190:
induce homomorphisms that are inverse to each other. A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations
191:
between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.
129:. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism. A homomorphism induced from a map
1190:
have isomorphic fundamental groups but are still not homeomorphic. Their fundamental groups are isomorphic because each space is simply connected. However, the two spaces cannot be homeomorphic because deleting a point from
1035:
1400:
695:
1784:
1556:
850:
1600:
1578:
1494:
1422:
1355:
465:
1472:
1122:
1083:
745:
588:
503:
296:
174:
1658:
1629:
1721:
1698:
1678:
147:
1175:
is homeomorphic to the circle). This also shows that the one-point compactification of a simply connected space need not be simply connected.
1850:
1898:
1825:
1496:(which is usually a continuous map, possibly preserving some other structure such as the base point) this functor induces an
1358:
1164:
248:
118:
44:
972:
1914:
1369:
1817:
1434:
1363:
or the category of pointed topological spaces (that is, topological spaces with a distinguished base point), and a
1234:
771:
1293:
1919:
1357:
of topological spaces (possibly with some additional structure) such as the category of all topological spaces
1216:
894:
it satisfies the definition of a functor, one has to further check that it is compatible with composition: for
478:
under homotopy, as in the definition of the fundamental group. It is easily checked from the definitions that
596:
1333:
1145:
179:
Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are
90:
17:
1140:
because their fundamental groups are not isomorphic (since their fundamental groups don’t have the same
1730:
1502:
1171:
has a fundamental group isomorphic to the group of integers (since the one-point compactification of
1317:
1285:
813:
560:
1583:
1561:
1477:
1405:
1338:
1425:
1313:
891:
122:
110:
59:
32:
1309:
410:
1894:
1856:
1846:
1821:
1445:
1289:
1258:
1087:
471:
214:
200:
126:
106:
98:
63:
48:
1439:
which then associates such an algebraic structure to every topological space, then for every
1098:
1059:
1787:
1300:
comes induced homomorphisms, though in the opposite direction (from a group associated with
1179:
723:
566:
481:
274:
152:
1634:
1605:
1281:
1219:
362:
78:
1706:
1683:
1663:
1277:
1273:
132:
102:
1320:
all have induced homomorphisms (IV.4.2–3, pp. 298–299). Generalizations such as
1908:
1811:
1807:
1431:
1242:
1149:
1045:
40:
1141:
180:
28:
1253:
induces an isomorphism between fundamental groups (so the fundamental group of
1203:, the space wouldn’t be simply connected any more. In fact this generalizes to
1091:
between fundamental groups (because the homomorphism induced by the inverse of
1297:
1124:, by the above equation). (See section III.5.4, p. 201, in H. Schubert.)
1860:
1321:
1296:
all have induced homomorphisms (IV.1.3, pp. 240–241) Similarly, any
1440:
187:
94:
563:
operation in fundamental groups (namely by concatenation of loops) that
1724:
1364:
1160:
82:
1156:
1727:
then by definition it induces morphisms in the opposite direction:
117:, meaning that their definition provides a functor from (e.g.) the
1148:
cannot be homeomorphic to a non-simply-connected space; one has a
1133:
184:
774:
to the category of groups: it associates the fundamental group
547:: loops in the same equivalence class, i.e. homotopic loops in
1199:
leaves a simply connected space (If we delete a line lying in
1195:
leaves a non-simply-connected space but deleting a point from
43:
derived in a canonical way from another map. For example, a
1893:
James
Munkres (1999). Topology, 2nd edition, Prentice Hall.
1602:
is a category of groups) between the algebraic structures
1424:
of algebraic structures such as the category of groups
1733:
1709:
1686:
1666:
1637:
1608:
1586:
1564:
1505:
1480:
1448:
1408:
1372:
1341:
1101:
1062:
975:
816:
726:
599:
569:
484:
413:
277:
155:
135:
1882:
Topologie, Eine Einführung (Mathematische Leitfäden)
1030:{\displaystyle \pi (k\circ h)=\pi (k)\circ \pi (h).}
559:
as well. It also follows from the definition of the
1272:Likewise there are induced homomorphisms of higher
1778:
1715:
1692:
1672:
1652:
1623:
1594:
1572:
1550:
1488:
1466:
1416:
1394:
1349:
1116:
1077:
1029:
844:
739:
689:
582:
497:
459:
290:
168:
141:
1284:comes with induced homomorphisms. For instance,
1884:. B. G. Teubner Verlagsgesellschaft, Stuttgart.
704:denotes concatenation of loops, with the first
1580:(which for example is a group homomorphism if
1395:{\displaystyle F:\mathbf {T} \to \mathbf {A} }
8:
1182:of the theorem need not hold. For example,
1044:is not only a continuous map but in fact a
810:and it associates the induced homomorphism
1152:fundamental group and the other does not.
