20:
4332:
4115:
1368:
4353:
4321:
4390:
4363:
4343:
1151:
270:
1135:
1363:{\displaystyle {\begin{aligned}0\rightarrow K_{1}\rightarrow {}&A_{1}\rightarrow K_{2}\rightarrow 0,\\0\rightarrow K_{2}\rightarrow {}&A_{2}\rightarrow K_{3}\rightarrow 0,\\&\ \,\vdots \\0\rightarrow K_{n-1}\rightarrow {}&A_{n-1}\rightarrow K_{n}\rightarrow 0,\\\end{aligned}}}
1855:
1577:
1731:
3213:
107:
977:
3861:
If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a
2823:
3302:(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the
799:
3377:
3297:
1785:
3085:
2991:
The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence
2731:
864:
661:
1516:
380:
2285:
3689:
1156:
2114:
1425:
2615:
3618:
1679:
2010:
2332:
1667:
955:
3553:
3474:
2467:
1639:
495:. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).
443:
2946:
For non-commutative groups, the splitting lemma does not apply, and one has only the equivalence between the two last conditions, with "the direct sum" replaced with "a
2529:
2429:
3507:
2364:
2182:
491:
To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the
2501:
2398:
2144:
1476:
411:
310:
51:
265:{\displaystyle G_{0}\;{\xrightarrow {\ f_{1}\ }}\;G_{1}\;{\xrightarrow {\ f_{2}\ }}\;G_{2}\;{\xrightarrow {\ f_{3}\ }}\;\cdots \;{\xrightarrow {\ f_{n}\ }}\;G_{n}}
3103:
2557:
3428:
2036:
4393:
1445:
1130:{\displaystyle A_{0}\;\xrightarrow {\ f_{1}\ } \;A_{1}\;\xrightarrow {\ f_{2}\ } \;A_{2}\;\xrightarrow {\ f_{3}\ } \;\cdots \;\xrightarrow {\ f_{n}\ } \;A_{n},}
2751:
3397:
3762:
732:
3329:
1850:{\displaystyle 0\to \mathbf {Z} \mathrel {\overset {2\times }{\longrightarrow }} \mathbf {Z} \longrightarrow \mathbf {Z} /2\mathbf {Z} \to 0}
3225:
2998:
1572:{\displaystyle \mathbf {Z} \mathrel {\overset {2\times }{\,\hookrightarrow }} \mathbf {Z} \twoheadrightarrow \mathbf {Z} /2\mathbf {Z} }
2630:
1744:
and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to 2
810:
4027:
608:
1768:
used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2
4381:
4376:
3996:
3966:
3219:
Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:
1953:
to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the
4371:
3718:
Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.
315:
3858:
Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.
2228:
4273:
3623:
4419:
4414:
2979:
gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the
2051:
3879:
972:
A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence
1378:
1726:{\displaystyle 2\mathbf {Z} \mathrel {\,\hookrightarrow } \mathbf {Z} \twoheadrightarrow \mathbf {Z} /2\mathbf {Z} }
4281:
2570:
19:
3726:
In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about
3558:
1954:
4352:
4080:
3766:
472:
1673:
of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as
1963:
507:. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from
4366:
2564:
These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence
4301:
4296:
4222:
4099:
4087:
4060:
4020:
3843:
2299:
468:
1652:
4143:
4070:
4331:
1779:
The first sequence may also be written without using special symbols for monomorphism and epimorphism:
925:
3512:
2434:
1624:
4291:
4243:
4217:
4065:
3882:
is another example. Long exact sequences induced by short exact sequences are also characteristic of
3433:
871:
78:
4342:
4138:
3988:
2965:
2213:
464:
452:
86:
70:
24:
416:
4336:
4286:
4207:
4197:
4075:
4055:
3871:
3303:
2947:
1895:
is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first
1884:
480:
276:
74:
4306:
3391:).) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:
2508:
2408:
4324:
4190:
4148:
4013:
3992:
3962:
3958:
3875:
3866:(that is, an exact sequence indexed by the natural numbers) on homology by application of the
3734:
3384:
3479:
2343:
1945:
The first and third sequences are somewhat of a special case owing to the infinite nature of
4104:
4050:
3980:
3950:
3396:
3208:{\displaystyle C_{k}\cong \ker(A_{k}\to A_{k+1})\cong \operatorname {im} (A_{k-1}\to A_{k})}
2980:
2157:
586:
90:
82:
2474:
2371:
2122:
1454:
389:
288:
29:
4163:
4158:
3918:
3883:
3851:
3700:
2742:
2621:
711:
3981:
2536:
2018:
1669:
indicates an epimorphism (the map mod 2). This is an exact sequence because the image 2
4253:
4185:
3976:
3406:
2185:
1430:
3403:
The only portion of this diagram that depends on the cokernel condition is the object
1497:
for details on how to re-form the long exact sequence from the short exact sequences.
4408:
4263:
4173:
4153:
3951:
3946:
3889:
3867:
3811:
2039:
900:
534:. Therefore the sequence is exact if and only if the image of the leftmost map (from
492:
54:
4356:
4248:
4168:
4114:
2147:
1950:
1873:
516:
456:
3691:
is ensured. Again taking the example of the category of groups, the fact that im(
4346:
4258:
2961:
2818:{\displaystyle 0\to A\;\xrightarrow {\ f\ } \;B\;\xrightarrow {\ g\ } \;C\to 0,}
574:
547:
4202:
4133:
4092:
3749:?" In the category of groups, this is equivalent to the question, what groups
2976:
2969:
2917:
2403:
908:
570:
460:
445:, i.e., if the image of each homomorphism is equal to the kernel of the next.
4227:
3727:
687:
66:
448:
The sequence of groups and homomorphisms may be either finite or infinite.
16:
Sequence of homomorphisms such that each kernel equals the preceding image
4212:
4180:
4129:
4036:
794:{\displaystyle C\cong B/\operatorname {im} (f)=B/\operatorname {ker} (g)}
476:
3745:
of a short exact sequence, what possibilities exist for the middle term
3893:
3372:{\displaystyle H/{\left\langle \operatorname {im} f\right\rangle }^{H}}
467:. More generally, the notion of an exact sequence makes sense in any
3761:
as the corresponding factor group? This problem is important in the
3292:{\displaystyle C_{k}\cong \operatorname {coker} (A_{k-2}\to A_{k-1})}
2790:
2769:
1094:
1066:
1031:
996:
840:
827:
638:
625:
231:
201:
164:
127:
3080:{\displaystyle A_{1}\to A_{2}\to A_{3}\to A_{4}\to A_{5}\to A_{6}}
2045:
As a more concrete example of an exact sequence on finite groups:
969:, to distinguish from the special case of a short exact sequence.
