2427:, and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational
1429:
3394:
1923:
978:
2135:
3229:
2409:
3160:
either as the zero section; i.e., a section at zero vector or as the infinity section; i.e., a section at infinity vector (topologically the difference is immaterial.) Using two ways of embedding we have the triple:
1297:
1631:
1186:
2636:
3270:
2701:
2896:
1494:
1312:
2810:
222:
2592:
1718:
2018:
If the
Steenrod operations are available, we can use them and the isomorphism of the theorem to construct the Stiefel–Whitney classes. Recall that the Steenrod operations (mod 2) are
458:
3158:
490:
322:
258:
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3016:
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1536:
801:
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111:
523:
3067:
1797:
988:
2497:
2270:
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1789:
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1112:
1080:
862:
847:
346:
2198:
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1655:
1042:
752:
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550:
169:
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583:
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2168:
3713:
3591:
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2549:
2466:
2730:
3689:
This textbook gives a detailed construction of the Thom class for trivial vector bundles, and also formulates the theorem in case of arbitrary vector bundles.
1005:
We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on
3491:
2027:
3166:
2278:
3467:
1212:
2706:
The embedded manifold together with the isomorphism class of the normal bundle actually encodes the same information as the cobordism class
1984:(among many others) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as
3718:
3682:
3626:
2012:
1553:
3844:
2432:
1981:
811:
3822:
1128:
2600:
3237:
2644:
1965:
1424:{\displaystyle {\begin{cases}H^{k}(E;\Lambda )\to H^{k+n}(E,E\setminus B;\Lambda )\\x\longmapsto x\smile u\end{cases}}}
3839:
3817:
3674:
1973:
2850:
1445:
3701:
2755:
1993:
225:
3105:
1937:
189:
1663:
3484:
1052:.) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:
428:
3131:
463:
295:
231:
3019:
2557:
3403:
3074:
2976:
2919:
2019:
2004:
1724:
3634:
3389:{\displaystyle H^{n}(Sph(E),B)\simeq H^{n}(\operatorname {Sph} (E),\operatorname {Sph} (E)\setminus B),}
2008:
766:
The significance of this construction begins with the following result, which belongs to the subject of
648:
55:
3662:
is another standard reference for the Thom class and Thom isomorphism. See especially the paragraph 18.
3561:
1499:
777:
3095:
3078:
3460:
1918:{\displaystyle {\tilde {H}}^{n}(T(E))=H^{n}(\operatorname {Sph} (E),B)\simeq H^{n}(E,E\setminus B).}
1321:
84:
3753:
3638:
3027:
495:
142:
3531:
3033:
3788:
3762:
3748:
3735:
3693:
3510:
2428:
1941:
992:
973:{\displaystyle \Phi :H^{k}(B;\mathbb {Z} _{2})\to {\widetilde {H}}^{k+n}(T(E);\mathbb {Z} _{2}),}
51:
3812:
3678:
3622:
2243:
2207:
1774:
1730:
1085:
1065:
820:
127:
59:
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of each fiber and gluing them together to get the total space. Finally, from the total space
3772:
3727:
3600:
3100:
2173:
1969:
1634:
17:
3784:
2815:
2502:
1640:
1020:
725:
654:
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147:
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1953:
1115:
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559:
388:
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1985:
47:
2709:
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807:
586:
349:
184:
123:
63:
3666:
803:
771:
176:
3709:
3605:
3586:
2011:
and stable homotopy theory, and is in particular integral to our knowledge of the
999:
43:
3653:
3642:
2130:{\displaystyle Sq^{i}:H^{m}(-;\mathbb {Z} _{2})\to H^{m+i}(-;\mathbb {Z} _{2}),}
1977:
120:
67:
28:
3614:
3224:{\displaystyle (\operatorname {Sph} (E),\operatorname {Sph} (E)\setminus B,B)}
3077:. The lack of transversality prevents from computing cobordism rings of, say,
2424:
767:
3806:
2597:
Another technique to encode this kind of information is to take an embedding
3776:
3090:
2404:{\displaystyle w_{i}(p)=\Phi ^{-1}(Sq^{i}(\Phi (1)))=\Phi ^{-1}(Sq^{i}(u)).}
1949:
1928:
The standard reference for the Thom isomorphism is the book by Bott and Tu.
3509:
Greenlees, J. P. C. (2006-09-15). "Spectra for commutative algebraists".
3731:
1292:{\displaystyle u|_{(F,F\setminus 0)}\in H^{n}(F,F\setminus 0;\Lambda )}
3515:
3767:
3751:; Rezk, Charles (2014). "Units of ring spectra and Thom spectra".