555:, because a homotopy can be composed with
1732:
1708:
1685:
1665:
1636:
1607:
1587:
1585:
1565:
1563:
1504:
1481:
1479:
1447:
1409:
1407:
1387:
1379:
1371:
1342:
1340:
1257:can be described using only loops in the
1100:
1061:
974:
836:
815:
731:
725:
666:
638:
604:
598:
574:
568:
489:
483:
418:
412:
282:
276:
160:
154:
134:
18:Induced homomorphism (algebraic topology)
1799:
690:{\displaystyle h_{*}()=h_{*}()+h_{*}()}
1402:from that category into some category
1226:leaves a non-simply-connected space).
201:Fundamental group § Functoriality
1843:Introduction to topological manifolds
97:in the source category. For example,
7:
1875:
1873:
1871:
1845:(2nd ed.). New York: Springer.
1723:is not a (covariant) functor but a
551:, are mapped to homotopic loops in
113:are algebraic structures that are
25:
1779:{\displaystyle F(f):F(Y)\to F(X)}
1551:{\displaystyle F(f):F(X)\to F(Y)}
1324:also have induced homomorphisms.
1588:
1566:
1482:
1410:
1388:
1380:
1343:
1155:2. The fundamental group of the
1056:, then the induced homomorphism
1773:
1767:
1761:
1758:
1752:
1743:
1737:
1647:
1641:
1618:
1612:
1545:
1539:
1533:
1530:
1524:
1515:
1509:
1458:
1384:
1159:is isomorphic to the group of
1111:
1105:
1072:
1066:
1021:
1015:
1006:
1000:
991:
979:
826:
820:
720:). The resulting homomorphism
684:
681:
675:
672:
656:
653:
647:
644:
628:
625:
613:
610:
454:
442:
436:
433:
427:
424:
119:category of topological spaces
1:
845:{\displaystyle \pi (h)=h_{*}}
1595:{\displaystyle \mathbf {A} }
1573:{\displaystyle \mathbf {A} }
1489:{\displaystyle \mathbf {T} }
1417:{\displaystyle \mathbf {A} }
1350:{\displaystyle \mathbf {T} }
70:to the fundamental group of
1304:to a group associated with
1163:. Therefore, the one-point
898:preserving continuous maps
505:is a well-defined function
400:obtained by composing with
379:, is mapped to the loop in
340:as follows: any element of
298:from the fundamental group
271:. Then we can define a map
1936:
1865:pg. 197, Proposition 7.24.
1818:Cambridge University Press
1235:strong deformation retract
856:preserving continuous map
772:category of pointed spaces
758:It may also be denoted as
590:is a group homomorphism:
198:
85:by definition provides an
770:gives a functor from the
460:{\displaystyle h_{*}():=}
319:to the fundamental group
1467:{\displaystyle f:X\to Y}
1237:of a topological space
1136:is not homeomorphic to
1117:{\displaystyle \pi (h)}
1078:{\displaystyle \pi (h)}
54:to a topological space
1780:
1717:
1694:
1674:
1654:
1625:
1596:
1574:
1552:
1490:
1468:
1418:
1396:
1351:
1146:simply connected space
1118:
1079:
1031:
846:
795:to each pointed space
741:
691:
584:
499:
461:
292:
170:
143:
1880:Schubert, H. (1975).
1841:Lee, John M. (2011).
1781:
1725:contravariant functor
1718:
1695:
1675:
1655:
1626:
1597:
1575:
1553:
1491:
1469:
1419:
1397:
1352:
1144:). More generally, a
1119:
1080:
1040:This implies that if
1032:
847:
742:
740:{\displaystyle h_{*}}
692:
585:
583:{\displaystyle h_{*}}
500:
498:{\displaystyle h_{*}}
462:
293:
291:{\displaystyle h_{*}}
195:In fundamental groups
171:
169:{\displaystyle h_{*}}
144:
1731:
1707:
1684:
1664:
1653:{\displaystyle F(Y)}
1635:
1624:{\displaystyle F(X)}
1606:
1584:
1562:
1503:
1478:
1446:
1406:
1370:
1339:
1294:Borel–Moore homology
1099:
1060:
973:
814:
747:is the homomorphism
724:
597:
567:
482:
411:
275:
153:
133:
37:induced homomorphism
1318:singular cohomology
1286:simplicial homology
1207:whereby deleting a
361:, represented by a
77:More generally, in
1915:Algebraic topology
1813:Algebraic Topology
1776:
1713:
1690:
1670:
1650:
1621:
1592:
1570:
1548:
1486:
1464:
1414:
1392:
1347:
1328:General definition
1314:de Rham cohomology
1308:). For instance,
1114:
1095:is the inverse of
1075:
1027:
842:
737:
687:
580:
495:
470:Here denotes the
457:
288:
215:topological spaces
166:
139:
123:category of groups
111:De Rham cohomology
99:fundamental groups
60:group homomorphism
33:algebraic topology
1790:give an example.