18:
2726:{\displaystyle 0\to R/(I\cap J)\to R/I\oplus R/J\to R/(I+J)\to 0}
859:{\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0\,}
1752:
through this monomorphism is however exactly the same subset of
656:{\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0.}
4009:
2968:
with two exact rows gives rise to a longer exact sequence. The
3850:
Exact sequences are precisely those chain complexes which are
2983:
is a special case thereof applying to short exact sequences.
4005:
3983:
Commutative
Algebra: with a View Toward Algebraic Geometry
3703:, which coincides with its conjugate closure; thus coker(
2953:
In both cases, one says that such a short exact sequence
666:
As established above, for any such short exact sequence,
515:) has kernel {0}; that is, if and only if that map is a
375:{\displaystyle \operatorname {im} (f_{i})=\ker(f_{i+1})}
3436:
3409:
602:
Short exact sequences are exact sequences of the form
3896:
that transform exact sequences into exact sequences.
3626:
3561:
3515:
3482:
3332:
3228:
3106:
3001:
2754:
2633:
2573:
2539:
2511:
2477:
2437:
2411:
2374:
2346:
2302:
2280:{\displaystyle 0\to I\cap J\to I\oplus J\to I+J\to 0}
2231:
2160:
2125:
2054:
2021:
1966:
1788:
1682:
1655:
1627:
1519:
1457:
1433:
1381:
1154:
980:
928:
813:
735:
611:
419:
392:
318:
291:
110:
32:
455:. For example, one could have an exact sequence of
4272:
4236:
4122:
4043:
3846:of this chain complex is trivial. More succinctly:
3684:{\displaystyle 0\to C_{k}\to A_{k}\to C_{k+1}\to 0}
1510:Consider the following sequence of abelian groups:
569:is both a monomorphism and epimorphism (that is, a
3772:Notice that in an exact sequence, the composition
3683:
3612:
3547:
3501:
3468:
3422:
3371:
3291:
3207:
3079:
2817:
2725:
2609:
2551:
2523:
2495:
2461:
2423:
2392:
2358:
2326:
2279:
2176:
2138:
2109:{\displaystyle 1\to C_{n}\to D_{2n}\to C_{2}\to 1}
2108:
2030:
2004:
1849:
1725:
1661:
1633:
1571:
1470:
1439:
1419:
1362:
1129:
949:
858:
793:
655:
437:
405:
374:
304:
264:
45:
3737:is essentially the question "Given the end terms
1420:{\displaystyle K_{i}=\operatorname {im} (f_{i})}
1906:, and the kernel of reducing modulo 2 is also 2
1860:Here 0 denotes the trivial group, the map from
1146:2, we can split it up into the short sequences
965:A general exact sequence is sometimes called a
1776:even though the two are isomorphic as groups.
4021:
2610:{\displaystyle 0\to R\to R\oplus R\to R\to 0}
674:is an epimorphism. Furthermore, the image of
8:
1649:is a monomorphism, and the two-headed arrow
1598:. The second homomorphism maps each element
3613:{\displaystyle A_{k-1}\to A_{k}\to A_{k+1}}
1933:, so the sequence is exact at the position
451:A similar definition can be made for other
101:In the context of group theory, a sequence
4389:
4362:
4028:
4014:
4006:
2802:
2785:
2781:
2764:
1113:
1089:
1085:
1061:
1050:
1026:
1015:
991:
251:
225:
221:
195:
184:
158:
147:
121:
3663:
3650:
3637:
3625:
3598:
3585:
3566:
3560:
3533:
3520:
3514:
3487:
3481:
3454:
3441:
3435:
3414:
3408:
3363:
3342:
3336:
3331:
3274:
3255:
3233:
3227:
3196:
3177:
3146:
3133:
3111:
3105:
3071:
3058:
3045:
3032:
3019:
3006:
3000:
2753:
2745:states that, for a short exact sequence
2697:
2683:
2669:
2643:
2632:
2572:
2538:
2510:
2476:
2436:
2410:
2373:
2345:
2301:
2230:
2165:
2159:
2130:
2124:
2094:
2078:
2065:
2053:
2020:
1988:
1965:
1925:, and the kernel of the zero map is also
1910:, so the sequence is exact at the second
1868:is multiplication by 2, and the map from
1836:
1828:
1823:
1815:
1800:
1795:
1787:
1718:
1710:
1705:
1697:
1692:
1691:
1686:
1681:
1654:
1626:
1582:The first homomorphism maps each element
1564:
1556:
1551:
1543:
1528:
1525:
1520:
1518:
1462:
1456:
1432:
1408:
1386:
1380:
1341:
1322:
1314:
1299:
1281:
1258:
1245:
1237:
1228:
1199:
1186:
1178:
1169:
1155:
1153:
1118:
1102:
1074:
1055:
1039:
1020:
1004:
985:
979:
927:
855:
812:
771:
745:
734:
610:
561:→ 0 is exact if and only if the map from
546:; that is, if and only if that map is an
418:
397:
391:
357:
332:
317:
296:
290:
256:
239:
226:
209:
196:
189:
172:
159:
152:
135:
122:
115:
109:
37:
31:
3695:) is the kernel of some homomorphism on
2825:the following conditions are equivalent.
2296:-modules, where the module homomorphism
1486:is a long exact sequence if and only if
530:→ 0. The kernel of the rightmost map is
3910:
3090:which implies that there exist objects
2038:as these groups are not supposed to be
2005:{\displaystyle 1\to N\to G\to G/N\to 1}
710:as the corresponding factor object (or
1902:the image of multiplication by 2 is 2
1887:2. This is indeed an exact sequence:
81:, and, more generally, objects of an
23:Illustration of an exact sequence of
7:
3987:. Springer-Verlag New York. p.