2448:
There are two ways to think about bordism: one as considering two
1944:
were all related. He used these ideas to prove in the 1954 paper
2423:
of a smooth manifold, the conclusion of the above is called the
328:; that is, by identifying all the new points to a single point
3276:. Taking the long exact sequence of this triple, we then see:
1626:{\displaystyle p^{*}:H^{*}(B;\Lambda )\to H^{*}(E;\Lambda )}
1417:
718:
can be defined as the quotient of the unit disk bundle of
3714:
Quelques propriétés globales des variétés différentiables
1964:). The proof depends on and is intimately related to the
1946:
Quelques propriétés globales des variétés differentiables
1438:
In concise terms, the last part of the theorem says that
1936:
In his 1952 paper, Thom showed that the Thom class, the
2812:
which gives a homotopy class of maps to the Thom space
3747:
Ando, Matthew; Blumberg, Andrew J.; Gepner, David J.;
3406:
3284:
3240:
3169:
3134:
3036:
2979:
2922:
2853:
2818:
2758:
2738:
2712:
2647:
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2537:
2505:
2474:
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2210:
2176:
2150:
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1448:
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1131:
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823:
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622:
595:
562:
531:
498:
466:
431:
391:
358:
334:
298:
269:
234:
192:
150:
87:
3587:"René Thom's Work on Geometric Homology and Bordism"
1181:{\displaystyle u\in H^{n}(E,E\setminus B;\Lambda ),}
2631:{\displaystyle M\hookrightarrow \mathbb {R} ^{N+n}}
1049:
78:One way to construct this space is as follows. Let
3440:
3388:
3265:{\displaystyle \operatorname {Sph} (E)\setminus B}
3264:
3223:
3152:
3061:
3010:
2962:
2890:
2836:
2804:
2744:
2724:
2696:{\displaystyle \nu :N_{\mathbb {R} ^{N+n}/M}\to M}
2695:
2630:
2586:
2543:
2523:
2491:
2460:
2403:
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2129:
1917:
1783:
1755:
1712:
1649:
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1291:
1180:
1106:
1074:
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841:
795:
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635:
608:
577:
544:
517:
484:
452:
406:
373:
340:
316:
284:
252:
216:
163:
105:
2200:coincides with the cup square. We can define the
3537:. Notes by O. Gwilliam. Northwestern University.
3073:; the proof of this theorem relies crucially on
3567:. Notes by I. Bobovka. Northwestern University.
2891:{\displaystyle \pi _{n}MO\cong \Omega _{n}^{O}}
3649:, Thom classes and Thom isomorphism, and more.
1771:is treated as an element of (we drop the ring
1723:In particular, the Thom isomorphism sends the
1489:{\displaystyle H^{*}(E,E\setminus B;\Lambda )}
3592:Bulletin of the American Mathematical Society
2805:{\displaystyle \mathbb {R} ^{N_{W}+n}\times }
2419:If we take the bundle in the above to be the
8:
217:{\displaystyle \operatorname {Sph} (E)\to B}
1302:is the class induced by the orientation of
1199:as a zero section, such that for any fiber
1056:
3807:http://ncatlab.org/nlab/show/Thom+spectrum
2844:defined below. Showing the isomorphism of
1713:{\displaystyle \Phi (b)=p^{*}(b)\smile u.}
998:This theorem was formulated and proved by
3766:
3604:
3514:
3411:
3405:
3332:
3289:
3283:
3239:
3168:
3133:
3041:
3035:
2984:
2978:
2951:
2921:
2882:
2877:
2858:
2852:
2817:
2770:
2765:
2761:
2760:
2757:
2737:
2732:. This can be shown by using a cobordism
2711:
2677:
2665:
2661:
2660:
2658:
2646:
2616:
2612:
2611:
2602:
2559:
2536:
2504:
2473:
2453:
2415:Consequences for differentiable manifolds
2380:
2361:
2327:
2308:
2286:
2280:
2245:
2215:
2209:
2184:
2175:
2149:
2115:
2111:
2110:
2088:
2072:
2068:
2067:
2051:
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1885:
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1814:
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1802:
1799:
1776:
1738:
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1602:
1574:
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1507:
1501:
1453:
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1328:
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1314:
1256:
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1220:
1214:
1142:
1130:
1087:
1067:
1028:
1022:
958:
954:
953:
922:
911:
910:
897:
893:
892:
876:
864:
822:
812:Orientation of a vector bundle#Thom space
787:
783:
782:
779:
774:. (We have stated the result in terms of
727:
694:
689:can be given a Euclidean metric and then
662:
656:
627:
621:
600:
594:
561:
536:
530:
509:
497:
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444:
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390:
357:
333:
297:
268:
233:
191:
155:
149:
86:
3619:Differential Forms in Algebraic Topology
2007:. Thom's construction thus also unifies
1767:. Note: for this formula to make sense,
853:. Then there is an isomorphism called a
453:{\displaystyle B\times \mathbb {R} ^{n}}
3429:
3374:
3256:
3206:
3153:{\displaystyle \operatorname {Sph} (E)}
3124:Proof of the isomorphism. We can embed
3117:
2003:, and the cobordism groups are in fact
1903:
1471:
1380:
1274:
1238:
1160:
485:{\displaystyle \operatorname {Sph} (E)}
317:{\displaystyle \operatorname {Sph} (E)}
253:{\displaystyle \operatorname {Sph} (E)}
3671:A Concise Course in Algebraic Topology
3398:the latter of which is isomorphic to:
2587:{\displaystyle \partial W=M\coprod M'}
987:greater than or equal to 0, where the
3562:"Math 465, lecture 4: transversality"
3547:
3441:{\displaystyle H^{n}(E,E\setminus B)}
2140:defined for all nonnegative integers
414:is the one-point compactification of
7:
3011:{\displaystyle \gamma ^{n}\to BO(n)}
2963:{\displaystyle MO(n)=T(\gamma ^{n})}
806:to avoid complications arising from
2874:
2638:and considering the normal bundle
2561:
2358:
2336:
2305:
1976:. By reversing this construction,
1778:
1667:
1644:
1617:
1589:
1522:
1480:
1389:
1343:
1283:
1169:
1069:
866:
335:
25:
3719:Commentarii Mathematici Helvetici
3497:from the original on 17 Jan 2021.
3473:from the original on 17 Jan 2021.
2752:and finding an embedding to some
2013:stable homotopy groups of spheres
1531:{\displaystyle H^{*}(E;\Lambda )}
1048:with a disjoint point added (cf.
