1978:
A central result of the theory is that the rational homotopy category can be described in a purely algebraic way; in fact, in two different algebraic ways. First, Quillen showed that the rational homotopy category is equivalent to the homotopy category of connected
3458:
The construction of
Sullivan minimal models for simply connected spaces extends to nilpotent spaces. For more general fundamental groups, things get more complicated; for example, the rational homotopy groups of a finite CW complex (such as the wedge
3114:, compatible with face and degeneracy maps. This algebra is usually very large (uncountable dimension) but can be replaced by a much smaller algebra. More precisely, any differential graded algebra with the same Sullivan minimal model as
229:
of simply connected spaces with respect to rational homotopy equivalences. The goal of rational homotopy theory is to understand this category (i.e. to determine the information that can be recovered from rational homotopy equivalences).
4986:
1909:
997:
667:
1306:
3051:: it is also necessary that the isomorphism of cohomology be induced by a homomorphism of differential graded algebras. There are examples of non-isomorphic minimal Sullivan models with isomorphic cohomology algebras.)
869:
3620:
need not be formal: the simplest example is the
Kodaira–Thurston manifold (the product of the Heisenberg manifold with a circle). There are also examples of non-formal, simply connected symplectic closed manifolds.
2144:
1141:
are two very different types of basic spaces from which all spaces can be built. In rational homotopy theory, these two types of spaces become much closer. In particular, Serre's calculation implies that
678:
1771:
1973:
3632:
is formal, then all (higher order) Massey products must vanish. The converse is not true: formality means, roughly speaking, the "uniform" vanishing of all Massey products. The complement of the
1628:
1717:
2376:
For spaces whose rational homology in each degree has finite dimension, Sullivan classified all rational homotopy types in terms of simpler algebraic objects, Sullivan algebras. By definition, a
2545:
4607:
cannot be a model for its cohomology algebra. The corresponding topological spaces are two spaces with isomorphic rational cohomology rings but different rational homotopy types. Notice that
4828:
1524:
2033:
2484:
4277:
4039:
1183:
4754:
4364:
2862:
1480:
587:
552:
471:
343:
1229:
4693:
3544:
3037:
3002:
2682:
2325:
2035:
is the homotopy Lie algebra.) Second, Quillen showed that the rational homotopy category is equivalent to the homotopy category of 1-connected differential graded cocommutative
4085:
2898:
1278:
372:
304:
5119:
3497:
4704:
Rational homotopy theory revealed an unexpected dichotomy among finite CW complexes: either the rational homotopy groups are zero in sufficiently high degrees, or they grow
3954:
2743:
163:
4515:. Then this algebra is a minimal Sullivan algebra that is not formal. The cohomology algebra has nontrivial components only in dimension 2, 3, 6, generated respectively by
2967:
2927:
2429:
4850:
4776:
4316:
2400:
2347:
2212:
2074:
1650:
1546:
1130:
This suggests the possibility of describing the whole rational homotopy category in a practically computable way. Rational homotopy theory has realized much of that goal.
207:
5179:
3990:
3197:
4219:
2364:. More generally, there are versions of these constructions for differential graded algebras. This duality between commutative algebras and Lie algebras is a version of
3682:
3236:
3151:
3100:
5281:
5246:
5077:
4481:
3880:
1577:
5015:
3784:
3369:
3269:
2638:
2245:
4884:
4111:
2352:
One may ask how to pass between the two algebraic descriptions of the rational homotopy category. In short, a Lie algebra determines a graded-commutative algebra by
2287:
831:
497:
4896:
4637:
4601:
4553:
4513:
3396:
3296:
2808:
2777:
2174:
1820:
4445:
4416:
4177:
4148:
3909:
3844:
3744:
3715:
2600:
1120:{\displaystyle \pi _{i}(S^{2a})\otimes \mathbb {Q} \cong {\begin{cases}\mathbb {Q} &{\text{if }}i=2a{\text{ or }}i=4a-1\\0&{\text{otherwise.}}\end{cases}}}
2571:
1801:
3815:
517:
427:
396:
275:
5914:
1415:{\displaystyle \operatorname {SU} (n)_{\mathbb {Q} }\simeq S_{\mathbb {Q} }^{3}\times S_{\mathbb {Q} }^{5}\times \cdots \times S_{\mathbb {Q} }^{2n-1}.}
595:
3306:; it is unique up to isomorphism. This gives an equivalence between rational homotopy types of such spaces and such algebras, with the properties:
981:{\displaystyle \pi _{i}(S^{2a-1})\otimes \mathbb {Q} \cong {\begin{cases}\mathbb {Q} &{\text{if }}i=2a-1\\0&{\text{otherwise}}\end{cases}}}
5819:
5649:
5619:
5218:
of a fiber sequence of simply-connected spaces with rationally elliptic fiber of non-zero Euler characteristic vanishes at the second page.
2086:
3043:
has a minimal
Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology algebra as
399:
5206:
should be rationally elliptic. Very little is known about the conjecture, although it holds for all known examples of such manifolds.
797:{\displaystyle H_{i}(X_{\mathbb {Q} },{\mathbb {Z} })\cong H_{i}(X,{\mathbb {Z} })\otimes {\mathbb {Q} }\cong H_{i}(X,{\mathbb {Q} })}
5781:
5723:
5682:
5581:
5532:
3616:
of real 3×3 upper triangular matrices with 1's on the diagonal by its subgroup of matrices with integral coefficients. Closed
5970:
2361:
1722:
81:
can be identified with (isomorphism classes of) certain algebraic objects called
Sullivan minimal models, which are commutative
1980:
1924:
1915:
1586:
3110:
with rational coefficients. An element of this algebra consists of (roughly) a polynomial form on each singular simplex of
1655:
5961:
5755:
2500:
1914:
Conversely, one can reconstruct the rational homotopy Lie algebra from the homology of the loop space as the subspace of
3103:
1811:
1138:
5956:
5811:
5670:
82:
4789:
1485:
409:
One obtains equivalent definitions using homology rather than homotopy groups. Namely, a simply connected CW complex
1986:
5186:
1240:
852:
222:
5302:
2437:
234:
184:
92:
A geometric application was the theorem of
Sullivan and Micheline Vigué-Poirrier (1976): every simply connected
6009:
4053:
5951:
3331:
Two spaces have the same rational homotopy type if and only if their minimal
Sullivan algebras are isomorphic.
2047:
is the rational cohomology ring.) These equivalences were among the first applications of
Quillen's theory of
4224:
3995:
3447:. Sullivan's "arithmetic square" reduces many problems in homotopy theory to the combination of rational and
1145:
5641:
5215:
5296:
4715:
4321:
3566:
model). Thus the rational homotopy type of a formal space is completely determined by its cohomology ring.
2824:
1441:
557:
522:
432:
313:
5968:
Sullivan, Dennis; Vigué-Poirrier, Micheline (1976), "The homology theory of the closed geodesic problem",
3432:
2357:
2353:
1188:
226:
166:
102:
whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct
78:
5869:
4642:
3554:
has a model with vanishing differential. This is equivalent to requiring that the cohomology algebra of
3514:
3007:
2972:
2643:
2292:
5287:. All known Riemannian manifolds with nonnegative sectional curvature are in fact integrally elliptic.
4058:
2867:
1254:
348:
280:
5129:
5085:
5080:
3462:
307:
3914:
2694:
1046:
924:
136:
5210:
5203:
3617:
3324:
The
Whitehead product on rational homotopy is the dual of the "quadratic part" of the differential
2943:
2903:
2405:
2248:
96:
4833:
4759:
4282:
2383:
2330:
2195:
2057:
1719:, this just amounts to a shift of the grading by 1.) For example, Serre's theorem above says that
1633:
1529:
190:
5886:
5759:
5139:
4705:
3959:
3415:
3179:
1549:
1236:
4182:
3578:
3652:
3202:
3117:
3066:
2184:. (By contrast, there are many restrictions, not completely understood, on the integral or mod
5836:
Pavlov, Aleksandr V. (2002), "Estimates for the Betti numbers of rationally elliptic spaces",
5815:
5777:
5719:
5678:
5666:
5645:
5615:
5577:
5263:
5228:
5046:
5021:
4981:{\displaystyle \sum _{i=1}^{n}\dim _{\mathbb {Q} }\pi _{i}(X)\otimes {\mathbb {Q} }\geq C^{n}}
4450:
3849:
3407:
2044:
1904:{\displaystyle H_{*}(\Omega X,{\mathbb {Q} })\cong U(\pi _{*}(\Omega X)\otimes \mathbb {Q} ).}
1559:
1553:
1285:
860:
841:
242:
214:
169:
51:
47:
4994:
3760:
3341:
3241:
2605:
2217:
345:
that induces an isomorphism on homotopy groups tensored with the rational numbers. The space
5979:
5931:
5923:
5878:
5845:
5769:
5711:
5607:
5603:
5569:
5541:
4863:
4090:
3613:
2262:
2048:
1281:
1248:
810:
476:
238:
5993:
5943:
5898:
5857:
5829:
5802:
5791:
5733:
5692:
5659:
5629:
5591:
5553:
4610:
4574:
4526:
4486:
3374:
3274:
2786:
2755:
2152:
1806:
Another way to think of the homotopy Lie algebra is that the homology of the loop space of
5989:
5939:
5905:
5894:
5853:
5825:
5787:
5729:
5688:
5655:
5625:
5587:
5565:
5549:
4421:
4392:
4153:
4124:
3885:
3820:
3720:
3691:
3633:
3574:
2576:
2252:
1244:
845:
837:
111:
103:
93:
86:
59:
43:
3176:
with all rational homology groups of finite dimension, there is a minimal
Sullivan model
2550:
1780:
5193:
There are many other restrictions on the rational cohomology ring of an elliptic space.
3797:
5909:
5864:
5715:
3625:
2365:
502:
412:
381:
260:
181:
131:
119:
67:
55:
5744:
5028:
are elliptic. On the other hand, "most" finite complexes are hyperbolic. For example:
6003:
3586:
3310:
The rational cohomology of the space is the cohomology of its
Sullivan minimal model.