3919:"exact sequence in nLab, Remark 2.3"
2327:{\displaystyle I\cap J\to I\oplus J}
2015:(here the trivial group is denoted
1736:In this case the monomorphism is 2
1662:{\displaystyle \twoheadrightarrow }
1478:'s (regardless of the exactness of
1917:the image of reducing modulo 2 is
1494:
14:
1447:. By construction, the sequences
950:{\displaystyle B\cong A\oplus C.}
4388:
4361:
4351:
4341:
4330:
4320:
4319:
4113:
3620:is exact, then the exactness of
3548:{\displaystyle A_{k}\to A_{k+1}}
3469:{\textstyle A_{6}\to C_{7}\to 0}
3430:and the final pair of morphisms
3395:
2462:{\displaystyle I\oplus J\to I+J}
2192:, which is a non-abelian group.
1837:
1824:
1816:
1796:
1719:
1706:
1698:
1687:
1634:{\displaystyle \hookrightarrow }
1610:in the quotient group; that is,
1565:
1552:
1544:
1521:
3810:, so every exact sequence is a
3722:Applications of exact sequences
2196:Intersection and sum of modules
1641:indicates that the map 2× from
1490:are all short exact sequences.
899:. It follows that if these are
875:if there exists a homomorphism
3675:
3656:
3643:
3630:
3591:
3578:
3526:
3476:. If there exists any object
3460:
3447:
3286:
3267:
3248:
3202:
3189:
3170:
3158:
3139:
3126:
3064:
3051:
3038:
3025:
3012:
2806:
2758:
2717:
2714:
2702:
2691:
2663:
2660:
2648:
2637:
2624:yields another exact sequence
2601:
2595:
2583:
2577:
2490:
2478:
2447:
2387:
2375:
2312:
2271:
2259:
2247:
2235:
2100:
2087:
2071:
2058:
1996:
1982:
1976:
1970:
1883:is given by reducing integers
1841:
1820:
1802:
1792:
1702:
1693:
1656:
1628:
1548:
1529:
1414:
1401:
1347:
1334:
1311:
1292:
1264:
1251:
1234:
1221:
1205:
1192:
1175:
1162:
849:
817:
788:
782:
762:
756:
647:
615:
369:
350:
338:
325:
69:between objects (for example,
1:
3707:) is isomorphic to the image
3957:. Berlin: Springer. p.
682:. It is helpful to think of
553:Therefore, the sequence 0 →
483:, where it is widely used.
438:{\displaystyle 1\leq i<n}
1949:. It is not possible for a
522:Consider the dual sequence
519:(injective, or one-to-one).
89:of one morphism equals the
4436:
4282:Banach fixed-point theorem
3097:in the category such that
887:such that the composition
678:is equal to the kernel of
499:Consider the sequence 0 →
4315:
4111:
3757:as a normal subgroup and
2524:{\displaystyle I\oplus J}
2424:{\displaystyle I\oplus J}
1955:first isomorphism theorem
1891:the image of the map 0 →
804:The short exact sequence
382:. The sequence is called
3767:Outer automorphism group
3763:classification of groups
3715:) of the next morphism.
2896:There exists a morphism
2863:There exists a morphism
2830:There exists a morphism
2290:is an exact sequence of
726:inducing an isomorphism
479:, and more specially in
3880:Mayer–Vietoris sequence
3823:-images of elements of
3502:{\displaystyle A_{k+1}}
2431:, and the homomorphism
2359:{\displaystyle I\cap J}
1772:is not the same set as
1586:in the set of integers
895:is the identity map on
386:if it is exact at each
4337:Mathematics portal
4237:Metrics and properties
4223:Second-countable space
3685:
3614:
3549:
3503:
3470:
3424:
3373:
3293:
3209:
3081:
2819:
2727:
2611:
2553:
2525:
2497:
2463:
2425:
2394:
2360:
2328:
2281:
2178:
2177:{\displaystyle D_{2n}}
2140:
2110:
2032:
2006:
1851:
1727:
1663:
1635:
1621:. Here the hook arrow
1573:
1472:
1441:
1421:
1364:
1131:
951:
860:
795:
670:is a monomorphism and
657:
585:(this always holds in
550:(surjective, or onto).
439:
407:
376:
306:
266:
58:
47:
3699:implies that it is a
3686:
3615:
3550:
3504:
3471:
3425:
3374:
3294:
3210:
3082:
2820:
2728:
2612:
2554:
2526:
2498:
2496:{\displaystyle (x,y)}
2464:
2426:
2395:
2393:{\displaystyle (x,x)}
2361:
2329:
2282:
2179:
2141:
2139:{\displaystyle C_{n}}
2111:
2033:
2007:
1852:
1728:
1664:
1636:
1574:
1473:
1471:{\displaystyle K_{i}}
1442:
1422:
1365:
1132:
952:
907:is isomorphic to the
861:
796:
658:
573:), and so usually an
440:
408:
406:{\displaystyle G_{i}}
377:
307:
305:{\displaystyle G_{i}}
267:
48:
46:{\displaystyle G_{i}}
22:
4292:Invariance of domain
4244:Euler characteristic
4218:Bundle (mathematics)
3947:Spanier, Edwin Henry
3814:. Furthermore, only
3730:and factor objects.
3624:
3559:
3513:
3480:
3434:
3407:
3330:
3226:
3104:
2999:
2752:
2631:
2571:
2537:
2509:
2475:
2435:
2409:
2372:
2344:
2300:
2229:
2158:
2123:
2052:
2019:
1964:
1786:
1680:
1653:
1625:
1517:
1455:
1431:
1379:
1152:
978:
926:
811:
733:
609:
598:Short exact sequence
465:module homomorphisms
463:, or of modules and
453:algebraic structures
417:
390:
316:
289:
108:
30:
4420:Additive categories
4415:Homological algebra
4302:Tychonoff's theorem
4297:Poincaré conjecture
4051:General (point-set)
3864:long exact sequence
3832:are mapped to 0 by
2972:is a special case.
