3470:
3019:
3465:{\displaystyle E_{2}=E_{3}=E_{4}={\begin{array}{c|ccccc}3&H^{0}(X;\mathbb {Z} )&H^{1}(X;\mathbb {Z} )&H^{2}(X;\mathbb {Z} )&H^{3}(X;\mathbb {Z} )&H^{4}(X;\mathbb {Z} )\\2&0&0&0&0&0\\1&0&0&0&0&0\\0&H^{0}(X;\mathbb {Z} )&H^{1}(X;\mathbb {Z} )&H^{2}(X;\mathbb {Z} )&H^{3}(X;\mathbb {Z} )&H^{4}(X;\mathbb {Z} )\\\hline &0&1&2&3&4\end{array}}}
1601:
1934:
1399:
1746:
3779:
3878:
We know the homology of the base and total space, so our intuition tells us that the Serre spectral sequence should be able to tell us the homology of the loop space. This is an example of a case where we can study the homology of a fibration by using the
3653:
2530:
4769:
This particular example illustrates a systematic technique which one can use in order to deduce information about the higher homotopy groups of spheres. Consider the following fibration which is an isomorphism on
2253:
1596:{\displaystyle {\begin{array}{c|ccc}1&H^{0}(S^{2};\mathbb {Z} )&0&H^{2}(S^{2};\mathbb {Z} )\\0&H^{0}(S^{2};\mathbb {Z} )&0&H^{2}(S^{2};\mathbb {Z} )\\\hline &0&1&2\end{array}}}
1929:{\displaystyle {\begin{array}{c|ccc}1&\ker d_{2}^{0,1}&0&H^{2}(S^{2};\mathbb {Z} )\\0&H^{0}(S^{2};\mathbb {Z} )&0&\operatorname {coker} d_{2}^{0,1}\\\hline &0&1&2\end{array}}}
1198:
236:
707:
4457:
4002:
3003:
973:
5518:
5446:
4997:
4767:
5055:
4143:
3873:
3664:
1298:
383:
4241:
5313:
2850:
2164:
1364:
2078:
4856:
2717:
594:
3525:
5580:
5138:
3063:
2443:
4657:
2377:
5375:
4602:
4387:
2913:
2405:
1989:
837:
484:
2937:
2640:
108:
4693:
4496:
4566:
4343:
4285:
4263:
4042:
5253:
5173:
2321:
4931:
4894:
2609:
4795:
5223:
5104:. Since there is nothing in degree 3 in the total cohomology, we know this must be killed by an isomorphism. But the only element that can map to it is the generator
1637:
1047:
773:
5102:
2880:
2666:
2283:
867:
5200:
5082:
3552:
2752:
1391:
1074:
1010:
740:
511:
266:
1689:
1663:
1751:
1404:
3560:
4933:
to a fibration; it is general knowledge that the iterated fiber is the loop space of the base space so in our example we get that the fiber is
2448:
2169:
1090:
128:
601:
4395:
3893:
2942:
877:
4572: + 1,0 coordinate must be 0. Reading off the horizontal bottom row of the spectral sequence gives us the cohomology ring of
5456:
5384:
4936:
4705:
5002:
3774:{\displaystyle H^{k}(X;\mathbb {Z} )={\begin{cases}\mathbb {Z} &k\in \{0,4\}\\\mathbb {Z} ^{\oplus 22}&k=2\end{cases}}}
4048:+1 and value 0 everywhere else. However, since the path space is contractible, we know that by the time the sequence gets to
4063:
3798:
1210:
780:
295:
4164:
5258:
2757:
1013:
2083:
1316:
5057:
Now we look at the cohomological Serre spectral sequence: we suppose we have a generator for the degree 3 cohomology of
4897:
1994:
5695:
5690:
4802:
2671:
531:
273:
5651:
389:
3478:
5532:
5111:
2410:
36:
4607:
2342:
412:. In particular, their cohomology is isomorphic, so the choice of "the" fiber does not give any ambiguity.
5322:
4469:
page, in the 0,0 coordinate we have the identity of the ring. In the 0,1 coordinate, we have an element
4575:
4360:
2885:
2382:
3790:
1942:
790:
437:
409:
3703:
2918:
2621:
81:
4664:
4476:
281:
119:
44:
4549:
4326:
4268:
4246:
4025:
5228:
5143:
2288:
40:
4903:
4864:
2538:
4773:
3554:
is the square of the
Lefschetz class. In this case, the only non-trivial differential is then
522:
427:
416:
67:
23:
5205:
1609:
1019:
745:
5378:
5087:
4498:
However, we know that by the limit page, there can only be nontrivial generators in degree 2
2855:
2645:
2258:
842:
401:
5178:
5060:
3530:
2730:
1369:
1052:
988:
718:
489:
244:
5634:
1668:
1642:
5319:(i.e., the groups in degrees less than 4) by using the iterated fibration, we know that
5591:
5527:
5523:
431:
405:
1203:
where the notations are dual to the ones above, in particular the differential on the
5684:
5617:
5606:
5602:
The Serre spectral sequence is covered in most textbooks on algebraic topology, e.g.
4145:
to another group. However, the only places a group can be nonzero are in the columns
5611:
5670:
4702:
A more sophisticated application of the Serre spectral sequence is the computation
3658:
We can finish this computation by noting the only nontrivial cohomology groups are
985:-term with (â1) times the cup product, and with respect to which the differentials
241:
Here, at least under standard simplifying conditions, the coefficient group in the
4661:
In the case of infinite complex projective space, taking limits gives the answer
2642:
which is a finite whitney sum of vector bundles we can construct a sphere bundle
5225:
by multiplication by 2 and that the generator of cohomology in degree 6 maps to
2720:
17:
3010:
56:
32:
111:
52:
3648:{\displaystyle d_{4}^{0,3}:H^{0}(X;\mathbb {Z} )\to H^{4}(X;\mathbb {Z} )}
1084:
Similarly to the cohomology spectral sequence, there is one for homology:
396:
given by the cohomology of the various fibers. Assuming for example, that
3883:
page (the homology of the total space) to control what can happen on the
4510:
in the 2,0 coordinate. Now, this tells us that there must be an element
4011:= 0, we are just looking at the regular integer valued homology groups
2525:{\displaystyle s_{0},\ldots ,s_{m}\in \Gamma (X,{\mathcal {O}}_{X}(1))}
5175:. Therefore by the cup product structure, the generator in degree 4,
4060:= 0. The only way this can happen is if there is an isomorphism from
2248:{\displaystyle \operatorname {coker} d_{2}^{0,1}=Gr^{0}H^{2}(S^{3}).}
1193:{\displaystyle E_{p,q}^{2}=H_{p}(B,H_{q}(F))\Rightarrow H_{p+q}(X).}
231:{\displaystyle E_{2}^{p,q}=H^{p}(B,H^{q}(F))\Rightarrow H^{p+q}(X).}
702:{\displaystyle E_{1}^{p,q}=C=\bigoplus _{p,q}H^{q}(X_{p},X_{p-1}),}
388:
Strictly speaking, what is meant is cohomology with respect to the
4452:{\displaystyle S^{1}\hookrightarrow S^{2n+1}\to \mathbb {CP} ^{n}}
2852:. We see that the only non-trivial differentials are given on the
3997:{\displaystyle E_{p,q}^{2}=H_{p}(S^{n+1};H_{q}(\Omega S^{n+1})).}
2998:{\displaystyle {\mathcal {O}}_{X}(a)\oplus {\mathcal {O}}_{X}(b)}
968:{\displaystyle E_{r}^{p,q}\times E_{r}^{s,t}\to E_{r}^{p+s,q+t},}
5658:
4526:
by the
Leibniz rule telling us that the 4,0 coordinate must be
2719:. Then, we can use the Serre spectral sequence along with the
5638:
gives many nice applications of the Serre spectral sequence.