5136:
is nonnegative. If the Euler characteristic is positive, then all odd Betti numbers
5798:
5740:
5699:
1919:
403:
254:
107:
5530:
FĂ©lix, Yves; Halperin, Stephen; Thomas, Jean-Claude (1993), "Elliptic spaces II",
1284:
of Eilenberg–MacLane spaces. The hypothesis on the cohomology ring applies to any
74:). This simplification of homotopy theory makes certain calculations much easier.
5202:
predicts that every simply connected closed Riemannian manifold with nonnegative
662:{\displaystyle \pi _{i}(X_{\mathbb {Q} })\cong \pi _{i}(X)\otimes {\mathbb {Q} }}
17:
3593:
1774:
177:
31:
5804:
More Concise Algebraic Topology. Localization, Completion, and Model Categories
5773:
213:
is a rational homotopy equivalence if and only if it induces an isomorphism on
5935:
5849:
5703:
5573:
5225:
is rationally elliptic if and only if the rational homology of the loop space
5197:
5039:
is an elliptic space whose top nonzero rational cohomology group is in degree
2054:
In particular, the second description implies that for any graded-commutative
1435:
1289:
250:
5984:
5032:
The rational cohomology ring of an elliptic space satisfies Poincaré duality.
5025:
3608:
of some dimension. The simplest example of a non-formal nilmanifold is the
3582:
3558:(viewed as a differential algebra with trivial differential) is a model for
2036:
245:
of topological spaces, the subcategory of rational spaces. By definition, a
4778:-vector space (for example, a finite CW complex has this property). Define
3165:
is simply connected, such a model determines the rational homotopy type of
2259:; in that case, one also needs to assume that the intersection pairing on
1134:
35:
5545:
3636:
is a non-formal space: it supports a nontrivial triple Massey product.
5927:
5890:
3570:
836:
These results for simply connected spaces extend with little change to
5764:
2139:{\displaystyle A=\mathbb {Q} \oplus A^{2}\oplus A^{3}\oplus \cdots ,}
5882:
2489:
satisfying the following "nilpotence condition" on its differential
5373:
FĂ©lix, Halperin & Thomas (2001), Corollary to Proposition 16.7.
2192:.) In the same spirit, Sullivan showed that any graded-commutative
3605:
2402:, whose underlying algebra is the free commutative graded algebra
1293:
5611:
2380:
is a commutative differential graded algebra over the rationals
589:
that induces an isomorphism on rational homology. Thus, one has
2360:
commutative algebra determines a graded Lie algebra by reduced
1526:. The rational cohomology is a graded-commutative algebra over
257:
over the rational numbers. For any simply connected CW complex
118:
are unbounded. The theorem then follows from a 1969 result of
3063:, Sullivan defined a commutative differential graded algebra
2688:, "commutative" is used to mean graded-commutative; that is,
5598:
FĂ©lix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2015),
5560:
FĂ©lix, Yves; Halperin, Stephen; Thomas, Jean-Claude (2001),
3317:
are the duals of the rational homotopy groups of the space
1113:
974:
855:
is a central open problem in homotopy theory. However, the
398:. This is a special case of Sullivan's construction of the
233:
One basic result is that the rational homotopy category is
106:. The proof used rational homotopy theory to show that the
3406:
is a smooth manifold, the differential algebra of smooth
2497:
is the union of an increasing series of graded subspaces,
2188:
cohomology rings of topological spaces, for prime numbers
1766:{\displaystyle \pi _{*}(\Omega S^{n})\otimes \mathbb {Q} }
5299:– analogue of rational homotopy theory in p-adic settings
3422:; more precisely it is the tensor product of a model for
2039:. (The associated coalgebra is the rational homology of
5490:
FĂ©lix, Halperin & Thomas (2001), Proposition 32.10.
5418:
FĂ©lix, Halperin & Thomas (2001), Proposition 12.10.
5391:
FĂ©lix, Halperin & Thomas (2001), Theorem 21.5(iii).
2176:
of finite dimension, there is a simply connected space
1968:{\displaystyle H_{*}(\Omega S^{n})\otimes \mathbb {Q} }
1251:
on generators of odd degree). Then the rationalization
499:. The rationalization of a simply connected CW complex
429:
is a rational space if and only if its homology groups
5472:
FĂ©lix, Halperin & Thomas (2001), Proposition 38.3.
4860:
is rationally hyperbolic, then there is a real number
2900:
is the direct sum of the positive-degree subspaces of
1623:{\displaystyle \pi _{*}(\Omega X)\otimes \mathbb {Q} }
1235:
be any space whose rational cohomology ring is a free
5382:
FĂ©lix, Halperin & Thomas (2001), Theorem 21.5(i).
5321:
5319:
5317:
5266:
5231:
5142:
5088:
5049:
5020:
For example, spheres, complex projective spaces, and
4997:
4899:
4866:
4836:
4792:
4762:
4718:
4645:
4613:
4577:
4529:
4489:
4453:
4424:
4395:
4324:
4285:
4227:
4185:
4156:
4127:
4093:
4061:
3998:
3962:
3917:
3888:
3852:
3823:
3800:
3763:
3723:
3694:
3655:
3517:
3465:
3377:
3344:
3338:
corresponding to each possible Sullivan algebra with
3277:
3244:
3205:
3182:
3120:
3069:
3010:
2975:
2946:
2906:
2870:
2827:
2789:
2758:
2697:
2646:
2608:
2579:
2553:
2503:
2440:
2408:
2386:
2333:
2295:
2265:
2220:
2198:
2155:
2089:
2060:
1989:
1927:
1823:
1783:
1725:
1712:{\displaystyle \pi _{i}(X)\cong \pi _{i-1}(\Omega X)}
1658:
1636:
1589:
1562:
1532:
1488:
1444:
1309:
1257:
1191:
1148:
1000:
872:
848:
and acts nilpotently on the higher homotopy groups).
813:
681:
598:
560:
525:
505:
479:
435:
415:
384:
351:
316:
283:
263:
193:
139:
5702:(1999), "A history of rational homotopy theory", in
5283:grows at most polynomially, for every prime number
3298:have finite dimension. This is called the Sullivan
2540:{\displaystyle V(0)\subseteq V(1)\subseteq \cdots }
5463:FĂ©lix, Halperin & Thomas (2001), Theorem 33.2.
5275:
5240:
5173:
5113:
5071:
5009:
4980:
4878:
4844:
4822:
4770:
4748:
4687:
4631:
4595:
4547:
4507:
4475:
4439:
4410:
4358:
4310:
4271:
4213:
4171:
4142:
4105:
4079:
4033:
3984:
3948:
3903:
3874:
3838:
3809:
3778:
3738:
3709:
3676:
3538:
3491:
3390:
3363:
3290:
3263:
3230:
3191:
3145:
3094:
3031:
2996:
2961:
2921:
2892:
2856:
2802:
2771:
2737:
2676:
2632:
2594:
2565:
2539:
2478:
2423:
2394:
2341:
2319:
2281:
2239:
2206:
2168:
2138:
2068:
2027:
1967:
1903:
1795:
1765:
1711:
1644:
1622:
1571:
1540:
1518:
1474:
1414:
1272:
1223:
1177:
1119:
980:
825:
796:
661:
581:
546:
511:
491:
465:
421:
390:
366:
337:
298:
269:
221:(of simply connected spaces) is defined to be the
201:
157:
5508:Félix, Oprea & Tanré (2008), Conjecture 6.43.
5364:FĂ©lix, Halperin & Thomas (2001), Theorem 9.3.
5355:FĂ©lix, Halperin & Thomas (2001), Theorem 9.7.
5346:FĂ©lix, Halperin & Thomas (2001), Theorem 8.6.
5105:
5092:
3786:, its minimal Sullivan model has two generators
3055:The Sullivan minimal model of a topological space
2684:. In the context of differential graded algebras
5752:Interactions between Homotopy Theory and Algebra
5745:"Rational homotopy theory: a brief introduction"
5636:Félix, Yves; Oprea, John; Tanré, Daniel (2008),
5499:FĂ©lix, Halperin & Thomas (2001), section 32.
4113:, its minimal Sullivan model has two generators
2251:is the cohomology ring of some simply connected
2180:whose rational cohomology ring is isomorphic to
1434:in the rational homotopy category: the rational
5675:Rational Homotopy Theory and Differential Forms
5517:FĂ©lix, Halperin & Thomas (1993), section 3.
4823:{\displaystyle \pi _{*}(X)\otimes \mathbb {Q} }
3684:, its minimal Sullivan model has one generator
3004:which induces an isomorphism on cohomology. If
1519:{\displaystyle \pi _{*}(X)\otimes \mathbb {Q} }
5710:, Amsterdam: North-Holland, pp. 757–796,
5454:Félix, Oprea & Tanré (2008), Theorem 8.29.
5436:Félix, Oprea & Tanré (2008), Theorem 4.43.
5337:Félix, Oprea & Tanré (2008), Theorem 5.13.
5181:are zero, and the rational cohomology ring of
2936:for a commutative differential graded algebra
2028:{\displaystyle \ker(d)/\operatorname {im} (d)}
1777:graded Lie algebra on one generator of degree
5445:Félix, Oprea & Tanré (2008), Remark 3.21.
3499:) can be infinite-dimensional vector spaces.