3532:"Math 465, lecture 2: cobordism"
1952:groups could be computed as the
849:be a real vector bundle of rank
796:{\displaystyle \mathbb {Z} _{2}}
1050:#Construction of the Thom space
3435:
3417:
3380:
3371:
3365:
3353:
3347:
3338:
3322:
3313:
3307:
3295:
3253:
3247:
3218:
3203:
3197:
3185:
3179:
3170:
3147:
3141:
3056:
3047:
3030:. A theorem of Thom says that
3005:
2999:
2990:
2957:
2944:
2935:
2929:
2831:
2825:
2799:
2787:
2719:
2713:
2687:
2607:
2518:
2506:
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1909:
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1820:
1808:
1750:
1744:
1698:
1692:
1676:
1670:
1620:
1608:
1595:
1592:
1580:
1525:
1513:
1483:
1459:
1402:
1392:
1368:
1349:
1346:
1334:
1286:
1262:
1244:
1226:
1221:
1172:
1148:
1098:
964:
946:
940:
934:
906:
903:
882:
833:
741:
729:
705:
699:
572:
566:
479:
473:
401:
395:
368:
362:
311:
305:
279:
273:
247:
241:
208:
205:
199:
106:{\displaystyle p\colon E\to B}
97:
74:Construction of the Thom space
1:
3606:10.1090/S0273-0979-04-01026-2
3075:Thom’s transversality theorem
2913:is a sequence of Thom spaces
2901:requires a little more work.
2499:are cobordant if there is an
518:{\displaystyle B\times S^{n}}
3062:{\displaystyle \pi _{*}(MO)}
1122:. Then there exists a class
40:Pontryagin–Thom construction
18:Pontrjagin–Thom construction
3818:Encyclopedia of Mathematics
3675:University of Chicago Press
2905:Definition of Thom spectrum
1974:Thom transversality theorem
1932:Significance of Thom's work
1118:real vector bundle of rank
1002:in his famous 1952 thesis.
556:with a disjoint basepoint,
3861:
3702:Princeton University Press
1988:. In addition, the spaces
1657:is given by the equation:
226:one-point compactification
3698:Notes on cobordism theory
2204:th Stiefel–Whitney class
3633:A classic reference for
3272:deformation-retracts to
2531:-manifold with boundary
2265:{\displaystyle p:E\to B}
2233:{\displaystyle w_{i}(p)}
1784:{\displaystyle \Lambda }
1756:{\displaystyle H^{*}(B)}
1107:{\displaystyle p:E\to B}
1075:{\displaystyle \Lambda }
842:{\displaystyle p:E\to B}
3637:, treating the link to
3026:. The sequence forms a
3020:universal vector bundle
2020:natural transformations
1956:of certain Thom spaces
1938:Stiefel–Whitney classes
348:, which we take as the
341:{\displaystyle \infty }
3845:Characteristic classes
3658:Characteristic classes
3621:. New York: Springer.
3442:
3390:
3266:
3225:
3154:
3063:
3012:
2964:
2892:
2838:
2806:
2746:
2726:
2697:
2632:
2588:
2545:
2525:
2493:
2462:
2405:
2266:
2234:
2194:
2193:{\displaystyle Sq^{i}}
2164:
2131:
1919:
1785:
1757:
1714:
1651:
1627:
1542:is usually called the
1532:
1490:
1425:
1293:
1182:
1108:
1076:
1038:
974:
843:
797:
748:
712:
672:
637:
610:
579:
546:
519:
486:
454:
425:is the trivial bundle
408:
375:
342:
318:
286:
254:
218:
165:
133:. Then for each point
107:
3777:10.1112/jtopol/jtu009
3635:differential topology
3617:; Tu, Loring (1982).
3443:
3391:
3267:
3226:
3155:
3106:Hattori–Stong theorem
3079:topological manifolds
3064:
3013:
2965:
2893:
2839:
2837:{\displaystyle MO(n)}
2807:
2747:
2727:
2698:
2633:
2589:
2546:
2526:
2524:{\displaystyle (n+1)}
2494:
2463:
2406:
2267:
2240:of the vector bundle
2235:
2195:
2165:
2132:
2009:differential topology
1992:fit together to form
1920:
1786:
1758:
1715:
1652:
1650:{\displaystyle \Phi }
1628:
1550:. Since the pullback
1533:
1491:
1426:
1294:
1183:
1109:
1077:
1039:
1037:{\displaystyle B_{+}}
1013:is isomorphic to the
975:
844:
798:
749:
747:{\displaystyle (n-1)}
713:
681:Alternatively, since
673:
671:{\displaystyle B_{+}}
638:
636:{\displaystyle S^{n}}
611:
609:{\displaystyle B_{+}}
580:
547:
545:{\displaystyle B_{+}}
520:
487:
455:
409:
376:
343:
319:
287:
255:
219:
166:
164:{\displaystyle E_{b}}
108:
56:differential topology
3677:. pp. 183–198.
3404:
3282:
3238:
3167:
3132:
3096:Cohomology operation
3034:
2977:
2920:
2851:
2816:
2756:
2736:
2710:
2645:
2601:
2558:
2535:
2503:
2492:{\displaystyle M,M'}
2472:
2452:
2279:
2244:
2208:
2174:
2148:
2028:
1931:
1798:
1775:
1731:
1664:
1641:
1554:
1500:
1446:
1313:
1213:
1129:
1086:
1066:
1021:
863:
821:
778:
762:The Thom isomorphism
726:
711:{\displaystyle T(E)}
693:
655:
620:
593:
578:{\displaystyle T(E)}
560:
529:
496:
464:
429:
407:{\displaystyle T(E)}
389:
374:{\displaystyle T(E)}
356:
332:
296:
285:{\displaystyle T(E)}
267:
232:
190:
148:
85:
3754:Journal of Topology
3749:Hopkins, Michael J.
3081:from Thom spectra.