2966:commutative diagram
2887:is the identity on
2854:is the identity on
2800:
2779:
2552:{\displaystyle x-y}
1506:Integers modulo two
1111:
1083:
1048:
1013:
967:long exact sequence
961:Long exact sequence
844:
831:
642:
629:
277:group homomorphisms
248:
218:
181:
144:
4287:De Rham cohomology
4208:Polyhedral complex
4198:Simplicial complex
3953:Algebraic Topology
3872:algebraic topology
3681:
3610:
3545:
3499:
3466:
3423:{\textstyle C_{7}}
3420:
3379:, the quotient of
3369:
3304:category of groups
3289:
3205:
3077:
2948:semidirect product
2815:
2723:
2607:
2549:
2521:
2493:
2469:maps each element
2459:
2421:
2390:
2356:
2334:maps each element
2324:
2277:
2174:
2136:
2106:
2031:{\displaystyle 1,}
2028:
2002:
1847:
1723:
1659:
1631:
1569:
1468:
1437:
1417:
1360:
1358:
1127:
947:
856:
791:
653:
481:abelian categories
435:
403:
372:
302:
262:
59:
43:
4402:
4401:
4191:fundamental group
3876:relative homology
3870:. It comes up in
3735:extension problem
3385:conjugate closure
3306:, in which coker(
2801:
2799:
2793:
2780:
2778:
2772:
1813:
1541:
1451:are exact at the
1440:{\displaystyle i}
1280:
1112:
1110:
1097:
1084:
1082:
1069:
1049:
1047:
1034:
1014:
1012:
999:
845:
832:
643:
630:
249:
247:
234:
219:
217:
204:
182:
180:
167:
145:
143:
130:
65:is a sequence of
4427:
4392:
4391:
4365:
4364:
4355:
4345:
4335:
4334:
4323:
4322:
4117:
4030:
4023:
4016:
4007:
4002:
3986:
3972:
3956:
3933:
3932:
3930:
3929:
3915:
3884:derived functors
3874:in the study of
3690:
3688:
3687:
3682:
3674:
3673:
3655:
3654:
3642:
3641:
3619:
3617:
3616:
3611:
3609:
3608:
3590:
3589:
3577:
3576:
3554:
3552:
3551:
3546:
3544:
3543:
3525:
3524:
3508:
3506:
3505:
3500:
3498:
3497:
3475:
3473:
3472:
3467:
3459:
3458:
3446:
3445:
3429:
3427:
3426:
3421:
3419:
3418:
3399:
3378:
3376:
3375:
3370:
3368:
3367:
3362:
3361:
3357:
3340:
3298:
3296:
3295:
3290:
3285:
3284:
3266:
3265:
3238:
3237:
3214:
3212:
3211:
3206:
3201:
3200:
3188:
3187:
3157:
3156:
3138:
3137:
3116:
3115:
3086:
3084:
3083:
3078:
3076:
3075:
3063:
3062:
3050:
3049:
3037:
3036:
3024:
3023:
3011:
3010:
2981:short five lemma
2941:
2930:
2915:
2909:
2892:
2886:
2876:
2859:
2853:
2843:
2824:
2822:
2821:
2816:
2797:
2791:
2786:
2776:
2770:
2765:
2732:
2730:
2729:
2724:
2701:
2687:
2673:
2647:
2622:quotient modules
2616:
2614:
2613:
2608:
2560:
2558:
2556:
2555:
2550:
2530:
2528:
2527:
2522:
2504:
2502:
2500:
2499:
2494:
2468:
2466:
2465:
2460:
2430:
2428:
2427:
2422:
2401:
2399:
2397:
2396:
2391:
2365:
2363:
2362:
2357:
2339:
2333:
2331:
2330:
2325:
2295:
2286:
2284:
2283:
2278:
2221:
2211:
2205:
2183:
2181:
2180:
2175:
2173:
2172:
2145:
2143:
2142:
2137:
2135:
2134:
2115:
2113:
2112:
2107:
2099:
2098:
2086:
2085:
2070:
2069:
2037:
2035:
2034:
2029:
2011:
2009:
2008:
2003:
1992:
1856:
1854:
1853:
1848:
1840:
1832:
1827:
1819:
1814:
1812:
1801:
1799:
1756:as the image of
1748:. The image of 2
1732:
1730:
1729:
1724:
1722:
1714:
1709:
1701:
1696:
1690:
1668:
1666:
1665:
1660:
1640:
1638:
1637:
1632:
1620:
1590:to the element 2
1578:
1576:
1575:
1570:
1568:
1560:
1555:
1547:
1542:
1540:
1532:
1526:
1524:
1482:). Furthermore,
1477:
1475:
1474:
1469:
1467:
1466:
1446:
1444:
1443:
1438:
1426:
1424:
1423:
1418:
1413:
1412:
1391:
1390:
1369:
1367:
1366:
1361:
1359:
1346:
1345:
1333:
1332:
1315:
1310:
1309:
1278:
1276:
1263:
1262:
1250:
1249:
1238:
1233:
1232:
1204:
1203:
1191:
1190:
1179:
1174:
1173:
1136:
1134:
1133:
1128:
1123:
1122:
1108:
1107:
1106:
1095:
1090:
1080:
1079:
1078:
1067:
1062:
1060:
1059:
1045:
1044:
1043:
1032:
1027:
1025:
1024:
1010:
1009:
1008:
997:
992:
990:
989:
956:
954:
953:
948:
865:
863:
862:
857:
836:
823:
800:
798:
797:
792:
775:
749:
662:
660:
659:
654:
634:
621:
587:exact categories
444:
442:
441:
436:
412:
410:
409:
404:
402:
401:
381:
379:
378:
373:
368:
367:
337:
336:
311:
309:
308:
303:
301:
300:
271:
269:
268:
263:
261:
260:
250:
245:
244:
243:
232:
227:
220:
215:
214:
213:
202:
197:
194:
193:
183:
178:
177:
176:
165:
160:
157:
156:
146:
141:
140:
139:
128:
123:
120:
119:
85:) such that the
83:abelian category
52:
50:
49:
44:
42:
41:
4435:
4434:
4430:
4429:
4428:
4426:
4425:
4424:
4405:
4404:
4403:
4398:
4329:
4311:
4307:Urysohn's lemma
4268:
4232:
4118:
4109:
4081:low-dimensional
4039:
4034:
3999:
3977:Eisenbud, David
3975:
3969:
3945:
3937:
3936:
3927:
3925:
3917:
3916:
3912:
3902:
3841:
3831:
3822:
3809:
3799:
3790:
3781:
3724:
3701:normal subgroup
3659:
3646:
3633:
3622:
3621:
3594:
3581:
3562:
3557:
3556:
3529:
3516:
3511:
3510:
3483:
3478:
3477:
3450:
3437:
3432:
3431:
3410:
3405:
3404:
3347:
3343:
3341:
3328:
3327:
3270:
3251:
3229:
3224:
3223:
3192:
3173:
3142:
3129:
3107:
3102:
3101:
3095:
3067:
3054:
3041:
3028:
3015:
3002:
2997:
2996:
2989:
2932:
2921:
2911:
2897:
2888:
2878:
2864:
2855:
2845:
2831:
2750:
2749:
2743:splitting lemma
2739:
2629:
2628:
2569:
2568:
2535:
2534:
2532:
2507:
2506:
2473:
2472:
2470:
2433:
2432:
2407:
2406:
2370:
2369:
2367:
2366:to the element
2342:
2341:
2335:
2298:
2297:
2291:
2227:
2226:
2217:
2207:
2201:
2198:
2161:
2156:
2155:
2126:
2121:
2120:
2090:
2074:
2061:
2050:
2049:
2017:
2016:
1962:
1961:
1805:
1784:
1783:
1678:
1677:
1651:
1650:
1623:
1622:
1611:
1533:
1527:
1515:
1514:
1508:
1503:
1458:
1453:
1452:
1429:
1428:
1404:
1382:
1377:
1376:
1357:
1356:
1337:
1318:
1316:
1295:
1286:
1285:
1274:
1273:
1254:
1241:
1239:
1224:
1215:
1214:
1195:
1182:
1180:
1165:
1150:
1149:
1114:
1098:
1070:
1051:
1035:
1016:
1000:
981:
976:
975:
963:
924:
923:
809:
808:
731:
730:
607:
606:
600:
489:
415:
414:
393:
388:
387:
353:
328:
314:
313:
292:
287:
286:
252:
235:
205:
185:
168:
148:
131:
111:
106:
105:
99:
33:
28:
27:
17:
12:
11:
5:
4433:
4431:
4423:
4422:
4417:
4407:
4406:
4400:
4399:
4397:
4396:
4386:
4385:
4384:
4379:
4374:
4359:
4349:
4339:
4327:
4316:
4313:
4312:
4310:
4309:
4304:
4299:
4294:
4289:
4284:
4278:
4276:
4270:
4269:
4267:
4266:
4261:
4256:
4254:Winding number
4251:
4246:
4240:
4238:
4234:
4233:
4231:
4230:
4225:
4220:
4215:
4210:
4205:
4200:
4195:
4194:
4193:
4188:
4186:homotopy group
4178:
4177:
4176:
4171:
4166:
4161:
4156:
4146:
4141:
4136:
4126:
4124:
4120:
4119:
4112:
4110:
4108:
4107:
4102:
4097:
4096:
4095:
4085:
4084:
4083:
4073:
4068:
4063:
4058:
4053:
4047:
4045:
4041:
4040:
4035:
4033:
4032:
4025:
4018:
4010:
4004:
4003:
3997:
3973:
3967:
3942:
3941:
3935:
3934:
3909:
3908:
3907:
3906:
3901:
3898:
3890:Exact functors
3856:
3855:
3836:
3827:
3818:
3804:
3795:
3786:
3776:
3723:
3720:
3680:
3677:
3672:
3669:
3666:
3662:
3658:
3653:
3649:
3645:
3640:
3636:
3632:
3629:
3607:
3604:
3601:
3597:
3593:
3588:
3584:
3580:
3575:
3572:
3569:
3565:
3542:
3539:
3536:
3532:
3528:
3523:
3519:
3496:
3493:
3490:
3486:
3465:
3462:
3457:
3453:
3449:
3444:
3440:
3417:
3413:
3401:
3400:
3366:
3360:
3356:
3353:
3350:
3346:
3339:
3335:
3300:
3299:
3288:
3283:
3280:
3277:
3273:
3269:
3264:
3261:
3258:
3254:
3250:
3247:
3244:
3241:
3236:
3232:
3217:
3216:
3204:
3199:
3195:
3191:
3186:
3183:
3180:
3176:
3172:
3169:
3166:
3163:
3160:
3155:
3152:
3149:
3145:
3141:
3136:
3132:
3128:
3125:
3122:
3119:
3114:
3110:
3093:
3088:
3087:
3074:
3070:
3066:
3061:
3057:
3053:
3048:
3044:
3040:
3035:
3031:
3027:
3022:
3018:
3014:
3009:
3005:
2988:
2985:
2944:
2943:
2894:
2861:
2827:
2826:
2814:
2811:
2808:
2805:
2796:
2789:
2784:
2775:
2768:
2763:
2760:
2757:
2738:
2735:
2734:
2733:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2700:
2696:
2693:
2690:
2686:
2682:
2679:
2676:
2672:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2646:
2642:
2639:
2636:
2618:
2617:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2548:
2545:
2542:
2520:
2517:
2514:
2492:
2489:
2486:
2483:
2480:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2420:
2417:
2414:
2389:
2386:
2383:
2380:
2377:
2355:
2352:
2349:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2288:
2287:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2197:
2194:
2186:dihedral group
2171:
2168:
2164:
2133:
2129:
2117:
2116:
2105:
2102:
2097:
2093:
2089:
2084:
2081:
2077:
2073:
2068:
2064:
2060:
2057:
2027:
2024:
2013:
2012:
2001:
1998:
1995:
1991:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1943:
1942:
1915:
1900:
1858:
1857:
1846:
1843:
1839:
1835:
1831:
1826:
1822:
1818:
1811:
1808:
1804:
1798:
1794:
1791:
1734:
1733:
1721:
1717:
1713:
1708:
1704:
1700:
1695:
1689:
1685:
1658:
1630:
1606:to an element
1580:
1579:
1567:
1563:
1559:
1554:
1550:
1546:
1539:
1536:
1531:
1523:
1507:
1504:
1502:
1499:
1465:
1461:
1436:
1416:
1411:
1407:
1403:
1400:
1397:
1394:
1389:
1385:
1355:
1352:
1349:
1344:
1340:
1336:
1331:
1328:
1325:
1321:
1317:
1313:
1308:
1305:
1302:
1298:
1294:
1291:
1288:
1287:
1284:
1277:
1275:
1272:
1269:
1266:
1261:
1257:
1253:
1248:
1244:
1240:
1236:
1231:
1227:
1223:
1220:
1217:
1216:
1213:
1210:
1207:
1202:
1198:
1194:
1189:
1185:
1181:
1177:
1172:
1168:
1164:
1161:
1158:
1157:
1126:
1121:
1117:
1105:
1101:
1093:
1088:
1077:
1073:
1065:
1058:
1054:
1042:
1038:
1030:
1023:
1019:
1007:
1003:
995:
988:
984:
962:
959:
958:
957:
946:
943:
940:
937:
934:
931:
901:abelian groups
867:
866:
854:
851:
848:
843:
839:
835:
830:
826:
822:
819:
816:
802:
801:
790:
787:
784:
781:
778:
774:
770:
767:
764:
761:
758:
755:
752:
748:
744:
741:
738:
664:
663:
652:
649:
646:
641:
637:
633:
628:
624:
620:
617:
614:
599:
596:
595:
594:
551:
520:
488:
485:
434:
431:
428:
425:
422:
400:
396:
371:
366:
363:
360:
356:
352:
349:
346:
343:
340:
335:
331:
327:
324:
321:
299:
295:
279:is said to be
275:of groups and
273:
272:
259:
255:
242:
238:
230:
224:
212:
208:
200:
192:
188:
175:
171:
163:
155:
151:
138:
134:
126:
118:
114:
98:
95:
63:exact sequence
55:Euler diagrams
40:
36:
15:
13:
10:
9:
6:
4:
3:
2:
4432:
4421:
4418:
4416:
4413:
4412:
4410:
4395:
4387:
4383:
4380:
4378:
4375:
4373:
4370:
4369:
4368:
4360:
4358:
4354:
4350:
4348:
4344:
4340:
4338:
4333:
4328:
4326:
4318:
4317:
4314:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4279:
4277:
4275:
4271:
4265:
4264:Orientability
4262:
4260:
4257:
4255:
4252:
4250:
4247:
4245:
4242:
4241:
4239:
4235:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4192:
4189:
4187:
4184:
4183:
4182:
4179:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4157:
4155:
4152:
4151:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4131:
4128:
4127:
4125:
4121:
4116:
4106:
4103:
4101:
4100:Set-theoretic
4098:
4094:
4091:
4090:
4089:
4086:
4082:
4079:
4078:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4061:Combinatorial
4059:
4057:
4054:
4052:
4049:
4048:
4046:
4042:
4038:
4031:
4026:
4024:
4019:
4017:
4012:
4011:
4008:
4000:
3998:0-387-94269-6
3994:
3990:
3985:
3984:
3978:
3974:
3970:
3968:0-387-94426-5
3964:
3960:
3955:
3954:
3948:
3944:
3943:
3939:
3938:
3924:
3920:
3914:
3911:
3904:
3903:
3899:
3897:
3895:
3891:
3887:
3885:
3881:
3877:
3873:
3869:
3868:zig-zag lemma
3865:
3859:
3853:
3849:
3848:
3847:
3845:
3839:
3835:
3830:
3826:
3821:
3817:
3813:
3812:chain complex
3807:
3803:
3798:
3794:
3789:
3785:
3779:
3775:
3770:
3768:
3764:
3760:
3756:
3752:
3748:
3744:
3740:
3736:
3731:
3729:
3721:
3719:
3716:
3714:
3710:
3706:
3702:
3698:
3694:
3678:
3670:
3667:
3664:
3660:
3651:
3647:
3638:
3634:
3627:
3605:
3602:
3599:
3595:
3586:
3582:
3573:
3570:
3567:
3563:
3540:
3537:
3534:
3530:
3521:
3517:
3509:and morphism
3494:
3491:
3488:
3484:
3463:
3455:
3451:
3442:
3438:
3415:
3411:
3398:
3394:
3393:
3392:
3390:
3386:
3382:
3364:
3358:
3354:
3351:
3348:
3344:
3337:
3333:
3325:
3321:
3317:
3313:
3309:
3305:
3281:
3278:
3275:
3271:
3262:
3259:
3256:
3252:
3245:
3242:
3239:
3234:
3230:
3222:
3221:
3220:
3197:
3193:
3184:
3181:
3178:
3174:
3167:
3164:
3161:
3153:
3150:
3147:
3143:
3134:
3130:
3123:
3120:
3117:
3112:
3108:
3100:
3099:
3098:
3096:
3072:
3068:
3059:
3055:
3046:
3042:
3033:
3029:
3020:
3016:
3007:
3003:
2995:
2994:
2993:
2987:Weaving lemma
2986:
2984:
2982:
2978:
2973:
2971:
2967:
2963:
2958:
2956:
2951:
2949:
2939:
2935:
2928:
2924:
2919:
2914:
2908:
2904:
2900:
2895:
2891:
2885:
2881:
2875:
2871:
2867:
2862:
2858:
2852:
2848:
2842:
2838:
2834:
2829:
2828:
2812:
2809:
2803:
2794:
2787:
2782:
2773:
2766:
2761:
2755:
2748:
2747:
2746:
2744:
2736:
2720:
2711:
2708:
2705:
2698:
2694:
2688:
2684:
2680:
2677:
2674:
2670:
2666:
2657:
2654:
2651:
2644:
2640:
2634:
2627:
2626:
2625:
2623:
2604:
2598:
2592:
2589:
2586:
2580:
2574:
2567:
2566:
2565:
2562:
2546:
2543:
2540:
2518:
2515:
2512:
2487:
2484:
2481:
2456:
2453:
2450:
2444:
2441:
2438:
2418:
2415:
2412:
2405:
2384:
2381:
2378:
2353:
2350:
2347:
2338:
2321:
2318:
2315:
2309:
2306:
2303:
2294:
2274:
2268:
2265:
2262:
2256:
2253:
2250:
2244:
2241:
2238:
2232:
2225:
2224:
2223:
2220:
2215:
2210:
2204:
2195:
2193:
2191:
2187:
2169:
2166:
2162:
2153:
2149:
2131:
2127:
2103:
2095:
2091:
2082:
2079:
2075:
2066:
2062:
2055:
2048:
2047:
2046:
2043:
2041:
2025:
2022:
1999:
1993:
1989:
1985:
1979:
1973:
1967:
1960:
1959:
1958:
1956:
1952:
1948:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1913:
1909:
1905:
1901:
1898:
1894:
1890:
1889:
1888:
1886:
1882:
1878:
1875:
1871:
1867:
1863:
1844:
1833:
1829:
1809:
1806:
1789:
1782:
1781:
1780:
1777:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1715:
1711:
1683:
1676:
1675:
1674:
1672:
1648:
1644:
1618:
1614:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1561:
1557:
1537:
1534:
1513:
1512:
1511:
1505:
1500:
1498:
1496:
1495:weaving lemma
1491:
1489:
1485:
1481:
1463:
1459:
1450:
1434:
1409:
1405:
1398:
1395:
1392:
1387:
1383:
1373:
1372:
1353:
1350:
1342:
1338:
1329:
1326:
1323:
1319:
1306:
1303:
1300:
1296:
1289:
1282:
1270:
1267:
1259:
1255:
1246:
1242:
1229:
1225:
1218:
1211:
1208:
1200:
1196:
1187:
1183:
1170:
1166:
1159:
1147:
1145:
1140:
1139:
1124:
1119:
1115:
1103:
1099:
1091:
1086:
1075:
1071:
1063:
1056:
1052:
1040:
1036:
1028:
1021:
1017:
1005:
1001:
993:
986:
982:
973:
970:
968:
960:
944:
941:
938:
935:
932:
929:
922:
921:
920:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
873:
852:
846:
841:
837:
833:
828:
824:
820:
814:
807:
806:
805:
785:
779:
776:
772:
768:
765:
759:
753:
750:
746:
742:
739:
736:
729:
728:
727:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
650:
644:
639:
635:
631:
626:
622:
618:
612:
605:
604:
603:
597:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
552:
549:
545:
541:
537:
533:
529:
525:
521:
518:
514:
510:
506:
502:
498:
497:
496:
494:
493:trivial group
486:
484:
482:
478:
474:
470:
466:
462:
458:
457:vector spaces
454:
449:
446:
432:
429:
426:
423:
420:
398:
394:
385:
364:
361:
358:
354:
347:
344:
341:
333:
329:
322:
319:
297:
293:
285:
282:
278:
257:
253:
240:
236:
228:
222:
210:
206:
198:
190:
186:
173:
169:
161:
153:
149:
136:
132:
124:
116:
112:
104:
103:
102:
96:
94:
93:of the next.