8:
5248:grows at most polynomially. More generally,
4682:
4646:
4389:of degrees 2, 3, 3 and 4 with differentials
3628:. Indeed, if a differential graded algebra
3581:. Formality is preserved under products and
3426:with the reals and therefore determines the
859:homotopy groups of spheres were computed by
5754:, Contemporary Mathematics, vol. 436,
3585:. For manifolds, formality is preserved by
3569:Examples of formal spaces include spheres,
3451:-completed homotopy theory, for all primes
2479:{\displaystyle V=\bigoplus _{n>0}V^{n},}
4041:, where the arrow indicates the action of
3507:A commutative differential graded algebra
1430:There are two basic invariants of a space
5983:
5763:
5265:
5230:
5147:
5141:
5104:
5091:
5089:
5087:
5054:
5048:
4996:
4972:
4960:
4959:
4958:
4940:
4927:
4926:
4925:
4915:
4904:
4898:
4865:
4838:
4837:
4835:
4816:
4815:
4797:
4791:
4764:
4763:
4761:
4739:
4738:
4723:
4717:
4644:
4612:
4576:
4528:
4488:
4467:
4452:
4423:
4394:
4338:
4323:
4296:
4284:
4263:
4244:
4226:
4199:
4184:
4155:
4126:
4092:
4071:
4067:
4064:
4063:
4060:
4019:
4003:
3997:
3976:
3961:
3940:
3916:
3887:
3866:
3851:
3822:
3799:
3762:
3722:
3693:
3654:
3532:
3531:
3522:
3516:
3483:
3470:
3464:
3382:
3376:
3349:
3343:
3282:
3276:
3249:
3243:
3210:
3204:
3181:
3125:
3119:
3074:
3068:
3047:need not be a minimal Sullivan model for
3025:
3024:
3015:
3009:
2974:
2945:
2905:
2875:
2869:
2848:
2832:
2826:
2794:
2788:
2763:
2757:
2720:
2696:
2645:
2607:
2578:
2552:
2502:
2467:
2451:
2439:
2407:
2388:
2387:
2385:
2335:
2334:
2332:
2311:
2306:
2294:
2270:
2264:
2225:
2219:
2200:
2199:
2197:
2160:
2154:
2121:
2108:
2097:
2096:
2088:
2062:
2061:
2059:
2005:
1988:
1961:
1960:
1948:
1932:
1926:
1891:
1890:
1869:
1848:
1847:
1846:
1828:
1822:
1782:
1759:
1758:
1746:
1730:
1724:
1685:
1663:
1657:
1638:
1637:
1635:
1616:
1615:
1594:
1588:
1561:
1534:
1533:
1531:
1512:
1511:
1493:
1487:
1465:
1464:
1449:
1443:
1394:
1389:
1388:
1387:
1368:
1363:
1362:
1361:
1348:
1343:
1342:
1341:
1328:
1327:
1326:
1308:
1264:
1263:
1262:
1256:
1199:
1198:
1190:
1160:
1155:
1154:
1153:
1147:
1105:
1073:
1056:
1050:
1049:
1041:
1034:
1033:
1018:
1005:
999:
966:
934:
928:
927:
919:
912:
911:
890:
877:
871:
812:
786:
785:
784:
769:
757:
756:
755:
744:
743:
742:
727:
712:
711:
710:
701:
700:
699:
686:
680:
654:
653:
652:
634:
618:
617:
616:
603:
597:
573:
572:
571:
559:
538:
537:
536:
524:
504:
478:
456:
455:
440:
434:
414:
383:
358:
357:
356:
350:
329:
328:
327:
315:
290:
289:
288:
282:
262:
195:
194:
192:
138:
27:Mathematical theory of topological spaces
5910:"Infinitesimal computations in topology"
1426:Cohomology ring and homotopy Lie algebra
554:(up to homotopy equivalence) with a map
63:
5313:
4272:{\displaystyle 1,u,u^{2},\ldots ,u^{n}}
4034:{\displaystyle a^{2}b\to a^{4},\ldots }
3624:Non-formality can often be detected by
1178:{\displaystyle S_{\mathbb {Q} }^{2a-1}}
217:groups with rational coefficients. The
71:
4712:be a simply connected space such that
2255:closed manifold, except in dimension 4
1292:). For example, for the unitary group
5427:May & Ponto (2012), section 13.1.
4856:. Then FĂ©lix and Halperin showed: if
4749:{\displaystyle H_{*}(X,\mathbb {Q} )}
4359:{\displaystyle xu\to u^{n+2},\ldots }
2857:{\displaystyle \bigwedge ^{+}(V)^{2}}
1983:. (The associated graded Lie algebra
1475:{\displaystyle H^{*}(X,\mathbb {Q} )}
582:{\displaystyle X\to X_{\mathbb {Q} }}
547:{\displaystyle X\to X_{\mathbb {Q} }}
466:{\displaystyle H_{i}(X,\mathbb {Z} )}
338:{\displaystyle X\to X_{\mathbb {Q} }}
7:
5915:Publications Mathématiques de l'IHÉS
5867:(1969), "Rational homotopy theory",
5325:
4559:to its cohomology algebra would map
3430:. One can go further and define the
1247:on generators of even degree and an
1224:{\displaystyle K(\mathbb {Q} ,2a-1)}
3562:(though it does not have to be the
3172:To any simply connected CW complex
473:are rational vector spaces for all
5267:
5232:
5221:A simply connected finite complex
5096:
4688:{\displaystyle \langle ,,\rangle }
3539:{\displaystyle A^{0}=\mathbb {Q} }
3334:There is a simply connected space
3032:{\displaystyle A^{0}=\mathbb {Q} }
2997:{\displaystyle \bigwedge (V)\to A}
2677:{\displaystyle \bigwedge (V(k-1))}
2320:{\displaystyle \sum \pm x_{i}^{2}}
1941:
1878:
1837:
1739:
1700:
1603:
1563:
25:
5400:Quillen (1969), Corollary II.6.2.
4080:{\displaystyle \mathbb {CP} ^{n}}
3313:The spaces of indecomposables in
2893:{\displaystyle \bigwedge ^{+}(V)}
1548:, and the homotopy groups form a
253:all of whose homotopy groups are
5971:Journal of Differential Geometry
5716:10.1016/B978-044482375-5/50028-6
1981:differential graded Lie algebras
1273:{\displaystyle X_{\mathbb {Q} }}
367:{\displaystyle X_{\mathbb {Q} }}
299:{\displaystyle X_{\mathbb {Q} }}
5114:{\displaystyle {\binom {n}{i}}}
3492:{\displaystyle S^{1}\vee S^{2}}
2813:The Sullivan algebra is called
1185:is the Eilenberg–MacLane space
89:satisfying certain conditions.
5409:Sullivan (1977), Theorem 13.2.
5168:
5162:
5066:
5060:
4952:
4946:
4809:
4803:
4743:
4729:
4700:Elliptic and hyperbolic spaces
4679:
4673:
4667:
4661:
4655:
4649:
4331:
4289:
4012:
3969:
3949:{\displaystyle 1,a,b\to a^{2}}
3933:
3757:is a sphere of even dimension
3600:is a formal nilmanifold, then
3238:, which has the property that
3225:
3219:
3140:
3134:
3089:
3083:
2988:
2985:
2979:
2956:
2950:
2916:
2910:
2887:
2881:
2845:
2838:
2738:{\displaystyle ab=(-1)^{ij}ba}
2717:
2707:
2671:
2668:
2656:
2650:
2627:
2624:
2618:
2612:
2589:
2583:
2528:
2522:
2513:
2507:
2418:
2412:
2022:
2016:
2002:
1996:
1954:
1938:
1895:
1884:
1875:
1862:
1853:
1834:
1752:
1736:
1706:
1697:
1675:
1669:
1609:
1600:
1505:
1499:
1469:
1455:
1323:
1316:
1218:
1195:
1027:
1011:
905:
883:
791:
775:
749:
733:
717:
692:
646:
640:
624:
609:
564:
529:
460:
446:
320:
158:{\displaystyle f\colon X\to Y}
149:
58:is ignored. It was founded by
1:
5838:Siberian Mathematical Journal
5756:American Mathematical Society
4852:-vector space, and otherwise
4830:is also a finite-dimensional
4221:. It has a basis of elements
3746:, and a basis of elements 1,
3649:is a sphere of odd dimension
3104:polynomial differential forms
2962:{\displaystyle \bigwedge (V)}
2922:{\displaystyle \bigwedge (V)}
2424:{\displaystyle \bigwedge (V)}
1814:of the homotopy Lie algebra:
1652:. In view of the isomorphism
1630:is a graded Lie algebra over
1482:and the homotopy Lie algebra
519:is the unique rational space
402:of a space at a given set of
174:rational homotopy equivalence
5638:Algebraic Models in Geometry
4845:{\displaystyle \mathbb {Q} }
4771:{\displaystyle \mathbb {Q} }
4311:{\displaystyle x\to u^{n+1}}
3596:are almost never formal: if
2395:{\displaystyle \mathbb {Q} }
2342:{\displaystyle \mathbb {Q} }
2207:{\displaystyle \mathbb {Q} }
2069:{\displaystyle \mathbb {Q} }
1812:universal enveloping algebra
1645:{\displaystyle \mathbb {Q} }
1541:{\displaystyle \mathbb {Q} }
277:, there is a rational space
202:{\displaystyle \mathbb {Q} }
83:differential graded algebras
5957:Encyclopedia of Mathematics
5812:University of Chicago Press
5600:Rational Homotopy Theory II
5533:L'Enseignement mathématique
5174:{\displaystyle b_{2i+1}(X)}
3985:{\displaystyle ab\to a^{3}}
3192:{\displaystyle \bigwedge V}
1556:. (More precisely, writing
77:Rational homotopy types of
42:is a simplified version of
6026:
5952:"Rational homotopy theory"
5950:Sullivan, Dennis (2001) ,
5801:; Ponto, Kathleen (2012),
5214:asserts that the rational
5187:complete intersection ring
4214:{\displaystyle dx=u^{n+1}}
3911:, and a basis of elements
3592:On the other hand, closed
3059:For any topological space
853:homotopy groups of spheres
219:rational homotopy category
187:with the rational numbers
5574:10.1007/978-1-4613-0105-9
5481:Pavlov (2002), Theorem 1.