2975:
2949:
2924:
2897:
2627:
2499:
2349:
5513:{\displaystyle \pi _{4}(S^{2})=\mathbb {Z} /2\mathbb {Z} .}
5441:{\displaystyle \pi _{4}(S^{3})=\mathbb {Z} /2\mathbb {Z} .}
4992:{\displaystyle \Omega K(\mathbb {Z} ,3)=K(\mathbb {Z} ,2).}
4762:{\displaystyle \pi _{4}(S^{3})=\mathbb {Z} /2\mathbb {Z} .}
3767:
1691:
right. Thus the only differential which is not necessarily
122:. The Serre cohomology spectral sequence is the following:
5050:{\displaystyle K(\mathbb {Z} ,2)=\mathbb {CP} ^{\infty }.}
5315:
But now since we killed off the lower homotopy groups of
288:
with coefficients in that group. The differential on the
5255:
by multiplication by 3, etc. In particular we find that
4530:
since there can be no nontrivial homology until degree 2
4138:{\displaystyle H_{n+1}(S^{n+1};H_{0}(F))=\mathbb {Z} }
3868:{\displaystyle \Omega S^{n+1}\to PS^{n+1}\to S^{n+1}.}
2882:-page and are defined by cupping with the Euler class
1293:{\displaystyle d_{k}:E_{p,q}^{k}\to E_{p-k,q-1+k}^{k}}
378:{\displaystyle d_{k}:E_{k}^{p,q}\to E_{k}^{p+k,q+1-k}}
5535:
5459:
5387:
5325:
5261:
5231:
5208:
5181:
5146:
5114:
5090:
5063:
5005:
4939:
4906:
4867:
4805:
4776:
4708:
4667:
4610:
4578:
4552:
4479:
4398:
4363:
4329:
4271:
4249:
4236:{\displaystyle H_{0}(S^{n+1};H_{n}(F))=\mathbb {Z} .}
4167:
4066:
4028:
3896:
3801:
3667:
3563:
3533:
3481:
3022:
2945:
2921:
2915:. In this case it is given by the top chern class of
2888:
2858:
2760:
2733:
2674:
2648:
2624:
2541:
2451:
2413:
2385:
2345:
2291:
2261:
2172:
2086:
1997:
1945:
1749:
1671:
1645:
1612:
1402:
1372:
1319:
1213:
1093:
1055:
1022:
991:
880:
845:
793:
748:
721:
604:
534:
492:
440:
298:
247:
131:
84:
5308:{\displaystyle H_{4}(X)=\mathbb {Z} /2\mathbb {Z} .}
2845:{\displaystyle E_{2}^{p,q}=H^{p}(X;H^{q}(S^{2r-1}))}
1719:-page). In particular, this sequence degenerates at
2159:{\displaystyle \ker d_{2}^{0,1}=Gr^{1}H^{1}(S^{3})}
1359:{\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2}}
5574:
5512:
5440:
5369:
5307:
5247:
5217:
5194:
5167:
5132:
5096:
5076:
5049:
4991:
4925:
4888:
4850:
4789:
4761:
4687:
4651:
4596:
4560:
4490:
4451:
4381:
4337:
4312:)). Inductively repeating this process shows that
4279:
4257:
4235:
4137:
4036:
3996:
3867:
3789:We begin first with a basic example; consider the
3773:
3647:
3546:
3519:
3464:
2997:
2931:
2907:
2874:
2844:
2746:
2711:
2660:
2634:
2603:
2524:
2437:
2399:
2371:
2315:
2277:
2247:
2158:
2072:
1983:
1928:
1683:
1657:
1631:
1595:
1385:
1358:
1292:
1192:
1068:
1041:
1004:
967:
861:
831:
767:
734:
701:
588:
505:
478:
377:
260:
230:
102:
2073:{\displaystyle E_{3}^{p,q}=Gr^{p}H^{p+q}(S^{3}).}
1393:-page of the LerayâSerre Spectral sequence reads
4851:{\displaystyle X\to S^{3}\to K(\mathbb {Z} ,3),}
2712:{\displaystyle S^{2r-1}\subset \mathbb {C} ^{r}}
404:, this collapses to the usual cohomology. For a
4534:+1. Repeating this argument inductively until 2
4052:, everything becomes 0 except for the group at
589:{\displaystyle A=\bigoplus _{p,q}H^{q}(X_{p}),}
47:, the singular (co)homology of the total space
4157:+1 so this isomorphism must occur on the page
426:This spectral sequence can be derived from an
2327:Sphere bundle on a complex projective variety
2080:Evaluating at the interesting parts, we have
513:is the restriction of the fibration over the
419:means integral cohomology of the total space
8:
4546:,1 which must then be the only generator of
3731:
3719:
2339:there is a canonical family of line bundles
5665:The case of simplicial sets is treated in
4353:Cohomology ring of complex projective space
1704:, because the rest have domain or codomain
1313:Recall that the Hopf fibration is given by
779:is defined using the coboundary map in the
2939:. For example, consider the vector bundle
5566:
5553:
5540:
5534:
5503:
5502:
5494:
5490:
5489:
5477:
5464:
5458:
5431:
5430:
5422:
5418:
5417:
5405:
5392:
5386:
5352:
5330:
5324:
5298:
5297:
5289:
5285:
5284:
5266:
5260:
5239:
5230:
5207:
5186:
5180:
5145:
5124:
5120:
5117:
5116:
5113:
5089:
5068:
5062:
5038:
5034:
5031:
5030:
5013:
5012:
5004:
4973:
4972:
4950:
4949:
4938:
4917:
4905:
4866:
4832:
4831:
4816:
4804:
4781:
4775:
4752:
4751:
4743:
4739:
4738:
4726:
4713:
4707:
4698:Fourth homotopy group of the three-sphere
4669:
4668:
4666:
4634:
4625:
4612:
4611:
4609:
4588:
4584:
4581:
4580:
4577:
4568:in that degree thus telling us that the 2
4554:
4553:
4551:
4481:
4480:
4478:
4443:
4439:
4436:
4435:
4416:
4403:
4397:
4373:
4369:
4366:
4365:
4362:
4331:
4330:
4328:
4273:
4272:
4270:
4251:
4250:
4248:
4226:
4225:
4204:
4185:
4172:
4166:
4131:
4130:
4109:
4090:
4071:
4065:
4030:
4029:
4027:
3973:
3957:
3938:
3925:
3912:
3901:
3895:
3850:
3831:
3809:
3800:
3744:
3740:
3739:
3707:
3706:
3698:
3688:
3687:
3672:
3666:
3638:
3637:
3622:
3608:
3607:
3592:
3573:
3568:
3562:
3538:
3532:
3511:
3486:
3480:
3423:
3422:
3407:
3394:
3393:
3378:
3365:
3364:
3349:
3336:
3335:
3320:
3307:
3306:
3291:
3207:
3206:
3191:
3178:
3177:
3162:
3149:
3148:
3133:
3120:
3119:
3104:
3091:
3090:
3075:
3062:
3053:
3040:
3027:
3021:
2980:
2974:
2973:
2954:
2948:
2947:
2944:
2923:
2922:
2920:
2896:
2895:
2887:
2863:
2857:
2821:
2808:
2789:
2770:
2765:
2759:
2738:
2732:
2703:
2699:
2698:
2679:
2673:
2647:
2626:
2625:
2623:
2583:
2555:
2540:
2504:
2498:
2497:
2475:
2456:
2450:
2429:
2425:
2422:
2421:
2412:
2393:
2392:
2384:
2354:
2348:
2347:
2344:
2301:
2296:
2290:
2266:
2260:
2233:
2220:
2210:
2188:
2183:
2171:
2147:
2134:
2124:
2102:
2097:
2085:
2058:
2039:
2029:
2007:
2002:
1996:
1969:
1950:
1944:
1892:
1887:
1863:
1862:
1853:
1840:
1820:
1819:
1810:
1797:
1774:
1769:
1750:
1748:
1670:
1644:
1617:
1611:
1564:
1563:
1554:
1541:
1523:
1522:
1513:
1500:
1480:
1479:
1470:
1457:
1439:
1438:
1429:
1416:
1403:
1401:
1377:
1371:
1350:
1337:
1324:
1318:
1284:
1255:
1242:
1231:
1218:
1212:
1166:
1141:
1122:
1109:
1098:
1092:
1060:
1054:
1027:
1021:
996:
990:
938:
933:
914:
909:
890:
885:
879:
850:
844:
814:
801:
792:
753:
747:
726:
720:
681:
668:
655:
639:
614:
609:
603:
574:
561:
545:
533:
497:
491:
461:
448:
439:
345:
340:
321:
316:
303:
297:
252:
246:
204:
179:
160:
141:
136:
130:
83:
4514:in the 2,1 coordinate. We then see that