2909:By definition, the
2887:
2163:{\displaystyle i=m}
1942:Steenrod operations
1538:-module. The class
1434:is an isomorphism.
1203:the restriction of
1060: —
292:as the quotient of
3840:Algebraic topology
3732:10.1007/BF02566923
3438:
3386:
3262:
3221:
3150:
3069:is the unoriented
3059:
3008:
2960:
2888:
2873:
2834:
2802:
2742:
2722:
2693:
2628:
2584:
2541:
2521:
2489:
2458:
2429:Pontryagin classes
2401:
2262:
2230:
2190:
2160:
2127:
1915:
1781:
1753:
1710:
1647:
1623:
1528:
1486:
1421:
1416:
1289:
1178:
1104:
1072:
1058:
1034:
993:reduced cohomology
970:
839:
793:
754:-sphere bundle of
744:
708:
668:
633:
606:
575:
542:
515:
482:
450:
404:
371:
338:
314:
282:
250:
214:
175:-dimensional real
161:
103:
52:algebraic topology
2745:{\displaystyle W}
2544:{\displaystyle W}
2461:{\displaystyle n}
1811:
1442:freely generates
1195:is embedded into
1017:th suspension of
919:
385:is compact, then
179:. We can form an
128:paracompact space
60:topological space
16:(Redirected from
3852:
3826:
3796:
3770:
3761:(4): 1077–1117.
3743:
3705:
3694:Stong, Robert E.
3688:
3661:
3639:Poincaré duality
3632:
3610:
3608:
3583:Sullivan, Dennis
3569:
3568:
3566:
3557:
3551:
3545:
3539:
3538:
3536:
3527:
3521:
3520:
3518:
3505:
3499:
3498:
3496:
3489:
3485:"Transversality"
3481:
3475:
3474:
3472:
3465:
3461:"Thom's theorem"
3457:
3451:
3447:
3445:
3444:
3439:
3416:
3415:
3395:
3393:
3392:
3387:
3337:
3336:
3294:
3293:
3271:
3269:
3268:
3263:
3230:
3228:
3227:
3222:
3159:
3157:
3156:
3151:
3122:
3101:Steenrod problem
3068:
3066:
3065:
3060:
3046:
3045:
3017:
3015:
3014:
3009:
2989:
2988:
2969:
2967:
2966:
2961:
2956:
2955:
2897:
2895:
2894:
2889:
2886:
2881:
2863:
2862:
2843:
2841:
2840:
2835:
2811:
2809:
2808:
2803:
2783:
2782:
2775:
2774:
2764:
2751:
2749:
2748:
2743:
2731:
2729:
2728:
2725:{\displaystyle }
2723:
2702:
2700:
2699:
2694:
2686:
2685:
2681:
2676:
2675:
2664:
2637:
2635:
2634:
2629:
2627:
2626:
2615:
2593:
2591:
2590:
2585:
2583:
2550:
2548:
2547:
2542:
2530:
2528:
2527:
2522:
2498:
2496:
2495:
2490:
2488:
2467:
2465:
2464:
2459:
2410:
2408:
2407:
2402:
2385:
2384:
2369:
2368:
2332:
2331:
2316:
2315:
2291:
2290:
2271:
2269:
2268:
2263:
2239:
2237:
2236:
2231:
2220:
2219:
2199:
2197:
2196:
2191:
2189:
2188:
2169:
2167:
2166:
2161:
2136:
2134:
2133:
2128:
2120:
2119:
2114:
2099:
2098:
2077:
2076:
2071:
2056:
2055:
2043:
2042:
1970:smooth manifolds
1924:
1922:
1921:
1916:
1890:
1889:
1850:
1849:
1819:
1818:
1813:
1812:
1804:
1790:
1788:
1787:
1782:
1762:
1760:
1759:
1754:
1743:
1742:
1719:
1717:
1716:
1711:
1691:
1690:
1656:
1654:
1653:
1648:
1635:ring isomorphism
1632:
1630:
1629:
1624:
1607:
1606:
1579:
1578:
1566:
1565:
1537:
1535:
1534:
1529:
1512:
1511:
1495:
1493:
1492:
1487:
1458:
1457:
1430:
1428:
1427:
1422:
1420:
1419:
1367:
1366:
1333:
1332:
1298:
1296:
1295:
1290:
1261:
1260:
1248:
1247:
1224:
1187:
1185:
1184:
1179:
1147:
1146:
1113:
1111:
1110:
1105:
1081:
1079:
1078:
1073:
1061:
1057:Thom isomorphism
1043:
1041:
1040:
1035:
1033:
1032:
979:
977:
976:
971:
963:
962:
957:
933:
932:
921:
920:
912:
902:
901:
896:
881:
880:
855:Thom isomorphism
848:
846:
845:
840:
802:
800:
799:
794:
792:
791:
786:
753:
751:
750:
745:
717:
715:
714:
709:
685:is paracompact,
677:
675:
674:
669:
667:
666:
642:
640:
639:
634:
632:
631:
615:
613:
612:
607:
605:
604:
584:
582:
581:
576:
551:
549:
548:
543:
541:
540:
524:
522:
521:
516:
514:
513:
491:
489:
488:
483:
459:
457:
456:
451:
449:
448:
443:
421:For example, if
413:
411:
410:
405:
380:
378:
377:
372:
347:
345:
344:
339:
323:
321:
320:
315:
291:
289:
288:
283:
259:
257:
256:
251:
223:
221:
220:
215:
170:
168:
167:
162:
160:
159:
112:
110:
109:
104:
62:associated to a
21:
3860:
3859:
3855:
3854:
3853:
3851:
3850:
3849:
3830:
3829:
3811:
3803:
3746:
3708:
3692:
3685:
3665:
3652:
3629:
3613:
3581:
3578:
3573:
3572:
3564:
3559:
3558:
3554:
3546:
3542:
3534:
3529:
3528:
3524:
3508:
3507:See pp. 8-9 in
3506:
3502:
3494:
3487:
3483:
3482:
3478:
3470:
3463:
3459:
3458:
3454:
3407:
3402:
3401:
3328:
3285:
3280:
3279:
3236:
3235:
3165:
3164:
3130:
3129:
3123:
3119:
3114:
3087:
3037:
3032:
3031:
2980:
2975:
2974:
2973:where we wrote
2947:
2918:
2917:
2907:
2854:
2849:
2848:
2814:
2813:
2766:
2759:
2754:
2753:
2734:
2733:
2708:
2707:
2659:
2654:
2643:
2642:
2610:
2599:
2598:
2576:
2556:
2555:
2533:
2532:
2501:
2500:
2481:
2470:
2469:
2450:
2449:
2446:
2441:
2417:
2376:
2357:
2323:
2304:
2282:
2277:
2276:
2242:
2241:
2211:
2206:
2205:
2180:
2172:
2171:
2146:
2145:
2109:
2084:
2066:
2047:
2034:
2026:
2025:
1954:homotopy groups
1934:
1881:
1841:
1801:
1796:
1795:
1773:
1772:
1734:
1729:
1728:
1682:
1662:
1661:
1639:
1638:
1598:
1570:
1557:
1552:
1551:
1503:
1498:
1497:
1449:
1444:
1443:
1436:
1415:
1414:
1396:
1395:
1352:
1324:
1317:
1311:
1310:
1252:
1219:
1211:
1210:
1138:
1127:
1126:
1084:
1083:
1064:
1063:
1059:
1024:
1019:
1018:
989:right hand side
952:
909:
891:
872:
861:
860:
819:
818:
781:
776:
775:
764:
724:
723:
691:
690:
658:
653:
652:
643:; that is, the
623:
618:
617:
596:
591:
590:
558:
557:
532:
527:
526:
505:
494:
493:
462:
461:
438:
427:
426:
387:
386:
354:
353:
330:
329:
294:
293:
265:
264:
230:
229:
188:
187:
151:
146:
145:
83:
82:
76:
23:
22:
15:
12:
11:
5:
3858:
3856:
3848:
3847:
3842:
3832:
3831:
3828:
3827:
3809:
3802:
3801:External links
3799:
3798:
3797:
3744:
3706:
3690:
3683:
3663:
3650:
3647:Sphere bundles
3627:
3611:
3599:(3): 341–350.
3577:
3574:
3571:
3570:
3552:
3540:
3522:
3500:
3476:
3452:
3449:
3448:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3414:
3410:
3397:
3396:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3335:
3331:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3292:
3288:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3233:
3232:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3149:
3146:
3143:
3140:
3137:
3116:
3115:
3113:
3110:
3109:
3108:
3103:
3098:
3093:
3086:
3083:
3071:cobordism ring
3058:
3055:
3052:
3049:
3044:
3040:
3007:
3004:
3001:
2998:
2995:
2992:
2987:
2983:
2971:
2970:
2959:
2954:
2950:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2906:
2903:
2899:
2898:
2885:
2880:
2876:
2872:
2869:
2866:
2861:
2857:
2833:
2830:
2827:
2824:
2821:
2801:
2798:
2795:
2792:
2789:
2786:
2781:
2778:
2773:
2769:
2763:
2741:
2721:
2718:
2715:
2704:
2703:
2692:
2689:
2684:
2680:
2674:
2671:
2668:
2663:
2657:
2653:
2650:
2625:
2622:
2619:
2614:
2609:
2606:
2595:
2594:
2582:
2579:
2575:
2572:
2569:
2566:
2563:
2540:
2520:
2517:
2514:
2511:
2508:
2487:
2484:
2480:
2477:
2457:
2445:
2444:Real cobordism
2442:
2440:
2437:
2433:Sergei Novikov
2421:tangent bundle
2416:
2413:
2412:
2411:
2400:
2397:
2394:
2391:
2388:
2383:
2379:
2375:
2372:
2367:
2364:
2360:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2330:
2326:
2322:
2319:
2314:
2311:
2307:
2303:
2300:
2297:
2294:
2289:
2285:
2261:
2258:
2255:
2252:
2249:
2229:
2226:
2223:
2218:
2214:
2187:
2183:
2179:
2159:
2156:
2153:
2138:
2137:
2126:
2123:
2118:
2113:
2108:
2105:
2102:
2097:
2094:
2091:
2087:
2083:
2080:
2075:
2070:
2065:
2062:
2059:
2054:
2050:
2046:
2041:
2037:
2033:
1986:surgery theory
1982:Sergei Novikov
1968:properties of
1966:transversality
1933:
1930:
1926:
1925:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1888:
1884:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1848:
1844:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1817:
1810:
1807:
1780:
1752:
1749:
1746:
1741:
1737:
1721:
1720:
1709:
1706:
1703:
1700:
1697:
1694:
1689:
1685:
1681:
1678:
1675:
1672:
1669:
1646:
1622:
1619:
1616:
1613:
1610:
1605:
1601:
1597:
1594:
1591:
1588:
1585:
1582:
1577:
1573:
1569:
1564:
1560:
1527:
1524:
1521:
1518:
1515:
1510:
1506:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1456:
1452:
1432:
1431:
1418:
1413:
1410:
1407:
1404:
1401:
1398:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1365:
1362:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1336:
1331:
1327:
1323:
1322:
1320:
1300:
1299:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1259:
1255:
1251:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1223:
1218:
1189:
1188:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1145:
1141:
1137:
1134:
1103:
1100:
1097:
1094:
1091:
1082:be a ring and
1071:
1054:
1031:
1027:
981:
980:
969:
966:
961:
956:
951:
948:
945:
942:
939:
936:
931:
928:
925:
918:
915:
908:
905:
900:
895:
890:
887:
884:
879:
875:
871:
868:
838:
835:
832:
829:
826:
790:
785:
763:
760:
743:
740:
737:
734:
731:
707:
704:
701:
698:
665:
661:
630:
626:
603:
599:
574:
571:
568:
565:
539:
535:
512:
508:
504:
501:
481:
478:
475:
472:
469:
447:
442:
437:
434:
403:
400:
397:
394:
370:
367:
364:
361:
337:
313:
310:
307:
304:
301:
281:
278:
275:
272:
260:we obtain the
249:
246:
243:
240:
237:
224:by taking the
213:
210:
207:
204:
201:
198:
195:
158:
154:
114:
113:
102:
99:
96:
93:
90:
75:
72:
48:Lev Pontryagin
24:
14:
13:
10:
9:
6:
4:
3:
2:
3857:
3846:
3843:
3841:
3838:
3837:
3835:
3824:
3820:
3819:
3814:
3810:
3808:
3805:
3804:
3800:
3794:
3790:
3786:
3782:
3778:
3774:
3769:
3764:
3760:
3756:
3755:
3750:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3720:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3686:
3684:0-226-51182-0
3680:
3676:
3672:
3668:
3667:May, J. Peter
3664:
3659:
3655:
3651:
3648:
3644:
3640:
3636:
3630:
3628:0-387-90613-4
3624:
3620:
3616:
3612:
3607:
3602:
3598:
3594:
3593:
3588:
3584:
3580:
3579:
3575:
3563:
3556:
3553:
3549:
3544:
3541:
3533:
3526:
3523:
3517:
3512:
3504:
3501:
3493:
3486:
3480:
3477:
3469:
3462:
3456:
3453:
3432:
3426:
3423:
3420:
3412:
3408:
3400:
3399:
3383:
3377:
3368:
3362:
3359:
3356:
3350:
3344:
3341:
3333:
3329:
3325:
3319:
3316:
3310:
3304:
3301:
3298:
3290:
3286:
3278:
3277:
3275:
3259:
3250:
3244:
3241:
3215:
3212:
3209:
3200:
3194:
3191:
3188:
3182:
3176:
3173:
3163:
3162:
3144:
3138:
3135:
3127:
3121:
3118:
3111:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3088:
3084:
3082:
3080:
3076:
3072:
3053:
3050:
3042:
3038:
3029:
3025:
3021:
3002:
2996:
2993:
2985:
2981:
2952:
2948:
2941:
2938:
2932:
2926:
2923:
2916:
2915:
2914:
2912:
2911:Thom spectrum
2904:
2902:
2883:
2878:
2870:
2867:
2864:
2859:
2855:
2847:
2846:
2845:
2828:
2822:
2819:
2796:
2793:
2790:
2784:
2779:
2776:
2771:
2767:
2739:
2716:
2690:
2682:
2678:
2672:
2669:
2666:
2655:
2651:
2648:
2641:
2640:
2639:
2623:
2620:
2617:
2604:
2580:
2577:
2573:
2570:
2567:
2564:
2554:
2553:
2552:
2538:
2515:
2512:
2509:
2485:
2482:
2478:
2475:
2455:
2443:
2439:Thom spectrum
2438:
2436:
2434:
2430:
2426:
2422:
2414:
2398:
2389:
2381:
2377:
2373:
2365:
2362:
2354:
2342:
2328:
2324:
2320:
2312:
2309:
2301:
2295:
2287:
2283:
2275:
2274:
2273:
2259:
2253:
2250:
2247:
2224:
2216:
2212:
2203:
2185:
2181:
2177:
2157:
2154:
2151:
2143:
2124:
2116:
2106:
2103:
2095:
2092:
2089:
2085:
2073:
2063:
2060:
2052:
2048:
2044:
2039:
2035:
2031:
2024:
2023:
2022:
2021:
2016:
2014:
2010:
2006:
2002:
1999:now known as
1998:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1929:
1912:
1906:
1900:
1897:
1894:
1886:
1882:
1878:
1872:
1869:
1863:
1857:
1854:
1846:
1842:
1838:
1829:
1823:
1815:
1805:
1794:
1793:
1792:
1770:
1766:
1747:
1739:
1735:
1726:
1707:
1704:
1701:
1695:
1687:
1683:
1679:
1673:
1660:
1659:
1658:
1636:
1614:
1611:
1603:
1599:
1586:
1583:
1575:
1571:
1567:
1562:
1558:
1549:
1545:
1541:
1519:
1516:
1508:
1504:
1477:
1474:
1468:
1465:
1462:
1454:
1450:
1441:
1435:
1411:
1408:
1405:
1399:
1386:
1383:
1377:
1374:
1371:
1363:
1360:
1357:
1353:
1340:
1337:
1329:
1325:
1318:
1309:
1308:
1307:
1305:
1280:
1277:
1271:
1268:
1265:
1257:
1253:
1249:
1241:
1235:
1232:
1229:
1216:
1209:
1208:
1207:
1206:
1202:
1198:
1194:
1175:
1166:
1163:
1157:
1154:
1151:
1143:
1139:
1135:
1132:
1125:
1124:
1123:
1121:
1117:
1101:
1095:
1092:
1089:
1053:
1051:
1047:
1029:
1025:
1016:
1012:
1008:
1003:
1001:
996:
994:
990:
986:
967:
959:
949:
943:
937:
929:
926:
923:
916:
913:
898:
888:
885:
877:
873:
869:
859:
858:
857:
856:
852:
836:
830:
827:
824:
815:
813:
809:
808:orientability
805:
788:
773:
772:fiber bundles
769:
761:
759:
757:
738:
735:
732:
721:
702:
696:
688:
684:
679:
663:
659:
650:
646:
628:
624:
601:
597:
588:
587:smash product
569:
563:
555:
537:
533:
525:and, writing
510:
506:
502:
499:
476:
470:
467:
445:
435:
432:
424:
419:
417:
398:
392:
384:
365:
359:
351:
327:
308:
302:
299:
276:
270:
263:
244:
238:
235:
227:
211:
202:
196:
193:
186:
185:sphere bundle
182:
178:
174:
156:
152:
144:
140:
136:
132:
129:
125:
124:vector bundle
122:
119:
100:
94:
91:
88:
81:
80:
79:
73:
71:
69:
65:
64:vector bundle
61:
57:
53:
49:
45:
42:(named after
41:
37:
36:Thom complex,
34:
30:
19:
3816:
3813:"Thom space"
3758:
3752:
3723:
3717:
3697:
3670:
3657:
3654:Milnor, John
3618:
3596:
3590:
3560:Francis, J.