5303:Chromatic homotopy theory
5043:, then each Betti number
4639:is in the Massey product
3677:{\displaystyle 2n+1>1}
3231:{\displaystyle A_{PL}(X)}
3146:{\displaystyle A_{PL}(X)}
3095:{\displaystyle A_{PL}(X)}
2431:on a graded vector space
5562:Rational Homotopy Theory
5276:{\displaystyle \Omega X}
5241:{\displaystyle \Omega X}
5072:{\displaystyle b_{i}(X)}
4756:is a finite-dimensional
4555:. Any homomorphism from
4476:{\displaystyle dx=a^{2}}
4054:complex projective space
3875:{\displaystyle db=a^{2}}
3436:-completed homotopy type
3418:) is almost a model for
3102:, called the algebra of
2362:André–Quillen cohomology
1572:{\displaystyle \Omega X}
1288:(or more generally, any
1139:Eilenberg–MacLane spaces
40:rational homotopy theory
5850:10.1023/A:1021173418920
5642:Oxford University Press
5216:Serre spectral sequence
5121:(with equality for the
5010:{\displaystyle n\geq N}
3779:{\displaystyle 2n>0}
3364:{\displaystyle V^{1}=0}
3264:{\displaystyle V^{1}=0}
2633:{\displaystyle d(V(k))}
2240:{\displaystyle A^{1}=0}
2149:with each vector space
79:simply connected spaces
5985:10.4310/jdg/1214433729
5774:10.1090/conm/436/08409
5677:, Boston: Birkhäuser,
5277:
5242:
5175:
5115:
5073:
5011:
4982:
4920:
4880:
4879:{\displaystyle C>1}
4846:
4824:
4772:
4750:
4689:
4633:
4597:
4549:
4509:
4477:
4441:
4412:
4360:
4312:
4273:
4215:
4173:
4144:
4107:
4106:{\displaystyle n>0}
4081:
4035:
3986:
3950:
3905:
3876:
3840:
3811:
3780:
3740:
3711:
3678:
3612:, the quotient of the
3540:
3493:
3392:
3365:
3292:
3265:
3232:
3193:
3147:
3096:
3033:
2998:
2963:
2940:is a Sullivan algebra
2923:
2894:
2858:
2804:
2773:
2739:
2678:
2634:
2596:
2567:
2541:
2480:
2425:
2396:
2354:Lie algebra cohomology
2343:
2321:
2283:
2282:{\displaystyle A^{2a}}
2241:
2208:
2170:
2140:
2070:
2029:
1969:
1905:
1797:
1767:
1713:
1646:
1624:
1579:for the loop space of
1573:
1542:
1520:
1476:
1416:
1274:
1231:. More generally, let
1225:
1179:
1121:
982:
827:
826:{\displaystyle i>0}
798:
663:
583:
548:
513:
493:
492:{\displaystyle i>0}
467:
423:
392:
368:
339:
300:
271:
249:is a simply connected
203:
159:
5870:Annals of Mathematics
5667:Griffiths, Phillip A.
5278:
5243:
5211:Halperin's conjecture
5176:
5132:of an elliptic space
5116:
5074:
5012:
4983:
4900:
4881:
4854:rationally hyperbolic
4847:
4825:
4773:
4751:
4690:
4634:
4632:{\displaystyle xb-ay}
4598:
4596:{\displaystyle xb-ay}
4550:
4548:{\displaystyle xb-ay}
4510:
4508:{\displaystyle dy=ab}
4478:
4442:
4413:
4361:
4313:
4274:
4216:
4174:
4145:
4108:
4082:
4036:
3987:
3951:
3906:
3877:
3841:
3812:
3781:
3741:
3712:
3679:
3541:
3494:
3393:
3391:{\displaystyle V^{k}}
3366:
3293:
3291:{\displaystyle V^{k}}
3266:
3233:
3194:
3148:
3097:
3034:
2999:
2964:
2924:
2895:
2859:
2805:
2803:{\displaystyle A^{j}}
2774:
2772:{\displaystyle A^{i}}
2740:
2679:
2635:
2597:
2568:
2542:
2481:
2426:
2397:
2344:
2322:
2284:
2242:
2209:
2171:
2169:{\displaystyle A^{i}}
2141:
2071:
2030:
1970:
1906:
1798:
1768:
1714:
1647:
1625:
1574:
1543:
1521:
1477:
1417:
1275:
1226:
1180:
1122:
983:
828:
799:
664:
584:
549:
514:
494:
468:
424:
393:
369:
340:
301:
272:
204:
160:
5758:, pp. 175–202,
5264:
5229:
5140:
5130:Euler characteristic
5125:-dimensional torus).
5086:
5081:binomial coefficient
5047:
4995:
4897:
4864:
4834:
4790:
4760:
4716:
4643:
4611:
4575:
4527:
4487:
4451:
4440:{\displaystyle db=0}
4422:
4411:{\displaystyle da=0}
4393:
4322:
4283:
4225:
4183:
4172:{\displaystyle du=0}
4154:
4143:{\displaystyle 2n+1}
4125:
4091:
4059:
3996:
3960:
3915:
3904:{\displaystyle da=0}
3886:
3850:
3839:{\displaystyle 4n+1}
3821:
3798:
3761:
3739:{\displaystyle da=0}
3721:
3710:{\displaystyle 2n+1}
3692:
3653:
3618:symplectic manifolds
3515:
3463:
3398:of finite dimension.
3375:
3342:
3275:
3242:
3203:
3180:
3118:
3067:
3008:
2973:
2969:with a homomorphism
2944:
2904:
2868:
2825:
2787:
2756:
2695:
2644:
2606:
2595:{\displaystyle V(0)}
2577:
2551:
2501:
2438:
2406:
2384:
2331:
2293:
2263:
2218:
2196:
2153:
2087:
2058:
2043:as a coalgebra; the
1987:
1925:
1821:
1781:
1723:
1656:
1634:
1587:
1560:
1530:
1486:
1442:
1307:
1255:
1189:
1146:
1133:In homotopy theory,
998:
870:
811:
679:
596:
558:
523:
503:
477:
433:
413:
382:
349:
314:
308:homotopy equivalence
281:
261:
191:
137:
122:and Wolfgang Meyer.
34:and specifically in
5708:History of Topology
5546:10.5169/seals-60412
5254:integrally elliptic
5204:sectional curvature
4784:rationally elliptic
3610:Heisenberg manifold
3443:for a prime number
2566:{\displaystyle d=0}
2316:
1796:{\displaystyle n-1}
1408:
1373:
1353:
1174:
97:Riemannian manifold
60:Dennis Sullivan
5936:10338.dmlcz/128041
5928:10.1007/bf02684341
5273:
5238:
5171:
5111:
5069:
5022:homogeneous spaces
5007:
4978:
4876:
4842:
4820:
4768:
4746:
4685:
4629:
4593:
4571:; so it would map
4545:
4505:
4473:
4437:
4408:
4356:
4308:
4269:
4211:
4169:
4140:
4103:
4077:
4031:
3982:
3946:
3901:
3872:
3836:
3810:{\displaystyle 2n}
3807:
3776:
3736:
3707:
3674:
3536:
3489:
3428:real homotopy type
3408:differential forms
3388:
3361:
3288:
3261:
3228:
3189:
3143:
3092:
3029:
2994:
2959:
2919:
2890:
2854:
2800:
2769:
2735:
2674:
2630:
2592:
2563:
2537:
2476:
2462:
2421:
2392:
2339:
2317:
2302:
2279:
2237:
2204:
2166:
2136:
2066:
2025:
1965:
1916:primitive elements
1901:
1793:
1763:
1709:
1642:
1620:
1569:
1550:graded Lie algebra
1538:
1516:
1472:
1412:
1383:
1357:
1337:
1270:
1237:graded-commutative
1221:
1175:
1149:
1117:
1112:
978:
973:
823:
794:
659:
579:
544:
509:
489:
463:
419:
388:
364:
335:
296:
267:
199:
170:topological spaces
155:
68:Daniel Quillen
48:topological spaces
5821:978-0-226-51178-8
5651:978-0-19-920651-3
5621:978-981-4651-42-4
5297:Mandell's theorem
5103:
4567:to a multiple of
4121:of degrees 2 and
2880:
2837:
2447:
2372:Sullivan algebras
2045:dual vector space
1554:Whitehead product
1286:compact Lie group
1108:
1076:
1059:
969:
937:
861:Jean-Pierre Serre
842:fundamental group
512:{\displaystyle X}
422:{\displaystyle X}
391:{\displaystyle X}
270:{\displaystyle X}
243:homotopy category
215:singular homology
176:if it induces an
18:Rational homotopy
16:(Redirected from
6017:
5996:
5987:
5964:
5946:
5906:Sullivan, Dennis
5901:
5860:
5844:(6): 1080–1085,
5832:
5809:
5794:
5767:
5749:
5736:
5695:
5662:
5632:
5604:World Scientific
5594:
5556:
5518:
5515:
5509:
5506:
5500:
5497:
5491:
5488:
5482:
5479:
5473:
5470:
5464:
5461:
5455:
5452:
5446:
5443:
5437:
5434:
5428:
5425:
5419:
5416:
5410:
5407:
5401:
5398:
5392:
5389:
5383:
5380:
5374:
5371:
5365:
5362:
5356:
5353:
5347:
5344:
5338:
5335:
5329:
5323:
5282:
5280:
5279:
5274:
5247:
5245:
5244:
5239:
5180:
5178:
5177:
5172:
5161:
5160:
5120:
5118:
5117:
5112:
5110:
5109:
5108:
5095:
5078:
5076:
5075:
5070:
5059:
5058:
5016:
5014:
5013:
5008:
4987:
4985:
4984:
4979:
4977:
4976:
4964:
4963:
4945:
4944:
4932:
4931:
4930:
4919:
4914:
4885:
4883:
4882:
4877:
4851:
4849:
4848:
4843:
4841:
4829:
4827:
4826:
4821:
4819:
4802:
4801:
4777:
4775:
4774:
4769:
4767:
4755:
4753:
4752:
4747:
4742:
4728:
4727:
4694:
4692:
4691:
4686:
4638:
4636:
4635:
4630:
4602:
4600:
4599:
4594:
4554:
4552:
4551:
4546:
4514:
4512:
4511:
4506:
4482:
4480:
4479:
4474:
4472:
4471:
4446:
4444:
4443:
4438:
4417:
4415:
4414:
4409:
4365:
4363:
4362:
4357:
4349:
4348:
4317:
4315:
4314:
4309:
4307:
4306:
4278:
4276:
4275:
4270:
4268:
4267:
4249:
4248:
4220:
4218:
4217:
4212:
4210:
4209:
4178:
4176:
4175:
4170:
4149:
4147:
4146:
4141:
4112:
4110:
4109:
4104:
4086:
4084:
4083:
4078:
4076:
4075:
4070:
4040:
4038:
4037:
4032:
4024:
4023:
4008:
4007:
3991:
3989:
3988:
3983:
3981:
3980:
3955:
3953:
3952:
3947:
3945:
3944:
3910:
3908:
3907:
3902:
3881:
3879:
3878:
3873:
3871:
3870:
3845:
3843:
3842:
3837:
3816:
3814:
3813:
3808:
3785:
3783:
3782:
3777:
3745:
3743:
3742:
3737:
3716:
3714:
3713:
3708:
3683:
3681:
3680:
3675:
3614:Heisenberg group
3579:Kähler manifolds
3575:symmetric spaces
3545:
3543:
3542:
3537:
3535:
3527:
3526:
3498:
3496:
3495:
3490:
3488:
3487:
3475:
3474:
3397:
3395:
3394:
3389:
3387:
3386:
3370:
3368:
3367:
3362:
3354:
3353:
3297:
3295:
3294:
3289:
3287:
3286:
3270:
3268:
3267:
3262:
3254:
3253:
3237:
3235:
3234:
3229:
3218:
3217:
3198:
3196:
3195:
3190:
3152:
3150:
3149:
3144:
3133:
3132:
3101:
3099:
3098:
3093:
3082:
3081:
3038:
3036:
3035:
3030:
3028:
3020:
3019:
3003:
3001:
3000:
2995:
2968:
2966:
2965:
2960:
2928:
2926:
2925:
2920:
2899:
2897:
2896:
2891:
2879:
2871:
2863:
2861:
2860:
2855:
2853:
2852:
2836:
2828:
2821:is contained in
2817:if the image of
2809:
2807:
2806:
2801:
2799:
2798:
2778:
2776:
2775:
2770:
2768:
2767:
2744:
2742:
2741:
2736:
2728:
2727:
2683:
2681:
2680:
2675:
2640:is contained in
2639:
2637:
2636:
2631:
2601:
2599:
2598:
2593:
2572:
2570:
2569:
2564:
2546:
2544:
2543:
2538:
2485:
2483:
2482:
2477:
2472:
2471:
2461:
2430:
2428:
2427:
2422:
2401:
2399:
2398:
2393:
2391:
2378:Sullivan algebra
2348:
2346:
2345:
2340:
2338:
2326:
2324:
2323:
2318:
2315:
2310:
2288:
2286:
2285:
2280:
2278:
2277:
2249:Poincaré duality
2246:
2244:
2243:
2238:
2230:
2229:
2213:
2211:
2210:
2205:
2203:
2175:
2173:
2172:
2167:
2165:
2164:
2145:
2143:
2142:
2137:
2126:
2125:
2113:
2112:
2100:
2075:
2073:
2072:
2067:
2065:
2049:model categories
2034:
2032:
2031:
2026:
2009:
1974:
1972:
1971:
1966:
1964:
1953:
1952:
1937:
1936:
1910:
1908:
1907:
1902:
1894:
1874:
1873:
1852:
1851:
1833:
1832:
1802:
1800:
1799:
1794:
1772:
1770:
1769:
1764:
1762:
1751:
1750:
1735:
1734:
1718:
1716:
1715:
1710:
1696:
1695:
1668:
1667:
1651:
1649:
1648:
1643:
1641:
1629:
1627:
1626:
1621:
1619:
1599:
1598:
1578:
1576:
1575:
1570:
1547:
1545:
1544:
1539:
1537:
1525:
1523:
1522:
1517:
1515:
1498:
1497:
1481:
1479:
1478:
1473:
1468:
1454:
1453:
1421:
1419:
1418:
1413:
1407:
1393:
1392:
1372:
1367:
1366:
1352:
1347:
1346:
1333:
1332:
1331:
1279:
1277:
1276:
1271:
1269:
1268:
1267:
1249:exterior algebra
1230:
1228:
1227:
1222:
1202:
1184:
1182:
1181:
1176:
1173:
1159:
1158:
1126:
1124:
1123:
1118:
1116:
1115:
1109:
1106:
1077:
1074:
1060:
1057:
1053:
1037:
1026:
1025:
1010:
1009:
987:
985:
984:
979:
977:
976:
970:
967:
938:
935:
931:
915:
904:
903:
882:
881:
838:nilpotent spaces
832:
830:
829:
824:
803:
801:
800:
795:
790:
789:
774:
773:
761:
760:
748:
747:
732:
731:
716:
715:
706:
705:
704:
691:
690:
668:
666:
665:
660:
658:
657:
639:
638:
623:
622:
621:
608:
607:
588:
586:
585:
580:
578:
577:
576:
553:
551:
550:
545:
543:
542:
541:
518:
516:
515:
510:
498:
496:
495:
490:
472:
470:
469:
464:
459:
445:
444:
428:
426:
425:
420:
397:
395:
394:
389:
373:
371:
370:
365:
363:
362:
361:
344:
342:
341:
336:
334:
333:
332:
305:
303:
302:
297:
295:
294:
293:
276:
274:
273:
268:
239:full subcategory
209:. Equivalently:
208:
206:
205:
200:
198:
167:simply connected
164:
162:
161:
156:
104:closed geodesics
87:rational numbers
21:
6025:
6024:
6020:
6019:
6018:
6016:
6015:
6014:
6010:Homotopy theory
6000:
5999:
5967:
5949:
5904:
5883:10.2307/1970725
5865:Quillen, Daniel
5863:
5835:
5822:
5807:
5797:
5784:
5747:
5739:
5726:
5698:
5685:
5671:Morgan, John W.
5665:
5652:
5635:
5622:
5597:
5584:
5566:Springer Nature
5559:
5529:
5526:
5521:
5516:
5512:
5507:
5503:
5498:
5494:
5489:
5485:
5480:
5476:
5471:
5467:
5462:
5458:
5453:
5449:
5444:
5440:
5435:
5431:
5426:
5422:
5417:
5413:
5408:
5404:
5399:
5395:
5390:
5386:
5381:
5377:
5372:
5368:
5363:
5359:
5354:
5350:
5345:
5341:
5336:
5332:
5324:
5315:
5311:
5293:
5262:
5261:
5227:
5226:
5143:
5138:
5137:
5090:
5084:
5083:
5079:is at most the
5050:
5045:
5044:
4993:
4992:
4968:
4936:
4921:
4895:
4894:
4886:and an integer
4862:
4861:
4832:
4831:
4793:
4788:
4787:
4758:
4757:
4719:
4714:
4713:
4702:
4641:
4640:
4609:
4608:
4573:
4572:
4525:
4524:
4485:
4484:
4463:
4449:
4448:
4420:
4419:
4391:
4390:
4373:has 4 elements
4334:
4320:
4319:
4292:
4281:
4280:
4259:
4240:
4223:
4222:
4195:
4181:
4180:
4152:
4151:
4123:
4122:
4089:
4088:
4062:
4057:
4056:
4015:
3999:
3994:
3993:
3972:
3958:
3957:
3936:
3913:
3912:
3884:
3883:
3862:
3848:
3847:
3819:
3818:
3796:
3795:
3759:
3758:
3719:
3718:
3690:
3689:
3651:
3650:
3642:
3634:Borromean rings
3626:Massey products
3518:
3513:
3512:
3505:
3479:
3466:
3461:
3460:
3416:de Rham complex
3378:
3373:
3372:
3345:
3340:
3339:
3278:
3273:
3272:
3245:
3240:
3239:
3206:
3201:
3200:
3178:
3177:
3121:
3116:
3115:
3070:
3065:
3064:
3057:
3011:
3006:
3005:
2971:
2970:
2942:
2941:
2902:
2901:
2866:
2865:
2844:
2823:
2822:
2790:
2785:
2784:
2759:
2754:
2753:
2716:
2693:
2692:
2642:
2641:
2604:
2603:
2575:
2574:
2549:
2548:
2499:
2498:
2463:
2436:
2435:
2404:
2403:
2382:
2381:
2374:
2329:
2328:
2291:
2290:
2289:is of the form
2266:
2261:
2260:
2247:that satisfies
2221:
2216:
2215:
2194:
2193:
2156:
2151:
2150:
2117:
2104:
2085:
2084:
2056:
2055:
1985:
1984:
1944:
1928:
1923:
1922:
1865:
1824:
1819:
1818:
1779:
1778:
1742:
1726:
1721:
1720:
1681:
1659:
1654:
1653:
1632:
1631:
1590:
1585:
1584:
1583:, we have that
1558:
1557:
1528:
1527:
1489:
1484:
1483:
1445:
1440:
1439:
1428:
1322:
1305:
1304:
1258:
1253:
1252:
1245:polynomial ring
1187:
1186:
1144:
1143:
1111:
1110:
1103:
1097:
1096:
1054:
1042:
1014:
1001:
996:
995:
972:
971:
964:
958:
957:
932:
920:
886:
873:
868:
867:
809:
808:
765:
723:
695:
682:
677:
676:
630:
612:
599:
594:
593:
567:
556:
555:
532:
521:
520:
501:
500:
475:
474:
436:
431:
430:
411:
410:
380:
379:
376:rationalization
352:
347:
346:
323:
312:
311:
306:, unique up to
284:
279:
278:
259:
258:
189:
188:
182:homotopy groups
135:
134:
128:
126:Rational spaces
112:free loop space
56:homotopy groups
50:, in which all
44:homotopy theory
28:
23:
22:
15:
12:
11:
5:
6023:
6021:
6013:
6012:
6002:
6001:
5998:
5997:
5978:(4): 633–644,
5965:
5947:
5902:
5877:(2): 205–295,
5861:
5833:
5820:
5795:
5782:
5737:
5724:
5704:James, Ioan M.
5696:
5683:
5663:
5650:
5633:
5620:
5595:
5582:
5557:
5540:(1–2): 25–32,
5525:
5522:
5520:
5519:
5510:
5501:
5492:
5483:
5474:
5465:
5456:
5447:
5438:
5429:
5420:
5411:
5402:
5393:
5384:
5375:
5366:
5357:
5348:
5339:
5330:
5328:, p. 757.