3520:{\displaystyle d_{4}=\cup a\cdot bH^{2}}
715:is defined by restricting each piece on
5575:{\displaystyle S^{1}\to S^{3}\to S^{2}}
5133:{\displaystyle \mathbb {CP} ^{\infty }}
3013:. Then, the spectral sequence reads as
2445:. This is given by the global sections
5524:long exact sequence of homotopy groups
2723:to compute the integral cohomology of
2438:{\displaystyle X\to \mathbb {CP} ^{m}}
4652:{\displaystyle \mathbb {Z} /x^{n+1}.}
2372:{\displaystyle {\mathcal {O}}_{X}(k)}
7:
5370:{\displaystyle H_{4}(X)=\pi _{4}(X)}
4265:in this group means there must be a
871:There is a multiplicative structure
55:in terms of the (co)homology of the
5635:Lecture notes in algebraic topology
4604:and it tells us that the answer is
2285:both are zero, so the differential
408:base, all the different fibers are
43:. It expresses, in the language of
5648:Zur Spektralsequenz einer Faserung
5642:An elegant construction is due to
5125:
5039:
4940:
4900:. We then further convert the map
3966:
3802:
2484:
14:
4597:{\displaystyle \mathbb {CP} ^{n}}
4502:+1 telling us that the generator
4382:{\displaystyle \mathbb {CP} ^{n}}
2908:{\displaystyle e({\mathcal {E}})}
2400:{\displaystyle k\in \mathbb {Z} }
4506:must transgress to some element
2335:-dimensional projective variety
1984:{\displaystyle H^{p+q}(S^{3}),}
1939:The spectral sequence abuts to
832:{\displaystyle (X_{p},X_{p-1})}
781:long exact sequence of the pair
479:{\displaystyle (X_{p},X_{p-1})}
114:of topological spaces, and let
31:to acknowledge earlier work of
5559:
5546:
5483:
5470:
5411:
5398:
5364:
5358:
5342:
5336:
5278:
5272:
5156:
5150:
5023:
5009:
4983:
4969:
4960:
4946:
4910:
4883:
4871:
4842:
4828:
4822:
4809:
4732:
4719:
4679:
4673:
4622:
4616:
4431:
4409:
4219:
4216:
4210:
4178:
4124:
4121:
4115:
4083:
3988:
3985:
3963:
3931:
3843:
3821:
3692:
3678:
3642:
3628:
3615:
3612:
3598:
3427:
3413:
3398:
3384:
3369:
3355:
3340:
3326:
3311:
3297:
3211:
3197:
3182:
3168:
3153:
3139:
3124:
3110:
3095:
3081:
2992:
2986:
2966:
2960:
2932:{\displaystyle {\mathcal {E}}}
2902:
2892:
2839:
2836:
2814:
2795:
2652:
2635:{\displaystyle {\mathcal {E}}}
2598:
2595:
2589:
2567:
2561:
2548:
2545:
2519:
2516:
2510:
2487:
2417:
2366:
2360:
2239:
2226:
2153:
2140:
2064:
2051:
1975:
1962:
1867:
1846:
1824:
1803:
1568:
1547:
1527:
1506:
1484:
1463:
1443:
1422:
1343:
1330:
1248:
1184:
1178:
1159:
1156:
1153:
1147:
1128:
926:
826:
794:
693:
661:
580:
567:
473:
441:
434:of the cohomology of the pair
333:
222:
216:
197:
194:
191:
185:
166:
103:{\displaystyle f\colon X\to B}
94:
70:in his doctoral dissertation.
1:
4688:{\displaystyle \mathbb {Z} .}
4491:{\displaystyle \mathbb {Z} .}
4357:We compute the cohomology of
2668:whose fibers are the spheres
280:, and the outer group is the
29:LerayâSerre spectral sequence
4561:{\displaystyle \mathbb {Z} }
4338:{\displaystyle \mathbb {Z} }
4280:{\displaystyle \mathbb {Z} }
4258:{\displaystyle \mathbb {Z} }
4037:{\displaystyle \mathbb {Z} }
1016:inducing the product on the
74:Cohomology spectral sequence
5248:{\displaystyle \iota a^{2}}
5168:{\displaystyle d(a)=\iota }
2316:{\displaystyle d_{2}^{0,1}}
5712:
5675:Simplicial homotopy theory
5108:of the cohomology ring of
4926:{\displaystyle X\to S^{3}}
2407:coming from the embedding
2255:Knowing the cohomology of
1080:Homology spectral sequence
1049:-page from the one on the
787:is defined by restricting
39:) is an important tool in
5632:James Davis, Paul Kirk,
4889:{\displaystyle K(\pi ,n)}
3785:Basic pathspace fibration
2604:{\displaystyle x\mapsto }
521:. More precisely, using
274:integral cohomology group
5652:Inventiones Mathematicae
5202:, maps to the generator
4790:{\displaystyle \pi _{3}}
4345:at integer multiples of
390:local coefficient system
118:be the (path-connected)
5218:{\displaystyle \iota a}
4898:EilenbergâMacLane space
4349:and 0 everywhere else.
2614:If we construct a rank
1632:{\displaystyle d_{2+i}}
1042:{\displaystyle E_{r+1}}
768:{\displaystyle X_{p-1}}
66:. The result is due to
37:Leray spectral sequence
5576:
5514:
5442:
5371:
5309:
5249:
5219:
5196:
5169:
5134:
5098:
5097:{\displaystyle \iota }
5078:
5051:
4993:
4927:
4890:
4852:
4791:
4763:
4689:
4653:
4598:
4562:
4492:
4453:
4383:
4339:
4281:
4259:
4237:
4139:
4038:
3998:
3887:page. So recall that
3869:
3775:
3649:
3548:
3521:
3466:
2999:
2933:
2909:
2876:
2875:{\displaystyle E_{2r}}
2846:
2748:
2713:
2662:
2661:{\displaystyle S\to X}
2636:
2605:
2526:
2439:
2401:
2373:
2317:
2279:
2278:{\displaystyle S^{3},}
2249:
2160:
2074:
1985:
1930:
1685:
1659:
1633:
1597:
1387:
1360:
1294:
1194:
1070:
1043:
1006:
969:
863:
862:{\displaystyle X_{p}.}
833:
769:
736:
703:
590:
507:
480:
379:
262:
232:
104:
5577:
5515:
5443:
5372:
5310:
5250:
5220:
5197:
5195:{\displaystyle a^{2}}
5170:
5135:
5099:
5079:
5077:{\displaystyle S^{3}}
5052:
4994:
4928:
4891:
4853:
4792:
4764:
4690:
4654:
4599:
4563:
4538: + 1 gives
4493:
4454:
4389:using the fibration:
4384:
4340:
4282:
4260:
4238:
4140:
4039:
3999:
3870:
3776:
3650:
3549:
3547:{\displaystyle H^{2}}
3522:
3467:
3000:
2934:
2910:
2877:
2847:
2749:
2747:{\displaystyle E_{2}}
2714:
2663:
2637:
2606:
2527:
2440:
2402:
2374:
2318:
2280:
2250:
2161:
2075:
1986:
1931:
1686:
1660:
1634:
1598:
1388:
1386:{\displaystyle E_{2}}
1361:
1295:
1195:
1071:
1069:{\displaystyle E_{r}}
1044:
1007:
1005:{\displaystyle d_{r}}
970:
864:
834:
770:
737:
735:{\displaystyle X_{p}}
704:
591:
508:
506:{\displaystyle X_{p}}
481:
380:
263:
261:{\displaystyle E_{2}}
233:
105:
5533:
5457:
5385:
5323:
5259:
5229:
5206:
5179:
5144:
5112:
5088:
5061:
5003:
4937:
4904:
4865:
4803:
4774:
4706:
4665:
4608:
4576:
4550:
4477:
4396:
4361:
4327:
4269:
4247:
4165:
4064:
4026:
3894:
3799:
3791:path space fibration
3665:
3561:
3531:
3479:
3020:
2943:
2919:
2886:
2856:
2758:
2731:
2672:
2646:
2622:
2539:
2449:
2411:
2383:
2343:
2289:
2259:
2170:
2084:
1995:
1943:
1747:
1669:
1643:
1610:
1400:
1370:
1317:
1304:Example computations
1211:
1091:
1053:
1020:
1014:(graded) derivations
989:
878:
843:
791:
746:
719:
602:
532:
490:
438:
432:long exact sequences
296:
245:
129:
82:
4243:However, putting a
3917:
3584:
2781:
2323:is an isomorphism.