3555:
3543:
3530:Francis, J.
3525:
3516:math/0609452
3503:
3479:
3455:
3450:by excision.
3273:
3125:
3120:
3023:
2972:
2910:
2908:
2900:
2705:
2596:
2447:
2418:
2201:
2141:
2139:
2017:
2001:Thom spectra
2000:
1996:
1989:
1961:
1957:
1945:
1935:
1927:
1768:
1764:
1722:
1547:
1543:
1539:
1439:
1437:
1433:
1306:. Moreover,
1303:
1301:
1204:
1200:
1196:
1192:
1190:
1119:
1055:
1045:
1014:
1010:
1006:
1004:
997:
984:
982:
854:
850:
816:
804:coefficients
765:
755:
722:by the unit
719:
686:
682:
680:
647:-th reduced
644:
553:
422:
420:
415:
382:
325:
261:
180:
177:vector space
172:
138:
134:
130:
117:
115:
77:
39:
35:
32:
26:
3643:Euler class
3615:Bott, Raoul
2468:-manifolds
1978:John Milnor
1727:element of
1496:as a right
810:; see also
68:paracompact
66:, over any
33:Thom space,
29:mathematics
3834:Categories
3710:Thom, René
3576:References
3548:Stong 1968
2551:such that
2425:Wu formula
1940:, and the
1544:Thom class
768:cohomology
649:suspension
262:Thom space
116:be a rank
3823:EMS Press
3793:119613530
3768:0810.4535
3740:120243638
3726:: 17–86.
3712:(1954). "
3430:∖
3375:∖
3363:
3345:
3326:≃
3257:∖
3245:
3234:Clearly,
3207:∖
3195:
3177:
3139:
3091:Cobordism
3043:∗
3039:π
2991:→
2982:γ
2949:γ
2875:Ω
2871:≅
2856:π
2785:×
2688:→
2649:ν
2608:↪
2574:∐
2562:∂
2431:, due to
2363:−
2359:Φ
2337:Φ
2310:−
2306:Φ
2257:→
2104:−
2082:→
2061:−
1950:cobordism
1948:that the
1904:∖
1879:≃
1858:
1809:~
1779:Λ
1740:∗
1702:⌣
1688:∗
1668:Φ
1645:Φ
1618:Λ
1604:∗
1596:→
1590:Λ
1576:∗
1563:∗
1523:Λ
1509:∗
1481:Λ
1472:∖
1455:∗
1409:⌣
1403:⟼
1390:Λ
1381:∖
1350:→
1344:Λ
1284:Λ
1275:∖
1250:∈
1239:∖
1170:Λ
1161:∖
1136:∈
1099:→
1070:Λ
1000:René Thom
917:~
907:→
867:Φ
834:→
736:−
503:×
471:
436:×
350:basepoint
336:∞
303:
239:
209:→
197:
126:over the
98:→
92::
44:René Thom
3696:(1968).
3669:(1999).
3585:(2004).
3492:Archived
3468:Archived
3085:See also
3028:spectrum
3022:of rank
3018:for the
2581:′
2486:′
1725:identity
1116:oriented
1009:of rank
983:for all
460:, then
3825:, 2001
3785:0286898
3550:, p. 18
2170:, then
1994:spectra
585:is the
70:space.
3791:
3783:
3738:
3681:
3625:
2005:stable
1191:where
1114:be an
171:is an
141:, the
31:, the
3789:S2CID
3763:arXiv
3736:S2CID
3565:(PDF)
3535:(PDF)
3511:arXiv
3495:(PDF)
3488:(PDF)
3471:(PDF)
3464:(PDF)
3128:into
3112:Notes
2144:. If
1990:MG(n)
1972:—see
1633:is a
381:. If
143:fiber
58:is a
50:) of
3679:ISBN
3623:ISBN
2272:by:
1980:and
1062:Let
817:Let
616:and
552:for
137:in
121:real
54:and
46:and
3773:doi
3728:doi
3716:".
3645:of
3601:doi
3360:Sph
3342:Sph
3242:Sph
3192:Sph
3174:Sph
3136:Sph
1855:Sph
1763:to
1546:of
991:is
814:.)
770:of
651:of
589:of
492:is
468:Sph
352:of
324:by
300:Sph
236:Sph
194:Sph
38:or
27:In
3836::
3821:,
3815:,
3787:.
3781:MR
3779:.