5312:
5310:
5307:
5306:
5305:
5300:
5292:
5289:
5272:
5269:
5237:
5234:
5191:
5190:
5170:
5167:
5164:
5159:
5156:
5153:
5150:
5146:
5126:
5107:
5102:
5099:
5094:
5068:
5065:
5062:
5057:
5053:
5033:
5006:
5003:
5000:
4989:
4988:
4975:
4971:
4967:
4962:
4957:
4954:
4951:
4948:
4943:
4939:
4935:
4929:
4924:
4918:
4913:
4910:
4907:
4903:
4875:
4872:
4869:
4840:
4818:
4814:
4811:
4808:
4805:
4800:
4796:
4766:
4745:
4741:
4737:
4734:
4731:
4726:
4722:
4708:. Namely, let
4701:
4698:
4697:
4696:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4628:
4625:
4622:
4619:
4616:
4592:
4589:
4586:
4583:
4580:
4544:
4541:
4538:
4535:
4532:
4504:
4501:
4498:
4495:
4492:
4470:
4466:
4462:
4459:
4456:
4436:
4433:
4430:
4427:
4407:
4404:
4401:
4398:
4367:
4355:
4352:
4347:
4344:
4341:
4337:
4333:
4330:
4327:
4305:
4302:
4299:
4295:
4291:
4288:
4266:
4262:
4258:
4255:
4252:
4247:
4243:
4239:
4236:
4233:
4230:
4208:
4205:
4202:
4198:
4194:
4191:
4188:
4168:
4165:
4162:
4159:
4139:
4136:
4133:
4130:
4102:
4099:
4096:
4074:
4069:
4066:
4046:
4030:
4027:
4022:
4018:
4014:
4011:
4006:
4002:
3979:
3975:
3971:
3968:
3965:
3943:
3939:
3935:
3932:
3929:
3926:
3923:
3920:
3900:
3897:
3894:
3891:
3869:
3865:
3861:
3858:
3855:
3835:
3832:
3829:
3826:
3806:
3803:
3775:
3772:
3769:
3766:
3751:
3735:
3732:
3729:
3726:
3706:
3703:
3700:
3697:
3673:
3670:
3667:
3664:
3661:
3658:
3641:
3638:
3587:connected sums
3577:, and compact
3534:
3530:
3525:
3521:
3504:
3501:
3486:
3482:
3478:
3473:
3469:
3400:
3399:
3385:
3381:
3360:
3357:
3352:
3348:
3332:
3329:
3322:
3311:
3285:
3281:
3260:
3257:
3252:
3248:
3227:
3224:
3221:
3216:
3213:
3209:
3188:
3185:
3157:for the space
3142:
3139:
3136:
3131:
3128:
3124:
3091:
3088:
3085:
3080:
3077:
3073:
3056:
3053:
3027:
3023:
3018:
3014:
2993:
2990:
2987:
2984:
2981:
2978:
2958:
2955:
2952:
2949:
2934:Sullivan model
2918:
2915:
2912:
2909:
2889:
2886:
2883:
2878:
2874:
2851:
2847:
2843:
2840:
2835:
2831:
2797:
2793:
2766:
2762:
2746:
2745:
2734:
2731:
2726:
2723:
2719:
2715:
2712:
2709:
2706:
2703:
2700:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2591:
2588:
2585:
2582:
2562:
2559:
2556:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2487:
2486:
2475:
2470:
2466:
2460:
2457:
2454:
2450:
2446:
2443:
2420:
2417:
2414:
2411:
2390:
2373:
2370:
2366:Koszul duality
2337:
2314:
2309:
2305:
2301:
2298:
2276:
2273:
2269:
2236:
2233:
2228:
2224:
2214:-algebra with
2202:
2163:
2159:
2147:
2146:
2135:
2132:
2129:
2124:
2120:
2116:
2111:
2107:
2103:
2099:
2095:
2092:
2064:
2024:
2021:
2018:
2015:
2012:
2008:
2004:
2001:
1998:
1995:
1992:
1963:
1959:
1956:
1951:
1947:
1943:
1940:
1935:
1931:
1912:
1911:
1900:
1897:
1893:
1889:
1886:
1883:
1880:
1877:
1872:
1868:
1864:
1861:
1858:
1855:
1850:
1845:
1842:
1839:
1836:
1831:
1827:
1792:
1789:
1786:
1761:
1757:
1754:
1749:
1745:
1741:
1738:
1733:
1729:
1708:
1705:
1702:
1699:
1694:
1691:
1688:
1684:
1680:
1677:
1674:
1671:
1666:
1662:
1640:
1618:
1614:
1611:
1608:
1605:
1602:
1597:
1593:
1568:
1565:
1536:
1514:
1510:
1507:
1504:
1501:
1496:
1492:
1471:
1467:
1463:
1460:
1457:
1452:
1448:
1427:
1424:
1423:
1422:
1411:
1406:
1403:
1400:
1397:
1391:
1386:
1382:
1379:
1376:
1371:
1365:
1360:
1356:
1351:
1345:
1340:
1336:
1330:
1325:
1321:
1318:
1315:
1312:
1266:
1261:
1241:tensor product
1220:
1217:
1214:
1211:
1208:
1205:
1201:
1197:
1194:
1172:
1169:
1166:
1163:
1157:
1152:
1128:
1127:
1114:
1104:
1102:
1099:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1075: or
1072:
1069:
1066:
1063:
1055:
1052:
1048:
1047:
1045:
1040:
1036:
1032:
1029:
1024:
1021:
1017:
1013:
1008:
1004:
989:
988:
975:
965:
963:
960:
959:
956:
953:
950:
947:
944:
941:
933:
930:
926:
925:
923:
918:
914:
910:
907:
902:
899:
896:
893:
889:
885:
880:
876:
851:Computing the
840:(spaces whose
822:
819:
816:
805:
804:
793:
788:
783:
780:
777:
772:
768:
764:
759:
754:
751:
746:
741:
738:
735:
730:
726:
722:
719:
714:
709:
703:
698:
694:
689:
685:
670:
669:
656:
651:
648:
645:
642:
637:
633:
629:
626:
620:
615:
611:
606:
602:
575:
570:
566:
563:
540:
535:
531:
528:
508:
488:
485:
482:
462:
458:
454:
451:
448:
443:
439:
418:
387:
374:is called the
360:
355:
331:
326:
322:
319:
292:
287:
266:
247:rational space
197:
154:
151:
148:
145:
142:
132:continuous map
127:
124:
120:Detlef Gromoll
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6022:
6011:
6008:
6007:
6005:
5995:
5991:
5986:
5981:
5977:
5973:
5972:
5966:
5963:
5959:
5958:
5953:
5948:
5945:
5941:
5937:
5933:
5929:
5925:
5921:
5917:
5916:
5911:
5907:
5903:
5900:
5896:
5892:
5888:
5884:
5880:
5876:
5872:
5871:
5866:
5862:
5859:
5855:
5851:
5847:
5843:
5839:
5834:
5831:
5827:
5823:
5817:
5813:
5806:
5805:
5800:
5799:May, J. Peter
5796:
5793:
5789:
5785:
5783:9780821838143
5779:
5775:
5771:
5766:
5761:
5757:
5753:
5746:
5742:
5741:Hess, Kathryn
5738:
5735:
5731:
5727:
5725:0-444-82375-1
5721:
5717:
5713:
5709:
5705:
5701:
5700:Hess, Kathryn
5697:
5694:
5690:
5686:
5684:3-7643-3041-4
5680:
5676:
5672:
5668:
5664:
5661:
5657:
5653:
5647:
5643:
5639:
5634:
5631:
5627:
5623:
5617:
5613:
5609:
5605:
5602:, Singapore:
5601:
5596:
5593:
5589:
5585:
5583:0-387-95068-0
5579:
5575:
5571:
5567:
5563:
5558:
5555:
5551:
5547:
5543:
5539:
5535:
5534:
5528:
5527:
5523:
5514:
5511:
5505:
5502:
5496:
5493:
5487:
5484:
5478:
5475:
5469:
5466:
5460:
5457:
5451:
5448:
5442:
5439:
5433:
5430:
5424:
5421:
5415:
5412:
5406:
5403:
5397:
5394:
5388:
5385:
5379:
5376:
5370:
5367:
5361:
5358:
5352:
5349:
5343:
5340:
5334:
5331:
5327:
5322:
5320:
5318:
5314:
5308:
5304:
5301:
5298:
5295:
5294:
5290:
5288:
5286:
5270:
5259:
5255:
5251:
5235:
5224:
5219:
5217:
5213:
5212:
5207:
5205:
5201:
5200:'s conjecture
5199:
5194:
5188:
5184:
5165:
5157:
5154:
5151:
5148:
5144:
5135:
5131:
5127:
5124:
5100:
5097:
5082:
5063:
5055:
5051:
5042:
5038:
5034:
5031:
5030:
5029:
5027:
5023:
5018:
5004:
5001:
4998:
4973:
4969:
4965:
4955:
4949:
4941:
4937:
4933:
4922:
4916:
4911:
4908:
4905:
4901:
4893:
4892:
4891:
4889:
4873:
4870:
4867:
4859:
4855:
4812:
4806:
4798:
4794:
4785:
4781:
4735:
4732:
4724:
4720:
4711:
4707:
4706:exponentially
4699:
4676:
4670:
4664:
4658:
4652:
4626:
4623:
4620:
4617:
4614:
4606:
4590:
4587:
4584:
4581:
4578:
4570:
4566:
4562:
4558:
4542:
4539:
4536:
4533:
4530:
4522:
4518:
4502:
4499:
4496:
4493:
4490:
4468:
4464:
4460:
4457:
4454:
4434:
4431:
4428:
4425:
4405:
4402:
4399:
4396:
4388:
4384:
4380:
4376:
4372:
4369:Suppose that
4368:
4353:
4350:
4345:
4342:
4339:
4335:
4328:
4325:
4303:
4300:
4297:
4293:
4286:
4264:
4260:
4256:
4253:
4250:
4245:
4241:
4237:
4234:
4231:
4228:
4206:
4203:
4200:
4196:
4192:
4189:
4186:
4166:
4163:
4160:
4157:
4137:
4134:
4131:
4128:
4120:
4116:
4100:
4097:
4094:
4072:
4055:
4051:
4047:
4044:
4028:
4025:
4020:
4016:
4009:
4004:
4000:
3977:
3973:
3966:
3963:
3941:
3937:
3930:
3927:
3924:
3921:
3918:
3898:
3895:
3892:
3889:
3867:
3863:
3859:
3856:
3853:
3833:
3830:
3827:
3824:
3804:
3801:
3793:
3789:
3773:
3770:
3767:
3764:
3756:
3752:
3749:
3733:
3730:
3727:
3724:
3704:
3701:
3698:
3695:
3687:
3671:
3668:
3665:
3662:
3659:
3656:
3648:
3644:
3643:
3639:
3637:
3635:
3631:
3627:
3622:
3619:
3615:
3611:
3607:
3603:
3599:
3595:
3590:
3588:
3584:
3580:
3576:
3572:
3567:
3565:
3561:
3557:
3553:
3549:
3528:
3523:
3519:
3511:, again with
3510:
3503:Formal spaces