2312:
2199:
2113:
2018:
1903:
1785:
1684:{\displaystyle 2+i}
1658:{\displaystyle 1+i}
1289:
1247:
1114:
961:
925:
901:
625:
410:homotopy equivalent
374:
332:
282:singular cohomology
152:
45:homological algebra
5696:Spectral sequences
5691:Algebraic topology
5622:Algebraic topology
5612:Spectral Sequences
5572:
5510:
5438:
5381:, telling us that
5367:
5305:
5245:
5215:
5192:
5165:
5130:
5094:
5074:
5047:
4989:
4923:
4886:
4848:
4787:
4759:
4685:
4649:
4594:
4558:
4488:
4449:
4379:
4335:
4277:
4255:
4233:
4135:
4034:
4022:) which has value
4007:Thus we know when
3994:
3897:
3865:
3771:
3766:
3645:
3564:
3544:
3517:
3462:
3460:
2995:
2929:
2905:
2872:
2842:
2761:
2754:-page is given by
2744:
2709:
2658:
2632:
2601:
2522:
2435:
2397:
2369:
2313:
2292:
2275:
2245:
2179:
2156:
2093:
2070:
1998:
1981:
1926:
1924:
1883:
1765:
1681:
1655:
1629:
1593:
1591:
1383:
1356:
1290:
1251:
1227:
1190:
1094:
1066:
1039:
1002:
978:coinciding on the
965:
929:
905:
881:
859:
829:
765:
732:
699:
650:
605:
586:
556:
503:
476:
375:
336:
312:
258:
228:
132:
100:
41:algebraic topology
5657:(1967), 172â178,
4999:But we know that
4044:in degrees 0 and
3475:The differential
1606:The differential
1207:th page is a map
635:
541:
430:built out of the
68:Jean-Pierre Serre
24:spectral sequence
5703:
5581:
5579:
5578:
5573:
5571:
5570:
5558:
5557:
5545:
5544:
5522:Proof: Take the
5519:
5517:
5516:
5511:
5506:
5498:
5493:
5482:
5481:
5469:
5468:
5447:
5445:
5444:
5439:
5434:
5426:
5421:
5410:
5409:
5397:
5396:
5379:Hurewicz theorem
5376:
5374:
5373:
5368:
5357:
5356:
5335:
5334:
5314:
5312:
5311:
5306:
5301:
5293:
5288:
5271:
5270:
5254:
5252:
5251:
5246:
5244:
5243:
5224:
5222:
5221:
5216:
5201:
5199:
5198:
5193:
5191:
5190:
5174:
5172:
5171:
5166:
5139:
5137:
5136:
5131:
5129:
5128:
5123:
5103:
5101:
5100:
5095:
5083:
5081:
5080:
5075:
5073:
5072:
5056:
5054:
5053:
5048:
5043:
5042:
5037:
5016:
4998:
4996:
4995:
4990:
4976:
4953:
4932:
4930:
4929:
4924:
4922:
4921:
4895:
4893:
4892:
4887:
4857:
4855:
4854:
4849:
4835:
4821:
4820:
4796:
4794:
4793:
4788:
4786:
4785:
4768:
4766:
4765:
4760:
4755:
4747:
4742:
4731:
4730:
4718:
4717:
4694:
4692:
4691:
4686:
4672:
4658:
4656:
4655:
4650:
4645:
4644:
4629:
4615:
4603:
4601:
4600:
4595:
4593:
4592:
4587:
4567:
4565:
4564:
4559:
4557:
4497:
4495:
4494:
4489:
4484:
4458:
4456:
4455:
4450:
4448:
4447:
4442:
4430:
4429:
4408:
4407:
4388:
4386:
4385:
4380:
4378:
4377:
4372:
4344:
4342:
4341:
4336:
4334:
4286:
4284:
4283:
4278:
4276:
4264:
4262:
4261:
4256:
4254:
4242:
4240:
4239:
4234:
4229:
4209:
4208:
4196:
4195:
4177:
4176:
4144:
4142:
4141:
4136:
4134:
4114:
4113:
4101:
4100:
4082:
4081:
4043:
4041:
4040:
4035:
4033:
4003:
4001:
4000:
3995:
3984:
3983:
3962:
3961:
3949:
3948:
3930:
3929:
3916:
3911:
3874:
3872:
3871:
3866:
3861:
3860:
3842:
3841:
3820:
3819:
3780:
3778:
3777:
3772:
3770:
3769:
3752:
3751:
3743:
3710:
3691:
3677:
3676:
3654:
3652:
3651:
3646:
3641:
3627:
3626:
3611:
3597:
3596:
3583:
3572:
3553:
3551:
3550:
3545:
3543:
3542:
3526:
3524:
3523:
3518:
3516:
3515:
3491:
3490:
3471:
3469:
3468:
3463:
3461:
3433:
3426:
3412:
3411:
3397:
3383:
3382:
3368:
3354:
3353:
3339:
3325:
3324:
3310:
3296:
3295:
3210:
3196:
3195:
3181:
3167:
3166:
3152:
3138:
3137:
3123:
3109:
3108:
3094:
3080:
3079:
3058:
3057:
3045:
3044:
3032:
3031:
3008:
3004:
3002:
3001:
2996:
2985:
2984:
2979:
2978:
2959:
2958:
2953:
2952:
2938:
2936:
2935:
2930:
2928:
2927:
2914:
2912:
2911:
2906:
2901:
2900:
2881:
2879:
2878:
2873:
2871:
2870:
2851:
2849:
2848:
2843:
2835:
2834:
2813:
2812:
2794:
2793:
2780:
2769:
2753:
2751:
2750:
2745:
2743:
2742:
2726:
2718:
2716:
2715:
2710:
2708:
2707:
2702:
2693:
2692:
2667:
2665:
2664:
2659:
2641:
2639:
2638:
2633:
2631:
2630:
2617:
2610:
2608:
2607:
2602:
2588:
2587:
2560:
2559:
2531:
2529:
2528:
2523:
2509:
2508:
2503:
2502:
2480:
2479:
2461:
2460:
2444:
2442:
2441:
2436:
2434:
2433:
2428:
2406:
2404:
2403:
2398:
2396:
2378:
2376:
2375:
2370:
2359:
2358:
2353:
2352:
2338:
2331:Given a complex
2322:
2320:
2319:
2314:
2311:
2300:
2284:
2282:
2281:
2276:
2271:
2270:
2254:
2252:
2251:
2246:
2238:
2237:
2225:
2224:
2215:
2214:
2198:
2187:
2165:
2163:
2162:
2157:
2152:
2151:
2139:
2138:
2129:
2128:
2112:
2101:
2079:
2077:
2076:
2071:
2063:
2062:
2050:
2049:
2034:
2033:
2017:
2006:
1990:
1988:
1987:
1982:
1974:
1973:
1961:
1960:
1935:
1933:
1932:
1927:
1925:
1907:
1902:
1891:
1866:
1858:
1857:
1845:
1844:
1823:
1815:
1814:
1802:
1801:
1784:
1773:
1711:
1708:(since they are
1703:
1694:
1690:
1688:
1687:
1682:
1664:
1662:
1661:
1656:
1638:
1636:
1635:
1630:
1628:
1627:
1602:
1600:
1599:
1594:
1592:
1574:
1567:
1559:
1558:
1546:
1545:
1526:
1518:
1517:
1505:
1504:
1483:
1475:
1474:
1462:
1461:
1442:
1434:
1433:
1421:
1420:
1392:
1390:
1389:
1384:
1382:
1381:
1365:
1363:
1362:
1357:
1355:
1354:
1342:
1341:
1329:
1328:
1299:
1297:
1296:
1291:
1288:
1283:
1246:
1241:
1223:
1222:
1199:
1197:
1196:
1191:
1177:
1176:
1146:
1145:
1127:
1126:
1113:
1108:
1075:
1073:
1072:
1067:
1065:
1064:
1048:
1046:
1045:
1040:
1038:
1037:
1011:
1009:
1008:
1003:
1001:
1000:
974:
972:
971:
966:
960:
937:
924:
913:
900:
889:
868:
866:
865:
860:
855:
854:
838:
836:
835:
830:
825:
824:
806:
805:
774:
772:
771:
766:
764:
763:
741:
739:
738:
733:
731:
730:
708:
706:
705:
700:
692:
691:
673:
672:
660:
659:
649:
624:
613:
598:
595:
593:
592:
587:
579:
578:
566:
565:
555:
512:
510:
509:
504:
502:
501:
485:
483:
482:
477:
472:
471:
453:
452:
402:simply connected
384:
382:
381:
376:
373:
344:
331:
320:
308:
307:
267:
265:
264:
259:
257:
256:
237:
235:
234:
229:
215:
214:
184:
183:
165:
164:
151:
140:
109:
107:
106:
101:
5711:
5710:
5706:
5705:
5704:
5702:
5701:
5700:
5681:
5680:
5646:Andreas Dress,
5600:
5588:
5562:
5549:
5536:
5531:
5530:
5473:
5460:
5455:
5454:
5401:
5388:
5383:
5382:
5348:
5326:
5321:
5320:
5262:
5257:
5256:
5235:
5227:
5226:
5204:
5203:
5182:
5177:
5176:
5142:
5141:
5115:
5110:
5109:
5086:
5085:
5064:
5059:
5058:
5029:
5001:
5000:
4935:
4934:
4913:
4902:
4901:
4863:
4862:
4812:
4801:
4800:
4777:
4772:
4771:
4722:
4709:
4704:
4703:
4700:
4663:
4662:
4630:
4606:
4605:
4579:
4574:
4573:
4548:
4547:
4542:in coordinate 2
4475:
4474:
4473:that generates
4468:
4434:
4412:
4399:
4394:
4393:
4364:
4359:
4358:
4355:
4325:
4324:
4317:
4306:
4296:
4267:
4266:
4245:
4244:
4200:
4181:
4168:
4163:
4162:
4105:
4086:
4067:
4062:
4061:
4024:
4023:
4016:
3969:
3953:
3934:
3921:
3892:
3891:
3846:
3827:
3805:
3797:
3796:
3787:
3765:
3764:
3753:
3738:
3735:
3734:
3711:
3699:
3668:
3663:
3662:
3618:
3588:
3559:
3558:
3534:
3529:
3528:
3507:
3482:
3477:
3476:
3459:
3458:
3453:
3448:
3443:
3438:
3431:
3430:
3403:
3401:
3374:
3372:
3345:
3343:
3316:
3314:
3287:
3285:
3279:
3278:
3273:
3268:
3263:
3258:
3253:
3247:
3246:
3241:
3236:
3231:
3226:
3221:
3215:
3214:
3187:
3185:
3158:
3156:
3129:
3127:
3100:
3098:
3071:
3069:
3049:
3036:
3023:
3018:
3017:
3006:
2972:
2946:
2941:
2940:
2917:
2916:
2884:
2883:
2859:
2854:
2853:
2817:
2804:
2785:
2756:
2755:
2734:
2729:
2728:
2724:
2697:
2675:
2670:
2669:
2644:
2643:
2620:
2619:
2615:
2579:
2551:
2537:
2536:
2496:
2471:
2452:
2447:
2446:
2420:
2409:
2408:
2381:
2380:
2346:
2341:
2340:
2336:
2329:
2287:
2286:
2262:
2257:
2256:
2229:
2216:
2206:
2168:
2167:
2143:
2130:
2120:
2082:
2081:
2054:
2035:
2025:
1993:
1992:
1965:
1946:
1941:
1940:
1923:
1922:
1917:
1912:
1905:
1904:
1875:
1870:
1849:
1836:
1834:
1828:
1827:
1806:
1793:
1791:
1786:
1757:
1745:
1744:
1739:
1732:
1725:
1718:
1709:
1702:
1696:
1692:
1667:
1666:
1641:
1640:
1613:
1608:
1607:
1590:
1589:
1584:
1579:
1572:
1571:
1550:
1537:
1535:
1530:
1509:
1496:
1494:
1488:
1487:
1466:
1453:
1451:
1446:
1425:
1412:
1410:
1398:
1397:
1373:
1368:
1367:
1346:
1333:
1320:
1315:
1314:
1311:
1306:
1214:
1209:
1208:
1162:
1137:
1118:
1089:
1088:
1082:
1056:
1051:
1050:
1023:
1018:
1017:
992:
987:
986:
984:
876:
875:
846:
841:
840:
810:
797:
789:
788:
749:
744:
743:
722:
717:
716:
677:
664:
651:
600:
599:
596:
570:
557:
530:
529:
493:
488:
487:
457:
444:
436:
435:
299:
294:
293:
248:
243:
242:
200:
175:
156:
127:
126:
112:Serre fibration
80:
79:
76:
12:
11:
5:
5709:
5707:
5699:
5698:
5693:
5683:
5682:
5679:
5678:
5663:
5662:
5640:
5639:
5626:
5625:
5615:
5599:
5596:
5595:
5594:
5592:Gysin sequence
5587:
5584:
5569:
5565:
5561:
5556:
5552:
5548:
5543:
5539:
5528:Hopf fibration
5509:
5505:
5501:
5497:
5492:
5488:
5485:
5480:
5476:
5472:
5467:
5463:
5437:
5433:
5429:
5425:
5420:
5416:
5413:
5408:
5404:
5400:
5395:
5391:
5366:
5363:
5360:
5355:
5351:
5347:
5344:
5341:
5338:
5333:
5329:
5304:
5300:
5296:
5292:
5287:
5283:
5280:
5277:
5274:
5269:
5265:
5242:
5238:
5234:
5214:
5211:
5189:
5185:
5164:
5161:
5158:
5155:
5152:
5149:
5127:
5122:
5119:
5093:
5071:
5067:
5046:
5041:
5036:
5033:
5028:
5025:
5022:
5019:
5015:
5011:
5008:
4988:
4985:
4982:
4979:
4975:
4971:
4968:
4965:
4962:
4959:
4956:
4952:
4948:
4945:
4942:
4920:
4916:
4912:
4909:
4885:
4882:
4879:
4876:
4873:
4870:
4859:
4858:
4847:
4844:
4841:
4838:
4834:
4830:
4827:
4824:
4819:
4815:
4811:
4808:
4784:
4780:
4758:
4754:
4750:
4746:
4741:
4737:
4734:
4729:
4725:
4721:
4716:
4712:
4699:
4696:
4684:
4681:
4678:
4675:
4671:
4648:
4643:
4640:
4637:
4633:
4628:
4624:
4621:
4618:
4614:
4591:
4586:
4583:
4556:
4487:
4483:
4466:
4460:
4459:
4446:
4441:
4438:
4433:
4428:
4425:
4422:
4419:
4415:
4411:
4406:
4402:
4376:
4371:
4368:
4354:
4351:
4333:
4315:
4304:
4291:
4275:
4253:
4232:
4228:
4224:
4221:
4218:
4215:
4212:
4207:
4203:
4199:
4194:
4191:
4188:
4184:
4180:
4175:
4171:
4161:with codomain
4133:
4129:
4126:
4123:
4120:
4117:
4112:
4108:
4104:
4099:
4096:
4093:
4089:
4085:
4080:
4077:
4074:
4070:
4032:
4014:
4005:
4004:
3993:
3990:
3987:
3982:
3979:
3976:
3972:
3968:
3965:
3960:
3956:
3952:
3947:
3944:
3941:
3937:
3933:
3928:
3924:
3920:
3915:
3910:
3907:
3904:
3900:
3876:
3875:
3864:
3859:
3856:
3853:
3849:
3845:
3840:
3837:
3834:
3830:
3826:
3823:
3818:
3815:
3812:
3808:
3804:
3786:
3783:
3782:
3781:
3768:
3763:
3760:
3757:
3754:
3750:
3747:
3742:
3737:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3705:
3704:
3702:
3697:
3694:
3690:
3686:
3683:
3680:
3675:
3671:
3656:
3655:
3644:
3640:
3636:
3633:
3630:
3625:
3621:
3617:
3614:
3610:
3606:
3603:
3600:
3595:
3591:
3587:
3582:
3579:
3576:
3571:
3567:
3541:
3537:
3514:
3510:
3506:
3503:
3500:
3497:
3494:
3489:
3485:
3473:
3472:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3425:
3421:
3418:
3415:
3410:
3406:
3402:
3400:
3396:
3392:
3389:
3386:
3381:
3377:
3373:
3371:
3367:
3363:
3360:
3357:
3352:
3348:
3344:
3342:
3338:
3334:
3331:
3328:
3323:
3319:
3315:
3313:
3309:
3305:
3302:
3299:
3294:
3290:
3286:
3284:
3281:
3280:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3249:
3248:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3216:
3213:
3209:
3205:
3202:
3199:
3194:
3190:
3186:
3184:
3180:
3176:
3173:
3170:
3165:
3161:
3157:
3155:
3151:
3147:
3144:
3141:
3136:
3132:
3128:
3126:
3122:
3118:
3115:
3112:
3107:
3103:
3099:
3097:
3093:
3089:
3086:
3083:
3078:
3074:
3070:
3068:
3065:
3064:
3061:
3056:
3052:
3048:
3043:
3039:
3035:
3030:
3026:
2994:
2991:
2988:
2983:
2977:
2971:
2968:
2965:
2962:
2957:
2951:
2926:
2904:
2899:
2894:
2891:
2869:
2866:
2862:
2841:
2838:
2833:
2830:
2827:
2824:
2820:
2816:
2811:
2807:
2803:
2800:
2797:
2792:
2788:
2784:
2779:
2776:
2773:
2768:
2764:
2741:
2737:
2706:
2701:
2696:
2691:
2688:
2685:
2682:
2678:
2657:
2654:
2651:
2629:
2618:vector bundle
2612:
2611:
2600:
2597:
2594:
2591:
2586:
2582:
2578:
2575:
2572:
2569:
2566:
2563:
2558:
2554:
2550:
2547:
2544:
2521:
2518:
2515:
2512:
2507:
2501:
2495:
2492:
2489:
2486:
2483:
2478:
2474:
2470:
2467:
2464:
2459:
2455:
2432:
2427:
2424:
2419:
2416:
2395:
2391:
2388:
2368:
2365:
2362:
2357:
2351:
2328:
2325:
2310:
2307:
2304:
2299:
2295:
2274:
2269:
2265:
2244:
2241:
2236:
2232:
2228:
2223:
2219:
2213:
2209:
2205:
2202:
2197:
2194:
2191:
2186:
2182:
2178:
2175:
2155:
2150:
2146:
2142:
2137:
2133:
2127:
2123:
2119:
2116:
2111:
2108:
2105:
2100:
2096:
2092:
2089:
2069:
2066:
2061:
2057:
2053:
2048:
2045:
2042:
2038:
2032:
2028:
2024:
2021:
2016:
2013:
2010:
2005:
2001:
1980:
1977:
1972:
1968:
1964:
1959:
1956:
1953:
1949:
1937:
1936:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1901:
1898:
1895:
1890:
1886:
1882:
1879:
1876:
1874:
1871:
1869:
1865:
1861:
1856:
1852:
1848:
1843:
1839:
1835:
1833:
1830:
1829:
1826:
1822:
1818:
1813:
1809:
1805:
1800:
1796:
1792:
1790:
1787:
1783:
1780:
1777:
1772:
1768:
1764:
1761:
1758:
1756:
1753:
1752:
1737:
1730:
1723:
1716:
1700:
1680:
1677:
1674:
1654:
1651:
1648:
1626:
1623:
1620:
1616:
1604:
1603:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1566:
1562:
1557:
1553:
1549:
1544:
1540:
1536:
1534:
1531:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1495:
1493:
1490:
1489:
1486:
1482:
1478:
1473:
1469:
1465:
1460:
1456:
1452:
1450:
1447:
1445:
1441:
1437:
1432:
1428:
1424:
1419:
1415:
1411:
1409:
1406:
1405:
1380:
1376:
1353:
1349:
1345:
1340:
1336:
1332:
1327:
1323:
1310:
1309:Hopf fibration
1307:
1305:
1302:
1287:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1254:
1250:
1245:
1240:
1237:
1234:
1230:
1226:
1221:
1217:
1201:
1200:
1189:
1186:
1183:
1180:
1175:
1172:
1169:
1165:
1161:
1158:
1155:
1152:
1149:
1144:
1140:
1136:
1133:
1130:
1125:
1121:
1117:
1112:
1107:
1104:
1101:
1097:
1081:
1078:
1063:
1059:
1036:
1033:
1030:
1026:
999:
995:
982:
976:
975:
964:
959:
956:
953:
950:
947:
944:
941:
936:
932:
928:
923:
920:
917:
912:
908:
904:
899:
896:
893:
888:
884:
858:
853:
849:
828:
823:
820:
817:
813:
809:
804:
800:
796:
762:
759:
756:
752:
729:
725:
710:
709:
698:
695:
690:
687:
684:
680:
676:
671:
667:
663:
658:
654:
648:
645:
642:
638:
634:
631:
628:
623:
620:
617:
612:
608:
585:
582:
577:
573:
569:
564:
560:
554:
551:
548:
544:
540:
537:
500:
496:
475:
470:
467:
464:
460:
456:
451:
447:
443:
406:path connected
372:
369:
366:
363:
360:
357:
354:
351:
348:
343:
339:
335:
330:
327:
324:
319:
315:
311:
306:
302:
255:
251:
239:
238:
227:
224:
221:
218:
213:
210:
207:
203:
199:
196:
193:
190:
187:
182:
178:
174:
171:
168:
163:
159:
155:
150:
147:
144:
139:
135:
99:
96:
93:
90:
87:
75:
72:
62:and the fiber
13:
10:
9:
6:
4:
3:
2:
5708:
5697:
5694:
5692:
5689:
5688:
5686:
5676:
5672:
5669:Paul Goerss,
5668:
5667:
5666:
5660:
5656:
5653:
5649:
5645:
5644:
5643:
5637:
5636:
5631:
5630:
5629:
5623:
5619:
5618:Edwin Spanier
5616:
5614:
5613:
5608:
5607:Allen Hatcher
5605:
5604:
5603:
5597:
5593:
5590:
5589:
5585:
5583:
5567:
5563:
5554:
5550:
5541:
5537:
5529:
5525:
5520:
5507:
5499:
5495:
5486:
5478:
5474:
5465:
5461:
5452:
5448:
5435:
5427:
5423:
5414:
5406:
5402:
5393:
5389:
5380:
5361:
5353:
5349:
5345:
5339:
5331:
5327:
5318:
5302:
5294:
5290:
5281:
5275:
5267:
5263:
5240:
5236:
5232:
5212:
5209:
5187:
5183:
5162:
5159:
5153:
5147:
5140:, so we have
5107:
5091:
5069:
5065:
5044:
5026:
5020:
5017:
5006:
4986:
4980:
4977:
4966:
4963:
4957:
4954:
4943:
4918:
4914:
4907:
4899:
4880:
4877:
4874:
4868:
4845:
4839:
4836:
4825:
4817:
4813:
4806:
4799:
4798:
4797:
4782:
4778:
4756:
4748:
4744:
4735:
4727:
4723:
4714:
4710:
4697:
4695:
4682:
4676:
4659:
4646:
4641:
4638:
4635:
4631:
4626:
4619:
4589:
4571:
4545:
4541:
4537:
4533:
4529:
4525:
4521:
4517:
4513:
4509:
4505:
4501:
4485:
4472:
4465:
4444:
4426:
4423:
4420:
4417:
4413:
4404:
4400:
4392:
4391:
4390:
4374:
4352:
4350:
4348:
4322:
4318:
4311:
4307:
4300:
4294:
4290:
4230:
4222:
4213:
4205:
4201:
4197:
4192:
4189:
4186:
4182:
4173:
4169:
4160:
4156:
4152:
4148:
4127:
4118:
4110:
4106:
4102:
4097:
4094:
4091:
4087:
4078:
4075:
4072:
4068:
4059:
4055:
4051:
4047:
4021:
4017:
4010:
3991:
3980:
3977:
3974:
3970:
3958:
3954:
3950:
3945:
3942:
3939:
3935:
3926:
3922:
3918:
3913:
3908:
3905:
3902:
3898:
3890:
3889:
3888:
3886:
3882:
3862:
3857:
3854:
3851:
3847:
3838:
3835:
3832:
3828:
3824:
3816:
3813:
3810:
3806:
3795:
3794:
3793:
3792:
3784:
3761:
3758:
3755:
3748:
3745:
3728:
3725:
3722:
3716:
3713:
3700:
3695:
3684:
3681:
3673:
3669:
3661:
3660:
3659:
3634:
3631:
3623:
3619:
3604:
3601:
3593:
3589:
3585:
3580:
3577:
3574:
3569:
3565:
3557:
3556:
3555:
3539:
3535:
3512:
3508:
3504:
3501:
3498:
3495:
3492:
3487:
3483:
3455:
3450:
3445:
3440:
3435:
3419:
3416:
3408:
3404:
3390:
3387:
3379:
3375:
3361:
3358:
3350:
3346:
3332:
3329:
3321:
3317:
3303:
3300:
3292:
3288:
3282:
3275:
3270:
3265:
3260:
3255:
3250:
3243:
3238:
3233:
3228:
3223:
3218:
3203:
3200:
3192:
3188:
3174:
3171:
3163:
3159:
3145:
3142:
3134:
3130:
3116:
3113:
3105:
3101:
3087:
3084:
3076:
3072:
3066:
3059:
3054:
3050:
3046:
3041:
3037:
3033:
3028:
3024:
3016:
3015:
3014:
3012:
2989:
2981:
2969:
2963:
2955:
2889:
2867:
2864:
2860:
2831:
2828:
2825:
2822:
2818:
2809:
2805:
2801:
2798:
2790:
2786:
2782:
2777:
2774:
2771:
2766:
2762:
2739:
2735:
2722:
2704:
2694:
2689:
2686:
2683:
2680:
2676:
2655:
2649:
2592:
2584:
2580:
2576:
2573:
2570:
2564:
2556:
2552:
2542:
2535:
2534:
2533:
2513:
2505:
2493:
2490:
2481:
2476:
2472:
2468:
2465:
2462:
2457:
2453:
2430:
2414:
2389:
2386:
2363:
2355:
2334:
2326:
2324:
2308:
2305:
2302:
2297:
2293:
2272:
2267:
2263:
2242:
2234:
2230:
2221:
2217:
2211:
2207:
2203:
2200:
2195:
2192:
2189:
2184:
2180:
2176:
2173:
2148:
2144:
2135:
2131:
2125:
2121:
2117:
2114:
2109:
2106:
2103:
2098:
2094:
2090:
2087:
2067:
2059:
2055:
2046:
2043:
2040:
2036:
2030:
2026:
2022:
2019:
2014:
2011:
2008:
2003:
1999:
1978:
1970:
1966:
1957:
1954:
1951:
1947:
1919:
1914:
1909:
1899:
1896:
1893:
1888:
1884:
1880:
1877:
1872:
1859:
1854:
1850:
1841:
1837:
1831:
1816:
1811:
1807:
1798:
1794:
1788:
1781:
1778:
1775:
1770:
1766:
1762:
1759:
1754:
1743:
1742:
1741:
1736:
1729:
1726: =
1722:
1715:
1707:
1699:
1678:
1675:
1672:
1652:
1649:
1646:
1624:
1621:
1618:
1614:
1586:
1581:
1576:
1560:
1555:
1551:
1542:
1538:
1532:
1519:
1514:
1510:
1501:
1497:
1491:
1476:
1471:
1467:
1458:
1454:
1448:
1435:
1430:
1426:
1417:
1413:
1407:
1396:
1395:
1394:
1378:
1374:
1351:
1347:
1338:
1334:
1325:
1321:
1308:
1303:
1301:
1285:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1252:
1243:
1238:
1235:
1232:
1228:
1224:
1219:
1215:
1206:
1187:
1181:
1173:
1170:
1167:
1163:
1150:
1142:
1138:
1134:
1131:
1123:
1119:
1115:
1110:
1105:
1102:
1099:
1095:
1087:
1086:
1085:
1079:
1077:
1061:
1057:
1034:
1031:
1028:
1024:
1015:
997:
993:
981:
962:
957:
954:
951:
948:
945:
942:
939:
934:
930:
921:
918:
915:
910:
906:
902:
897:
894:
891:
886:
882:
874:
873:
872:
869:
856:
851:
847:
821:
818:
815:
811:
807:
802:
798:
786:
782:
778:
760:
757:
754:
750:
727:
723:
714:
696:
688:
685:
682:
678:
674:
669:
665:
656:
652:
646:
643:
640:
636:
632:
629:
626:
621:
618:
615:
610:
606:
583:
575:
571:
562:
558:
552:
549:
546:
542:
538:
535:
528:
527:
526:
524:
523:this notation
520:
517:-skeleton of
516:
498:
494:
468:
465:
462:
458:
454:
449:
445:
433:
429:
424:
422:
418:
413:
411:
407:
403:
399:
395:
391:
386:
370:
367:
364:
361:
358:
355:
352:
349:
346:
341:
337:
328:
325:
322:
317:
313:
309:
304:
300:
291:
287:
283:
279:
275:
271:
268:-term is the
253:
249:
225:
219:
211:
208:
205:
201:
188:
180:
176:
172:
169:
161:
157:
153:
148:
145:
142:
137:
133:
125:
124:
123:
121:
117:
113:
97:
91:
88:
85:
73:
71:
69:
65:
61:
58:
54:
51:of a (Serre)
50:
46:
42:
38:
34:
30:
26:
25:
19:
5677:, BirkhÀuser
5674:
5671:Rick Jardine
5664:
5654:
5647:
5641:
5633:
5627:
5621:
5610:
5601:
5521:
5450:
5449:
5316:
5105:
4860:
4701:
4660:
4569:
4543:
4539:
4535:
4531:
4527:
4523:
4519:
4515:
4511:
4507:
4503:
4499:
4470:
4463:
4462:Now, on the
4461:
4356:
4346:
4323:) has value
4320:
4313:
4309:
4302:
4298:
4292:
4288:
4158:
4154:
4150:
4146:
4057:
4053:
4049:
4045:
4019:
4012:
4008:
4006:
3884:
3880:
3877:
3788:
3657:
3474:
2613:
2332:
2330:
1938:
1740:-page reads
1734:
1727:
1720:
1713:
1705:
1697:
1605:
1312:
1204:
1202:
1083:
979:
977:
870:
784:
776:
712:
711:
518:
514:
428:exact couple
425:
420:
414:
397:
393:
387:
289:
285:
277:
269:
240:
115:
77:
63:
59:
48:
28:
21:
15:
2721:Euler class
2532:which send
292:th page is
27:(sometimes
18:mathematics
5685:Categories
5624:, Springer
5598:References
3011:K3 surface
57:base space
33:Jean Leray
5560:→
5547:→
5462:π
5451:Corollary
5390:π
5350:π
5233:ι
5210:ι
5163:ι
5126:∞
5092:ι
5084:, called
5040:∞
4941:Ω
4911:→
4875:π
4823:→
4810:→
4779:π
4711:π
4432:→
4410:↪
3967:Ω
3844:→
3822:→
3803:Ω
3746:⊕
3717:∈
3616:→
3502:⋅
3496:∪
2970:⊕
2829:−
2695:⊂
2687:−
2653:→
2574:…
2546:↦
2485:Γ
2482:∈
2466:…
2418:→
2390:∈
2177:
2091:
1881:
1763:
1665:down and
1344:→
1331:↪
1272:−
1260:−
1249:→
1160:⇒
927:→
903:×
819:−
758:−
686:−
637:⨁
543:⨁
466:−
368:−
334:→
198:⇒
95:→
89::
53:fibration
5586:See also
5526:for the
486:, where
417:abutment
5377:by the
4149:= 0 or
1712:on the
1076:-page.
35:in the
4896:is an
4861:where
2727:. The
1733:. The
1366:. The
783:, and
597:
22:Serre
20:, the
5659:EuDML
5628:Also
2174:coker
1991:i.e.
1878:coker
1639:goes
120:fiber
110:be a
4522:) =
3527:for
3005:for
2379:for
2166:and
1012:are
415:The
272:-th
78:Let
4287:at
2088:ker
1760:ker
1695:is
839:to
742:to
400:is
392:on
284:of
276:of
16:In
5687::
5673:,
5650:,
5620:,
5609:,
5582:.
5453::
4540:ix
4520:ix
4512:ix
4319:(Ω
4301:;
4295:+1
4153:=
4056:=
3749:22
3009:a
1300:.
775:,
525:,
423:.
385:.
5661:.
5655:3
5568:2
5564:S
5555:3
5551:S
5542:1
5538:S
5508:.
5504:Z
5500:2
5496:/
5491:Z
5487:=
5484:)
5479:2
5475:S
5471:(
5466:4
5436:.
5432:Z
5428:2
5424:/
5419:Z
5415:=
5412:)
5407:3
5403:S
5399:(
5394:4
5365:)
5362:X
5359:(
5354:4
5346:=
5343:)
5340:X
5337:(
5332:4
5328:H
5317:X
5303:.
5299:Z
5295:2
5291:/
5286:Z
5282:=
5279:)
5276:X
5273:(
5268:4
5264:H
5241:2
5237:a
5213:a
5188:2
5184:a
5160:=
5157:)
5154:a
5151:(
5148:d
5121:P
5118:C
5106:a
5070:3
5066:S
5045:.
5035:P
5032:C
5027:=
5024:)
5021:2
5018:,
5014:Z
5010:(
5007:K
4987:.
4984:)
4981:2
4978:,
4974:Z
4970:(
4967:K
4964:=
4961:)
4958:3
4955:,
4951:Z
4947:(
4944:K
4919:3
4915:S
4908:X
4884:)
4881:n
4878:,
4872:(
4869:K
4846:,
4843:)
4840:3
4837:,
4833:Z
4829:(
4826:K
4818:3
4814:S
4807:X
4783:3
4757:.
4753:Z
4749:2
4745:/
4740:Z
4736:=
4733:)
4728:3
4724:S
4720:(
4715:4
4683:.
4680:]
4677:x
4674:[
4670:Z
4647:.
4642:1
4639:+
4636:n
4632:x
4627:/
4623:]
4620:x
4617:[
4613:Z
4590:n
4585:P
4582:C
4570:n
4555:Z
4544:n
4536:n
4532:n
4528:x
4524:x
4518:(
4516:d
4508:x
4504:i
4500:n
4486:.