92:
88:
84:
80:
76:
72:
68:
64:
56:
38:
34:
26:
21:
4394:Publications
4259:Chern number
4249:Betti number
4132: /
4123:Key concepts
4071:Differential
3982:
3952:
3926:. Retrieved
3922:
3913:
3888:
3863:
3860:
3857:
3837:
3833:
3828:
3824:
3819:
3815:
3805:
3801:
3796:
3792:
3787:
3783:
3777:
3773:
3771:
3758:
3754:
3750:
3746:
3742:
3738:
3732:
3725:
3717:
3712:
3708:
3704:
3696:
3692:
3402:
3388:
3380:
3323:
3319:
3315:
3311:
3307:
3301:
3218:
3091:
3089:
2990:
2974:
2964:shows how a
2959:
2954:
2952:
2945:
2937:
2933:
2926:
2922:
2912:
2906:
2902:
2898:
2889:
2883:
2879:
2873:
2869:
2865:
2856:
2850:
2846:
2840:
2836:
2832:
2740:
2619:
2563:
2336:
2292:
2289:
2218:
2208:
2202:
2199:
2189:
2151:
2148:cyclic group
2118:
2044:
2014:
1951:finite group
1946:
1944:
1938:
1934:
1930:
1926:
1922:
1918:
1911:
1907:
1903:
1896:
1892:
1880:
1876:
1874:factor group
1869:
1865:
1861:
1859:
1778:
1773:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1735:
1670:
1646:
1642:
1616:
1612:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1581:
1509:
1492:
1487:
1483:
1479:
1448:
1374:
1370:
1148:
1143:
1141:
1137:
974:
971:
966:
964:
916:
912:
904:
896:
892:
888:
884:
880:
876:
870:
868:
803:
723:
719:
715:
707:
703:
699:
695:
691:
683:
679:
675:
671:
667:
665:
601:
590:
582:
578:
566:
562:
558:
554:
543:
542:) is all of
539:
535:
531:
527:
523:
517:monomorphism
512:
508:
504:
500:
490:
487:Simple cases
450:
447:
383:
283:
280:
274:
100:
62:
60:
4357:Wikiversity
4274:Key results
3923:ncatlab.org
3765:. See also
2962:snake lemma
2620:Passing to
575:isomorphism
548:epimorphism
461:linear maps
4409:Categories
4203:CW complex
4144:Continuity
4134:Closed set
4093:cohomology
3928:2021-09-05
3900:References
3728:subobjects
3555:such that
2977:five lemma
2970:nine lemma
2918:direct sum
2910:such that
2877:such that
2844:such that
2737:Properties
2404:direct sum
2216:of a ring
2188:of order 2
1427:for every
909:direct sum
869:is called
698:embedding
571:bimorphism
97:Definition
4382:geometric
4377:algebraic
4228:Cobordism
4164:Hausdorff
4159:connected
4076:Geometric
4066:Continuum
4056:Algebraic
3905:Citations
3842:, so the
3676:→
3657:→
3644:→
3631:→
3592:→
3579:→
3571:−
3527:→
3461:→
3448:→
3352:
3310:) :
3279:−
3268:→
3260:−
3246:
3240:≅
3190:→
3182:−
3168:
3162:≅
3140:→
3124:
3118:≅
3065:→
3052:→
3039:→
3026:→
3013:→
2807:→
2759:→
2718:→
2692:→
2678:⊕
2664:→
2655:∩
2638:→
2602:→
2596:→
2590:⊕
2584:→
2578:→
2544:−
2516:⊕
2448:→
2442:⊕
2416:⊕
2351:∩
2319:⊕
2313:→
2307:∩
2272:→
2260:→
2254:⊕
2248:→
2242:∩
2236:→
2150:of order
2101:→
2088:→
2072:→
2059:→
1997:→
1983:→
1977:→
1971:→
1842:→
1821:⟶
1810:×
1803:⟶
1793:→
1703:↠
1694:↪
1657:↠
1629:↪
1549:↠
1538:×
1530:↪
1399:
1348:→
1335:→
1327:−
1312:→
1304:−
1293:→
1283:⋮
1265:→
1252:→
1235:→
1222:→
1206:→
1193:→
1176:→
1163:→
1087:⋯
939:⊕
933:≅
850:→
818:→
780:
754:
740:≅
706:, and of
688:subobject
648:→
616:→
477:cokernels
424:≤
348:
323:
223:⋯
67:morphisms
4347:Wikibook
4325:Category
4213:Manifold
4181:Homotopy
4139:Interior
4130:Open set
4088:Homology
4037:Topology
3979:(1995).
3949:(1995).