1788:Cohomology groups
1716:{\displaystyle F}
1693:{\displaystyle Y}
1673:{\displaystyle X}
1290:singular homology
472:equivalence class
149:is often denoted
142:{\displaystyle h}
107:singular homology
64:fundamental group
49:topological space
16:(Redirected from
1927:
1886:
1885:
1877:
1866:
1864:
1838:
1832:
1831:
1804:
1785:
1783:
1782:
1777:
1722:
1720:
1719:
1714:
1700:, respectively.
1699:
1697:
1696:
1691:
1679:
1677:
1676:
1671:
1659:
1657:
1656:
1651:
1630:
1628:
1627:
1622:
1601:
1599:
1598:
1593:
1591:
1579:
1577:
1576:
1571:
1569:
1557:
1555:
1554:
1549:
1498:induced morphism
1495:
1493:
1492:
1487:
1485:
1473:
1471:
1470:
1465:
1423:
1421:
1420:
1415:
1413:
1401:
1399:
1398:
1393:
1391:
1383:
1356:
1354:
1353:
1348:
1346:
1214:
1165:compactification
1123:
1121:
1120:
1115:
1084:
1082:
1081:
1076:
1036:
1034:
1033:
1028:
965:
950:
931:
916:
897:
889:
874:
855:
851:
849:
848:
843:
841:
840:
809:
794:
778:
769:
761:
746:
744:
743:
738:
736:
735:
696:
694:
693:
688:
671:
670:
643:
642:
609:
608:
589:
587:
586:
581:
579:
578:
546:
530:
525:
509:
504:
502:
501:
496:
494:
493:
466:
464:
463:
458:
423:
422:
399:
383:
360:
344:
339:
323:
318:
302:
297:
295:
294:
289:
287:
286:
270:
175:
173:
172:
167:
165:
164:
148:
146:
145:
140:
87:induced morphism
31:, especially in
21:
1935:
1934:
1930:
1929:
1928:
1926:
1925:
1924:
1920:Category theory
1905:
1904:
1890:
1889:
1879:
1878:
1869:
1853:
1840:
1839:
1835:
1828:
1806:
1805:
1801:
1796:
1729:
1728:
1705:
1704:
1682:
1681:
1662:
1661:
1633:
1632:
1604:
1603:
1582:
1581:
1560:
1559:
1501:
1500:
1476:
1475:
1444:
1443:
1404:
1403:
1368:
1367:
1337:
1336:
1330:
1310:Čech cohomology
1282:homology theory
1278:homology groups
1274:homotopy groups
1270:
1208:
1130:
1097:
1096:
1058:
1057:
971:
970:
963:
952:
948:
937:
929:
918:
914:
903:
895:
887:
876:
872:
861:
853:
832:
812:
811:
807:
796:
792:
781:
776:
775:
767:
759:
727:
722:
721:
712:and the second
662:
634:
600:
595:
594:
570:
565:
564:
544:
533:
528:
527:
523:
512:
507:
506:
485:
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414:
409:
408:
397:
386:
381:
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378:
358:
347:
342:
341:
337:
326:
321:
320:
316:
305:
300:
299:
278:
273:
272:
269:
262:
252:
234:
223:
203:
197:
156:
151:
150:
131:
130:
103:homotopy groups
79:category theory
23:
22:
15:
12:
11:
5:
1933:
1931:
1923:
1922:
1917:
1907:
1906:
1903:
1902:
1888:
1887:
1867:
1852:978-1441979391
1851:
1833:
1826:
1808:Hatcher, Allen
1798:
1797:
1795:
1792:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1712:
1689:
1669:
1660:associated to
1649:
1646:
1643:
1640:
1620:
1617:
1614:
1611:
1590:
1568:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1484:
1463:
1460:
1457:
1454:
1451:
1432:abelian groups
1412:
1390:
1386:
1382:
1378:
1375:
1345:
1329:
1326:
1269:
1268:Other examples
1266:
1129:
1126:
1113:
1110:
1107:
1104:
1074:
1071:
1068:
1065:
1038:
1037:
1026:
1023:
1020:
1017:
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1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
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946:
927:
912:
885:
870:
839:
835:
831:
828:
825:
822:
819:
805:
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734:
730:
698:
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577:
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542:
531:
521:
510:
492:
488:
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467:
456:
453:
450:
447:
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441:
438:
435:
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429:
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421:
417:
395:
384:
376:
356:
345:
335:
324:
314:
303:
285:
281:
267:
260:
249:continuous map