4482:Z
4471:i
4467:2
4464:E
4445:n
4440:P
4437:C
4427:1
4424:+
4421:n
4418:2
4414:S
4405:1
4401:S
4375:n
4370:P
4367:C
4347:n
4332:Z
4321:S
4316:i
4314:H
4310:F
4308:(
4305:n
4303:H
4299:S
4297:(
4293:n
4289:H
4274:Z
4252:Z
4231:.
4227:Z
4223:=
4220:)
4217:)
4214:F
4211:(
4206:n
4202:H
4198:;
4193:1
4190:+
4187:n
4183:S
4179:(
4174:0
4170:H
4159:E
4155:n
4151:p
4147:p
4132:Z
4128:=
4125:)
4122:)
4119:F
4116:(
4111:0
4107:H
4103:;
4098:1
4095:+
4092:n
4088:S
4084:(
4079:1
4076:+
4073:n
4069:H
4058:q
4054:p
4050:E
4046:n
4031:Z
4020:S
4018:(
4015:p
4013:H
4009:q
3992:.
3989:)
3986:)
3981:1
3978:+
3975:n
3971:S
3964:(
3959:q
3955:H
3951:;
3946:1
3943:+
3940:n
3936:S
3932:(
3927:p
3923:H
3919:=
3914:2
3909:q
3906:,
3903:p
3899:E
3885:E
3881:E
3863:.
3858:1
3855:+
3852:n
3848:S
3839:1
3836:+
3833:n
3829:S
3825:P
3817:1
3814:+
3811:n
3807:S
3762:2
3759:=
3756:k
3741:Z
3732:}
3729:4
3726:,
3723:0
3720:{
3714:k
3708:Z
3701:{
3696:=
3693:)
3689:Z
3685:;
3682:X
3679:(
3674:k
3670:H
3643:)
3639:Z
3635:;
3632:X
3629:(
3624:4
3620:H
3613:)
3609:Z
3605:;
3602:X
3599:(
3594:0
3590:H
3586::
3581:3
3578:,
3575:0
3570:4
3566:d
3540:2
3536:H
3513:2
3509:H
3505:b
3499:a
3493:=
3488:4
3484:d
3456:4
3451:3
3446:2
3441:1
3436:0
3428:)
3424:Z
3420:;
3417:X
3414:(
3409:4
3405:H
3399:)
3395:Z
3391:;
3388:X
3385:(
3380:3
3376:H
3370:)
3366:Z
3362:;
3359:X
3356:(
3351:2
3347:H
3341:)
3337:Z
3333:;
3330:X
3327:(
3322:1
3318:H
3312:)
3308:Z
3304:;
3301:X
3298:(
3293:0
3289:H
3283:0
3276:0
3271:0
3266:0
3261:0
3256:0
3251:1
3244:0
3239:0
3234:0
3229:0
3224:0
3219:2
3212:)
3208:Z
3204:;
3201:X
3198:(
3193:4
3189:H
3183:)
3179:Z
3175:;
3172:X
3169:(
3164:3
3160:H
3154:)
3150:Z
3146:;
3143:X
3140:(
3135:2
3131:H
3125:)
3121:Z
3117:;
3114:X
3111:(
3106:1
3102:H
3096:)
3092:Z
3088:;
3085:X
3082:(
3077:0
3073:H
3067:3
3060:=
3055:4
3051:E
3047:=
3042:3
3038:E
3034:=
3029:2
3025:E
3007:X
2993:)
2990:b
2987:(
2982:X
2976:O
2967:)
2964:a
2961:(
2956:X
2950:O
2925:E
2903:)
2898:E
2893:(
2890:e
2868:r
2865:2
2861:E
2840:)
2837:)
2832:1
2826:r
2823:2
2819:S
2815:(
2810:q
2806:H
2802:;
2799:X
2796:(
2791:p
2787:H
2783:=
2778:q
2775:,
2772:p
2767:2
2763:E
2740:2
2736:E
2725:S
2705:r
2700:C
2690:1
2684:r
2681:2
2677:S
2656:X
2650:S
2628:E
2616:r
2599:]
2596:)
2593:x
2590:(
2585:m
2581:s
2577::
2571::
2568:)
2565:x
2562:(
2557:0
2553:s
2549:[
2543:x
2520:)
2517:)
2514:1
2511:(
2506:X
2500:O
2494:,
2491:X
2488:(
2477:m
2473:s
2469:,
2463:,
2458:0
2454:s
2431:m
2426:P
2423:C
2415:X
2394:Z
2387:k
2367:)
2364:k
2361:(
2356:X
2350:O
2337:X
2333:n
2309:1
2306:,
2303:0
2298:2
2294:d
2273:,
2268:3
2264:S
2243:.
2240:)
2235:3
2231:S
2227:(
2222:2
2218:H
2212:0
2208:r
2204:G
2201:=
2196:1
2193:,
2190:0
2185:2
2181:d
2154:)
2149:3
2145:S
2141:(
2136:1
2132:H
2126:1
2122:r
2118:G
2115:=
2110:1
2107:,
2104:0
2099:2
2095:d
2068:.
2065:)
2060:3
2056:S
2052:(
2047:q
2044:+
2041:p
2037:H
2031:p
2027:r
2023:G
2020:=
2015:q
2012:,
2009:p
2004:3
2000:E
1979:,
1976:)
1971:3
1967:S
1963:(
1958:q
1955:+
1952:p
1948:H
1920:2
1915:1
1910:0
1900:1
1897:,
1894:0
1889:2
1885:d
1873:0
1868:)
1864:Z
1860:;
1855:2
1851:S
1847:(
1842:0
1838:H
1832:0
1825:)
1821:Z
1817:;
1812:2
1808:S
1804:(
1799:2
1795:H
1789:0
1782:1
1779:,
1776:0
1771:2
1767:d
1755:1
1738:3
1735:E
1731:â
1728:E
1724:2
1721:E
1717:2
1714:E
1710:0
1706:0
1701:2
1698:d
1693:0
1679:i
1676:+
1673:2
1653:i
1650:+
1647:1
1625:i
1622:+
1619:2
1615:d
1587:2
1582:1
1577:0
1569:)
1565:Z
1561:;
1556:2
1552:S
1548:(
1543:2
1539:H
1533:0
1528:)
1524:Z
1520:;
1515:2
1511:S
1507:(
1502:0
1498:H
1492:0
1485:)
1481:Z
1477:;
1472:2
1468:S
1464:(
1459:2
1455:H
1449:0
1444:)
1440:Z
1436:;
1431:2
1427:S
1423:(
1418:0
1414:H
1408:1
1379:2
1375:E
1352:2
1348:S
1339:3
1335:S
1326:1
1322:S
1286:k
1281:k
1278:+
1275:1
1269:q
1266:,
1263:k
1257:p
1253:E
1244:k
1239:q
1236:,
1233:p
1229:E
1225::
1220:k
1216:d
1205:k
1188:.
1185:)
1182:X
1179:(
1174:q
1171:+
1168:p
1164:H
1157:)
1154:)
1151:F
1148:(
1143:q
1139:H
1135:,
1132:B
1129:(
1124:p
1120:H
1116:=
1111:2
1106:q
1103:,
1100:p
1096:E
1062:r
1058:E
1035:1
1032:+
1029:r
1025:E
998:r
994:d
983:2
980:E
963:,
958:t
955:+
952:q
949:,
946:s
943:+
940:p
935:r
931:E
922:t
919:,
916:s
911:r
907:E
898:q
895:,
892:p
887:r
883:E
857:.
852:p
848:X
827:)
822:1
816:p
812:X
808:,
803:p
799:X
795:(
785:h
777:g
761:1
755:p
751:X
728:p
724:X
713:f
697:,
694:)
689:1
683:p
679:X
675:,
670:p
666:X
662:(
657:q
653:H
647:q
644:,
641:p
633:=
630:C
627:=
622:q
619:,
616:p
611:1
607:E
584:,
581:)
576:p
572:X
568:(
563:q
559:H
553:q
550:,
547:p
539:=
536:A
519:B
515:p
499:p
495:X
474:)
469:1
463:p
459:X
455:,
450:p
446:X
442:(
421:X
398:B
394:B
371:k
365:1
362:+
359:q
356:,
353:k
350:+
347:p
342:k
338:E
329:q
326:,
323:p
318:k
314:E
310::
305:k
301:d
290:k
286:B
278:F
270:q
254:2
250:E
226:.
223:)
220:X
217:(
212:q
209:+
206:p
202:H
195:)
192:)
189:F
186:(
181:q
177:H
173:,
170:B
167:(
162:p
158:H
154:=
149:q
146:,
143:p
138:2
134:E
116:F
98:B
92:X
86:f
64:F
60:B
49:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.