3502:
3500:
3484:
3480:
3476:
3471:
3467:
3456:
3454:
3450:
3446:
3442:
3438:
3437:
3435:
3429:
3425:
3421:
3417:
3413:
3409:
3405:
3383:
3379:
3358:
3355:
3350:
3346:
3337:
3333:
3330:
3327:
3323:
3320:
3316:
3312:
3309:
3308:
3307:
3305:
3301:
3300:minimal model
3283:
3279:
3258:
3255:
3250:
3246:
3222:
3214:
3211:
3207:
3186:
3183:
3175:
3170:
3168:
3164:
3160:
3156:
3137:
3129:
3126:
3122:
3113:
3109:
3105:
3086:
3078:
3075:
3071:
3062:
3054:
3052:
3050:
3046:
3042:
3021:
3016:
3012:
2991:
2982:
2976:
2953:
2947:
2939:
2935:
2930:
2913:
2907:
2884:
2876:
2872:
2849:
2841:
2833:
2829:
2820:
2816:
2811:
2795:
2791:
2782:
2764:
2760:
2751:
2732:
2729:
2724:
2721:
2713:
2710:
2704:
2701:
2698:
2691:
2690:
2689:
2687:
2665:
2662:
2659:
2653:
2647:
2621:
2615:
2609:
2586:
2580:
2560:
2557:
2554:
2534:
2531:
2525:
2519:
2516:
2510:
2504:
2496:
2492:
2473:
2468:
2464:
2458:
2455:
2452:
2448:
2444:
2441:
2434:
2433:
2432:
2415:
2409:
2379:
2371:
2369:
2367:
2363:
2359:
2355:
2350:
2312:
2307:
2303:
2299:
2296:
2274:
2271:
2267:
2258:
2254:
2250:
2234:
2231:
2226:
2222:
2191:
2187:
2183:
2179:
2161:
2157:
2133:
2130:
2127:
2122:
2118:
2114:
2109:
2105:
2101:
2093:
2090:
2083:
2082:
2081:
2079:
2052:
2050:
2046:
2042:
2038:
2019:
2013:
2010:
2006:
1999:
1993:
1990:
1982:
1976:
1957:
1949:
1945:
1933:
1929:
1921:
1917:
1898:
1887:
1881:
1870:
1866:
1859:
1856:
1843:
1840:
1829:
1825:
1817:
1816:
1815:
1813:
1809:
1804:
1790:
1787:
1784:
1776:
1755:
1747:
1743:
1731:
1727:
1703:
1692:
1689:
1686:
1682:
1678:
1672:
1664:
1660:
1612:
1606:
1595:
1591:
1582:
1566:
1555:
1551:
1508:
1502:
1494:
1490:
1461:
1458:
1450:
1446:
1437:
1433:
1425:
1409:
1404:
1401:
1398:
1395:
1384:
1380:
1377:
1374:
1369:
1358:
1354:
1349:
1338:
1334:
1319:
1313:
1310:
1303:
1302:
1301:
1299:
1297:
1291:
1287:
1283:
1259:
1250:
1246:
1242:
1238:
1234:
1215:
1212:
1209:
1206:
1203:
1192:
1170:
1167:
1164:
1161:
1150:
1140:
1136:
1131:
1100:
1093:
1090:
1087:
1084:
1081:
1078:
1070:
1067:
1064:
1061:
1043:
1038:
1030:
1022:
1019:
1015:
1006:
1002:
994:
993:
992:
961:
954:
951:
948:
945:
942:
939:
921:
916:
908:
900:
897:
894:
891:
887:
878:
874:
866:
865:
864:
862:
858:
854:
849:
847:
843:
839:
834:
820:
817:
814:
781:
778:
770:
766:
762:
752:
739:
736:
728:
724:
720:
707:
696:
687:
683:
675:
674:
673:
649:
643:
635:
631:
627:
613:
604:
600:
592:
591:
590:
568:
561:
533:
526:
506:
486:
483:
480:
452:
449:
441:
437:
416:
407:
405:
404:prime numbers
401:
385:
377:
353:
324:
317:
310:, with a map
309:
285:
264:
256:
255:vector spaces
252:
248:
244:
240:
236:
231:
228:
224:
220:
216:
212:
186:
183:
179:
175:
171:
168:
152:
146:
143:
140:
133:
125:
123:
121:
117:
113:
109:
108:Betti numbers
105:
101:
98:
95:
90:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5975:
5969:
5955:
5919:
5913:
5874:
5868:
5841:
5837:
5803:
5765:math/0604626
5751:
5707:
5674:
5637:
5612:10.1142/9473
5599:
5564:, New York:
5561:
5537:
5531:
5513:
5504:
5495:
5486:
5477:
5468:
5459:
5450:
5441:
5432:
5423:
5414:
5405:
5396:
5387:
5378:
5369:
5360:
5351:
5342:
5333:
5284:
5260:homology of
5257:
5253:
5249:
5222:
5220:
5209:
5208:
5196:
5195:
5192:
5182:
5133:
5122:
5040:
5036:
5024:for compact
5019:
4990:
4887:
4857:
4853:
4783:
4779:
4709:
4703:
4604:
4568:
4564:
4560:
4556:
4520:
4516:
4386:
4382:
4378:
4374:
4370:
4118:
4114:
4049:
4042:
3791:
3787:
3754:
3747:
3685:
3646:
3629:
3623:
3609:
3604:must be the
3601:
3597:
3594:nilmanifolds
3591:
3568:
3563:
3559:
3555:
3551:
3547:
3546:, is called
3508:
3506:
3457:
3452:
3448:
3444:
3440:
3433:
3431:
3427:
3423:
3419:
3411:
3403:
3401:
3371:and all the
3335:
3325:
3318:
3314:
3303:
3299:
3271:and all the
3173:
3171:
3166:
3162:
3158:
3154:
3153:is called a
3111:
3107:
3060:
3058:
3048:
3044:
3040:
2937:
2933:
2931:
2818:
2814:
2812:
2780:
2749:
2747:
2685:
2494:
2493:: the space
2490:
2488:
2377:
2375:
2351:
2256:
2189:
2185:
2181:
2177:
2148:
2080:of the form
2077:
2053:
2040:
1977:
1920:Hopf algebra
1913:
1807:
1805:
1580:
1431:
1429:
1295:
1232:
1132:
1129:
990:
856:
850:
835:
806:
671:
408:
400:localization
375:
246:
232:
223:localization
218:
210:
173:
172:is called a
129:
115:
99:
91:
76:
39:
29:
5922:: 269–331,
5256:if the mod
3794:of degrees
1239:algebra (a
178:isomorphism
32:mathematics
5640:, Oxford:
5524:References
5252:is called
5026:Lie groups
4890:such that
3688:of degree
3583:wedge sums
2037:coalgebras
1436:cohomology
1290:loop space
1107:otherwise.
251:CW complex
235:equivalent
5962:EMS Press
5326:Hess 1999
5268:Ω
5233:Ω
5002:≥
4966:≥
4956:⊗
4938:π
4934:
4902:∑
4813:⊗
4799:∗
4795:π
4725:∗
4683:⟩
4647:⟨
4621:−
4603:to 0. So
4585:−
4563:to 0 and
4537:−
4354:…
4332:→
4290:→
4254:…
4029:…
4013:→
3970:→
3934:→
3477:∨
3184:⋀
2989:→
2977:⋀
2948:⋀
2908:⋀
2873:⋀
2830:⋀
2711:−
2663:−
2648:⋀
2535:⋯
2532:⊆
2517:⊆
2449:⨁
2410:⋀
2358:augmented
2356:, and an
2300:±
2297:∑
2131:⋯
2128:⊕
2115:⊕
2102:⊕
2076:-algebra
2014:
1994:
1958:⊗
1942:Ω
1934:∗
1888:⊗
1879:Ω
1871:∗
1867:π
1857:≅
1838:Ω
1830:∗
1788:−
1756:⊗
1740:Ω
1732:∗
1728:π
1701:Ω
1690:−
1683:π
1679:≅
1661:π
1613:⊗
1604:Ω
1596:∗
1592:π
1564:Ω
1509:⊗
1495:∗
1491:π
1451:∗
1402:−
1381:×
1378:⋯
1375:×
1355:×
1335:≃
1314:
1213:−
1168:−
1091:−
1039:≅
1031:⊗
1003:π
968:otherwise
952:−
917:≅
909:⊗
898:−
875:π
863:in 1951:
846:nilpotent
763:≅
753:⊗
721:≅
650:⊗
632:π
628:≅
601:π
565:→
530:→
321:→
150:→
144::
85:over the
6004:Category
5908:(1977),
5743:(2007),
5673:(1981),
5291:See also
4991:for all
3640:Examples
3571:H-spaces
2864:, where
2547:, where
1552:via the
1058:if
936:if
857:rational
807:for all
227:category
185:tensored
36:topology
5994:0455028
5944:0646078
5899:0258031
5891:1970725
5858:1946233
5830:2884233
5792:2355774
5734:1721122
5706:(ed.),
5693:0641551
5660:2403898
5630:3379890
5592:1802847
5554:1225255
4150:, with
4052:is the
3846:, with
3564:minimal
3161:. When
3039:, then
2815:minimal
1918:in the
1810:is the
1773:is the
1282:product
1135:spheres
241:of the
225:of the
110:of the
70: (
62: (
54:in the
52:torsion
5992:
5942:
5897:
5889:
5856:
5828:
5818:
5790:
5780:
5732:
5722:
5691:
5681:
5658:
5648:
5628:
5618:
5590:
5580:
5552:
4782:to be
4523:, and
3548:formal
2253:smooth
94:closed
66:) and
5887:JSTOR
5808:(PDF)
5760:arXiv
5748:(PDF)
5309:Notes
5185:is a
4087:with
3717:with
3606:torus
3414:(the
3402:When
3155:model
2327:over
1438:ring
1280:is a
1243:of a
237:to a
5816:ISBN
5778:ISBN
5720:ISBN
5679:ISBN
5646:ISBN
5616:ISBN
5578:ISBN
5198:Bott
5128:The
4871:>
4179:and
4117:and
4098:>
3817:and
3790:and
3771:>
3669:>
3199:for
2783:in
2779:and
2748:for
2602:and
2456:>
1775:free
1137:and
991:and
818:>
672:and
484:>
72:1969
64:1977
46:for
5980:doi
5932:hdl
5924:doi
5879:doi
5846:doi
5770:doi
5712:doi
5608:doi
5570:doi
5542:doi
5035:If
4923:dim
4786:if
4048:If
3753:If
3645:If
3550:if
3439:of
3410:on
3302:of
3106:on
2752:in
2573:on
1991:ker
1294:SU(
844:is
378:of
180:on
165:of
114:of
30:In
6006::
5990:MR
5988:,
5976:11
5974:,
5960:,
5954:,
5940:MR
5938:,
5930:,
5920:47
5918:,
5912:,
5895:MR
5893:,
5885:,
5875:90
5873:,
5854:MR
5852:,
5842:43
5840:,
5826:MR
5824:,
5814:,
5810:,
5788:MR
5786:,
5776:,
5768:,
5750:,
5730:MR
5728:,
5718:,
5689:MR
5687:,
5669:;
5656:MR
5654:,
5644:,
5626:MR
5624:,
5614:,
5606:,
5588:MR
5586:,
5576:,
5568:,
5550:MR
5548:,
5538:39
5536:,
5316:^
5017:.