3894:functors
3844:homology
3800:to 0 in
3359:⟩
3345:⟨
2835: :
2788:→
2767:→
1760:through
1501:Examples
1092:→
1064:→
1029:→
994:→
879: :
838:→
825:→
712:quotient
636:→
623:→
469:category
413:for all
229:→
199:→
162:→
125:→
4372:general
4174:uniform
4154:compact
4105:Digital
3940:Sources
3852:acyclic
3383:by the
3318:is not
2916:is the
2559:
2533:
2503:
2471:
2402:of the
2400:
2368:
2222:. Then
2212:be two
2184:is the
2146:is the
2040:abelian
1872:to the
722:, with
473:kernels
79:modules
4367:Topics
4169:metric
4044:Fields
3995:
3965:
3878:; the
3387:of im(
3326:) but
2955:splits
2798:
2792:
2777:
2771:
2214:ideals
2119:where
1885:modulo
1375:where
1279:
1109:
1096:
1081:
1068:
1046:
1033:
1011:
998:
246:
233:
216:
203:
179:
166:
142:
129:
91:kernel
71:groups
53:using
25:groups
4149:Space
3791:maps
3753:have
3243:coker
1619:mod 2
1142:with
872:split
702:into
694:with
686:as a
589:like
577:from
471:with
384:exact
281:exact
87:image
75:rings
3993:ISBN
3963:ISBN
3892:are
3741:and
3733:The
3711:/im(
3322:/im(
2975:The
2960:The
2931:and
2741:The
2206:and
2200:Let
2154:and
1957:is
1493:See
915:and
475:and
459:and
430:<
3989:785
3959:179
3121:ker
2950:".
2920:of
2531:to
2505:of
2340:of
2042:).
1864:to
1764:↦ 2
1740:↦ 2
1645:to
1602:in
1594:in
1488:(2)
1484:(1)
1480:(1)
1449:(2)
1371:(2)
1144:n ≥
1138:(1)
911:of
777:ker
714:),
690:of
591:Set
581:to
565:to
538:to
511:to
345:ker
312:if
61:An
4411::
3991:.
3961:.
3921:.
3886:.
3840:+1
3808:+2
3782:∘
3780:+1
3769:.
3349:im
3314:→
3165:im
2957:.
2905:→
2901::
2882:∘
2872:→
2868::
2849:∘
2839:→
2561:.
1937:/2
1929:/2
1921:/2
1879:/2
1615:=
1396:im
919::
903:,
891:∘
883:→
751:im
651:0.
593:).
557:→
526:→
503:→
320:im
284:at
77:,
73:,
4029:e
4022:t
4015:v
4001:.
3971:.
3931:.
3854:.
3838:i
3834:f
3829:i
3825:A
3820:i
3816:f
3806:i
3802:A
3797:i
3793:A
3788:i
3784:f
3778:i
3774:f
3759:C
3755:A
3751:B
3747:B
3743:C
3739:A
3713:f
3709:H
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3697:H
3693:f
3679:0
3671:1
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3648:A
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3628:0
3606:1
3603:+
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3596:A
3587:k
3583:A
3574:1
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3541:1
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3464:0
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3355:f
3338:/
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3324:f
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3282:1
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3272:A
3263:2
3257:k
3253:A
3249:(
3235:k
3231:C
3215:.
3203:)
3198:k
3194:A
3185:1
3179:k
3175:A
3171:(
3159:)
3154:1
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3148:k
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3135:k
3131:A
3127:(
3113:k
3109:C
3094:k
3092:C
3073:6
3069:A
3060:5
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3043:A
3034:3
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3021:2
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2942:.
2940:)
2938:C
2936:(
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2874:B
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2756:0
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2712:J
2709:+
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2703:(
2699:/
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2685:/
2681:R
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2671:/
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2661:)
2658:J
2652:I
2649:(
2645:/
2641:R
2635:0
2605:0
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2593:R
2587:R
2581:R
2575:0
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2541:x
2519:J
2513:I
2491:)
2488:y
2485:,
2482:x
2479:(
2457:J
2454:+
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2382:,
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2348:I
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2266:+
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2257:J
2251:I
2245:J
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2233:0
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2209:J
2203:I
2190:n
2170:n
2167:2
2163:D
2152:n
2132:n
2128:C
2104:1
2096:2
2092:C
2083:n
2080:2
2076:D
2067:n
2063:C
2056:1
2026:,
2023:1
2000:1
1994:N
1990:/
1986:G
1980:G
1974:N
1968:1
1947:Z
1941:.
1939:Z
1935:Z
1931:Z
1927:Z
1923:Z
1919:Z
1914:.
1912:Z
1908:Z
1904:Z
1899:.
1897:Z
1893:Z
1881:Z
1877:Z
1870:Z
1866:Z
1862:Z
1845:0
1838:Z
1834:2
1830:/
1825:Z
1817:Z
1807:2
1797:Z
1790:0
1774:Z
1770:Z
1766:n
1762:n
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1754:Z
1750:Z
1746:Z
1742:n
1738:n
1720:Z
1716:2
1712:/
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1684:2
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1617:i
1613:j
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1600:i
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1562:2
1558:/
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1460:K
1435:i
1415:)
1410:i
1406:f
1402:(
1393:=
1388:i
1384:K
1354:,
1351:0
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1339:K
1330:1
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1320:A
1307:1
1301:n
1297:K
1290:0
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1268:0
1260:3
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1247:2
1243:A
1230:2
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1197:K
1188:1
1184:A
1171:1
1167:K
1160:0
1125:,
1120:n
1116:A
1104:n
1100:f
1076:3
1072:f
1057:2
1053:A
1041:2
1037:f
1022:1
1018:A
1006:1
1002:f
987:0
983:A
945:.
942:C
936:A
930:B
917:C
913:A
905:B
897:C
893:h
889:g
885:B
881:C
877:h
853:0
847:C
842:g
834:B
829:f
821:A
815:0
789:)
786:g
783:(
773:/
769:B
766:=
763:)
760:f
757:(
747:/
743:B
737:C
724:g
720:A
718:/
716:B
708:C
704:B
700:A
696:f
692:B
684:A
680:g
676:f
672:g
668:f
645:C
640:g
632:B
627:f
619:A
613:0
583:Y
579:X
567:Y
563:X
559:Y
555:X
544:C
540:C
536:B
532:C
528:C
524:B
513:B
509:A
505:B
501:A
433:n
427:i
421:1
399:i
395:G
370:)
365:1
362:+
359:i
355:f
351:(
342:=
339:)
334:i
330:f
326:(
298:i
294:G
258:n
254:G
241:n
237:f
211:3
207:f
191:2
187:G
174:2
170:f
154:1
150:G
137:1
133:f
117:0
113:G
57:.
39:i
35:G
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