3771:.
3757:.
3734:.
3724:28
3722:.
3700:.
3673:.
3656:.
3641:,
3597:41
3595:.
3589:.
3490:.
3466:.
2435:.
2015:.
1997:MG
1958:MG
1791:)
1637:,
1044:,
995:.
758:.
678:.
418:.
3795:.
3775::
3765::
3759:7
3742:.
3730::
3704:.
3687:.
3660:.
3631:.
3609:.
3603::
3519:.
3513::
3436:)
3433:B
3427:E
3424:,
3421:E
3418:(
3413:n
3409:H
3384:,
3381:)
3378:B
3372:)
3369:E
3366:(
3357:,
3354:)
3351:E
3348:(
3339:(
3334:n
3330:H
3323:)
3320:B
3317:,
3314:)
3311:E
3308:(
3305:h
3302:p
3299:S
3296:(
3291:n
3287:H
3274:B
3260:B
3254:)
3251:E
3248:(
3231:.
3219:)
3216:B
3213:,
3210:B
3204:)
3201:E
3198:(
3189:,
3186:)
3183:E
3180:(
3171:(
3148:)
3145:E
3142:(
3126:B
3057:)
3054:O
3051:M
3048:(
3024:n
3006:)
3003:n
3000:(
2997:O
2994:B
2986:n
2958:)
2953:n
2945:(
2942:T
2939:=
2936:)
2933:n
2930:(
2927:O
2924:M
2884:O
2879:n
2868:O
2865:M
2860:n
2832:)
2829:n
2826:(
2823:O
2820:M
2800:]
2797:1
2794:,
2791:0
2788:[
2780:n
2777:+
2772:W
2768:N
2762:R
2740:W
2720:]
2717:M
2714:[
2691:M
2683:M
2679:/
2673:n
2670:+
2667:N
2662:R
2656:N
2652::
2624:n
2621:+
2618:N
2613:R
2605:M
2578:M
2571:M
2568:=
2565:W
2539:W
2519:)
2516:1
2513:+
2510:n
2507:(
2483:M
2479:,
2476:M
2456:n
2399:.
2396:)
2393:)
2390:u
2387:(
2382:i
2378:q
2374:S
2371:(
2366:1
2355:=
2352:)
2349:)
2346:)
2343:1
2340:(
2334:(
2329:i
2325:q
2321:S
2318:(
2313:1
2302:=
2299:)
2296:p
2293:(
2288:i
2284:w
2260:B
2254:E
2251::
2248:p
2228:)
2225:p
2222:(
2217:i
2213:w
2202:i
2186:i
2182:q
2178:S
2158:m
2155:=
2152:i
2142:m
2125:,
2122:)
2117:2
2112:Z
2107:;
2101:(
2096:i
2093:+
2090:m
2086:H
2079:)
2074:2
2069:Z
2064:;
2058:(
2053:m
2049:H
2045::
2040:i
2036:q
2032:S
1962:n
1960:(
1913:.
1910:)
1907:B
1901:E
1898:,
1895:E
1892:(
1887:n
1883:H
1876:)
1873:B
1870:,
1867:)
1864:E
1861:(
1852:(
1847:n
1843:H
1839:=
1836:)
1833:)
1830:E
1827:(
1824:T
1821:(
1816:n
1806:H
1769:u
1765:u
1751:)
1748:B
1745:(
1736:H
1708:.
1705:u
1699:)
1696:b
1693:(
1684:p
1680:=
1677:)
1674:b
1671:(
1621:)
1615:;
1612:E
1609:(
1600:H
1593:)
1587:;
1584:B
1581:(
1572:H
1568::
1559:p
1548:E
1540:u
1526:)
1520:;
1517:E
1514:(
1505:H
1484:)
1478:;
1475:B
1469:E
1466:,
1463:E
1460:(
1451:H
1440:u
1412:u
1406:x
1400:x
1393:)
1387:;
1384:B
1378:E
1375:,
1372:E
1369:(
1364:n
1361:+
1358:k
1354:H
1347:)
1341:;
1338:E
1335:(
1330:k
1326:H
1319:{
1304:F
1287:)
1281:;
1278:0
1272:F
1269:,
1266:F
1263:(
1258:n
1254:H
1245:)
1242:0
1236:F
1233:,
1230:F
1227:(
1222:|
1217:u
1205:u
1201:F
1197:E
1193:B
1176:,
1173:)
1167:;
1164:B
1158:E
1155:,
1152:E
1149:(
1144:n
1140:H
1133:u
1120:n
1102:B
1096:E
1093::
1090:p
1046:B
1030:+
1026:B
1015:k
1011:k
1007:B
985:k
968:,
965:)
960:2
955:Z
950:;
947:)
944:E
941:(
938:T
935:(
930:n
927:+
924:k
914:H
904:)
899:2
894:Z
889:;
886:B
883:(
878:k
874:H
870::
851:n
837:B
831:E
828::
825:p
789:2
784:Z
756:E
742:)
739:1
733:n
730:(
720:E
706:)
703:E
700:(
697:T
687:E
683:B
664:+
660:B
645:n
629:n
625:S
602:+
598:B
573:)
570:E
567:(
564:T
554:B
538:+
534:B
511:n
507:S
500:B
480:)
477:E
474:(
446:n
441:R
433:B
423:E
416:E
402:)
399:E
396:(
393:T
383:B
369:)
366:E
363:(
360:T
326:B
312:)
309:E
306:(
280:)
277:E
274:(
271:T
248:)
245:E
242:(
212:B
206:)
203:E
200:(
183:-
181:n
173:n
157:b
153:E
139:B
135:b
131:B
118:n
101:B
95:E
89:p
20:)
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