4519:,
4483:,
4447:,
4418:,
4385:,
4381:,
4377:,
4318:,
4279:,
3992:,
3956:,
3882:,
3589:.
3573:,
3455:.
3169:.
2932:A
2929:.
2810:.
2368:.
2349:.
2051:.
2011:im
1975:.
1803:.
1311:SU
1300:,
833:.
406:.
130:A
38:,
5982::
5934::
5926::
5881::
5848::
5772::
5762::
5714::
5610::
5572::
5544::
5285:p
5271:X
5258:p
5250:X
5236:X
5223:X
5189:.
5183:X
5169:)
5166:X
5163:(
5158:1
5155:+
5152:i
5149:2
5145:b
5134:X
5123:n
5106:)
5101:i
5098:n
5093:(
5067:)
5064:X
5061:(
5056:i
5052:b
5041:n
5037:X
5005:N
4999:n
4974:n
4970:C
4961:Q
4953:)
4950:X
4947:(
4942:i
4928:Q
4917:n
4912:1
4909:=
4906:i
4888:N
4874:1
4868:C
4858:X
4839:Q
4817:Q
4810:)
4807:X
4804:(
4780:X
4765:Q
4744:)
4740:Q
4736:,
4733:X
4730:(
4721:H
4710:X
4695:.
4680:]
4677:b
4674:[
4671:,
4668:]
4665:a
4662:[
4659:,
4656:]
4653:a
4650:[
4627:y
4624:a
4618:b
4615:x
4605:V
4591:y
4588:a
4582:b
4579:x
4569:b
4565:x
4561:y
4557:V
4543:y
4540:a
4534:b
4531:x
4521:b
4517:a
4503:b
4500:a
4497:=
4494:y
4491:d
4469:2
4465:a
4461:=
4458:x
4455:d
4435:0
4432:=
4429:b
4426:d
4406:0
4403:=
4400:a
4397:d
4387:y
4383:x
4379:b
4375:a
4371:V
4366:.
4351:,
4346:2
4343:+
4340:n
4336:u
4329:u
4326:x
4304:1
4301:+
4298:n
4294:u
4287:x
4265:n
4261:u
4257:,
4251:,
4246:2
4242:u
4238:,
4235:u
4232:,
4229:1
4207:1
4204:+
4201:n
4197:u
4193:=
4190:x
4187:d
4167:0
4164:=
4161:u
4158:d
4138:1
4135:+
4132:n
4129:2
4119:x
4115:u
4101:0
4095:n
4073:n
4068:P
4065:C
4050:X
4045:.
4043:d
4026:,
4021:4
4017:a
4010:b
4005:2
4001:a
3978:3
3974:a
3967:b
3964:a
3942:2
3938:a
3931:b
3928:,
3925:a
3922:,
3919:1
3899:0
3896:=
3893:a
3890:d
3868:2
3864:a
3860:=
3857:b
3854:d
3834:1
3831:+
3828:n
3825:4
3805:n
3802:2
3792:b
3788:a
3774:0
3768:n
3765:2
3755:X
3750:.
3748:a
3734:0
3731:=
3728:a
3725:d
3705:1
3702:+
3699:n
3696:2
3686:a
3672:1
3666:1
3663:+
3660:n
3657:2
3647:X
3630:A
3602:M
3598:M
3560:A
3556:A
3552:A
3533:Q
3529:=
3524:0
3520:A
3509:A
3485:2
3481:S
3472:1
3468:S
3453:p
3449:p
3445:p
3441:X
3434:p
3424:X
3420:X
3412:X
3404:X
3384:k
3380:V
3359:0
3356:=
3351:1
3347:V
3336:X
3328:.
3326:d
3321:.
3319:X
3315:V
3304:X
3284:k
3280:V
3259:0
3256:=
3251:1
3247:V
3226:)
3223:X
3220:(
3215:L
3212:P
3208:A
3187:V
3174:X
3167:X
3163:X
3159:X
3141:)
3138:X
3135:(
3130:L
3127:P
3123:A
3112:X
3108:X
3090:)
3087:X
3084:(
3079:L
3076:P
3072:A
3061:X
3049:A
3045:A
3041:A
3026:Q
3022:=
3017:0
3013:A
2992:A
2986:)
2983:V
2980:(
2957:)
2954:V
2951:(
2938:A
2917:)
2914:V
2911:(
2888:)
2885:V
2882:(
2877:+
2850:2
2846:)
2842:V
2839:(
2834:+
2819:d
2796:j
2792:A
2781:b
2765:i
2761:A
2750:a
2733:a
2730:b
2725:j
2722:i
2718:)
2714:1
2708:(
2705:=
2702:b
2699:a
2686:A
2672:)
2669:)
2666:1
2660:k
2657:(
2654:V
2651:(
2628:)
2625:)
2622:k
2619:(
2616:V
2613:(
2610:d
2590:)
2587:0
2584:(
2581:V
2561:0
2558:=
2555:d
2529:)
2526:1
2523:(
2520:V
2514:)
2511:0
2508:(
2505:V
2495:V
2491:d
2474:,
2469:n
2465:V
2459:0
2453:n
2445:=
2442:V
2419:)
2416:V
2413:(
2389:Q
2336:Q
2313:2
2308:i
2304:x
2275:a
2272:2
2268:A
2257:a
2235:0
2232:=
2227:1
2223:A
2201:Q
2190:p
2186:p
2182:A
2178:X
2162:i
2158:A
2134:,
2123:3
2119:A
2110:2
2106:A
2098:Q
2094:=
2091:A
2078:A
2063:Q
2041:X
2023:)
2020:d
2017:(
2007:/
2003:)
2000:d
1997:(
1962:Q
1955:)
1950:n
1946:S
1939:(
1930:H
1899:.
1896:)
1892:Q
1885:)
1882:X
1876:(
1863:(
1860:U
1854:)
1849:Q
1844:,
1841:X
1835:(
1826:H
1808:X
1791:1
1785:n
1760:Q
1753:)
1748:n
1744:S
1737:(
1707:)
1704:X
1698:(
1693:1
1687:i
1676:)
1673:X
1670:(
1665:i
1639:Q
1617:Q
1610:)
1607:X
1601:(
1581:X
1567:X
1535:Q
1513:Q
1506:)
1503:X
1500:(
1470:)
1466:Q
1462:,
1459:X
1456:(
1447:H
1432:X
1410:.
1405:1
1399:n
1396:2
1390:Q
1385:S
1370:5
1364:Q
1359:S
1350:3
1344:Q
1339:S
1329:Q
1324:)
1320:n
1317:(
1298:)
1296:n
1265:Q
1260:X
1233:X
1219:)
1216:1
1210:a
1207:2
1204:,
1200:Q
1196:(
1193:K
1171:1
1165:a
1162:2
1156:Q
1151:S
1101:0
1094:1
1088:a
1085:4
1082:=
1079:i
1071:a
1068:2
1065:=
1062:i
1051:Q
1044:{
1035:Q
1028:)
1023:a
1020:2
1016:S
1012:(
1007:i
962:0
955:1
949:a
946:2
943:=
940:i
929:Q
922:{
913:Q
906:)
901:1
895:a
892:2
888:S
884:(
879:i
821:0
815:i
792:)
787:Q
782:,
779:X
776:(
771:i
767:H
758:Q
750:)
745:Z
740:,
737:X
734:(
729:i
725:H
718:)
713:Z
708:,
702:Q
697:X
693:(
688:i
684:H
655:Q
647:)
644:X
641:(
636:i
625:)
619:Q
614:X
610:(
605:i
574:Q
569:X
562:X
539:Q
534:X
527:X
507:X
487:0
481:i
461:)
457:Z
453:,
450:X
447:(
442:i
438:H
417:X
386:X
359:Q
354:X
330:Q
325:X
318:X
291:Q
286:X
265:X
211:f
196:Q
153:Y
147:X
141:f
116:X
100:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.