232:
221:
196:
193:
183:to each other
163:
159:
138:
121:to (e.g.) the
89:in the target
45:continuous map
24:
14:
13:
10:
9:
6:
4:
3:
2:
1932:
1921:
1918:
1916:
1913:
1912:
1910:
1900:
1899:0-13-181629-2
1896:
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1891:
1883:
1876:
1874:
1872:
1868:
1862:
1858:
1854:
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1844:
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1834:
1829:
1827:0-521-79540-0
1823:
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1506:
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1428:
1376:
1373:
1366:
1362:
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1335:
1327:
1325:
1323:
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1267:
1265:
1263:
1260:
1256:
1252:
1248:
1244:
1243:inclusion map
1240:
1236:
1232:
1227:
1225:
1221:
1218:
1212:
1206:
1202:
1198:
1194:
1189:
1185:
1181:
1176:
1174:
1170:
1166:
1162:
1158:
1153:
1151:
1147:
1143:
1139:
1135:
1127:
1125:
1108:
1102:
1094:
1090:
1089:
1069:
1063:
1055:
1051:
1047:
1046:homeomorphism
1043:
1024:
1018:
1012:
1009:
1003:
997:
994:
988:
985:
982:
976:
969:
968:
967:
960:
956:
945:
941:
935:
926:
922:
911:
907:
901:
893:
884:
880:
869:
865:
859:
837:
833:
829:
823:
817:
804:
800:
789:
785:
773:
765:
756:
754:
750:
732:
728:
719:
715:
711:
707:
703:
678:
667:
663:
659:
650:
639:
635:
631:
622:
619:
616:
605:
601:
593:
592:
591:
575:
571:
562:
558:
554:
550:
541:
537:
520:
516:
490:
486:
477:
473:
451:
448:
445:
439:
430:
419:
415:
407:
406:
405:
403:
394:
390:
375:
371:
367:
364:
355:
351:
334:
330:
313:
309:
283:
279:
266:
259:
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250:
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231:
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220:
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212:
208:
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194:
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182:
177:
161:
157:
136:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
50:
46:
42:
38:
34:
30:
19:
1881:
1842:
1836:
1812:
1802:
1702:
1497:
1435:
1426:
1359:
1331:
1305:
1301:
1271:
1261:
1254:
1250:
1246:
1238:
1230:
1228:
1223:
1210:
1204:
1200:
1196:
1192:
1187:
1183:
1177:
1172:
1168:
1154:
1137:
1131:
1128:Applications
1092:
1086:
1053:
1049:
1041:
1039:
958:
954:
943:
939:
933:
924:
920:
909:
905:
899:
882:
878:
867:
863:
857:
802:
798:
787:
783:
763:
757:
752:
748:
717:
713:
709:
705:
701:
699:
556:
552:
548:
539:
535:
518:
514:
475:
469:
401:
392:
388:
373:
369:
365:
353:
349:
332:
328:
311:
307:
264:
257:
253:
244:
240:
236:
229:
225:
218:
217:with points
210:
206:
204:
178:
114:
86:
76:
71:
67:
55:
51:
41:homomorphism
36:
26:
1332:Given some
1241:, then the
1217:dimensional
1142:cardinality
1088:isomorphism
966:, we have:
766:). Indeed,
29:mathematics
1909:Categories
1794:References
1298:cohomology
896:base-point
854:base-point
251:such that
199:See also:
115:functorial
58:induces a
1861:697506452
1762:→
1534:→
1459:→
1385:→
1322:cobordism
1103:π
1064:π
1013:π
1010:∘
998:π
986:∘
977:π
838:∗
818:π
733:∗
668:∗
640:∗
606:∗
576:∗
491:∗
449:∘
420:∗
372:based at
284:∗
162:∗
101:, higher
93:for each
62:from the
1810:(2002).
1441:morphism
1334:category
1259:subspace
1220:subspace
1180:converse
1161:integers
1048:between
936::
902::
860::
852:to each
243: :
188:homotopy
95:morphism
91:category
1365:functor
1178:3. The
1150:trivial
1132:1. The
749:induced
700:(where
181:inverse
83:functor
47:from a
1897:
1859:
1849:
1824:
1430:or of
1316:, and
1292:, and
1280:. Any
1229:4. If
1157:circle
1085:is an
239:. Let
109:, and
81:, any
1245:from
1233:is a
1222:from
1134:torus
892:prove
890:. To
751:from
561:group
247:be a
185:up to
127:rings
39:is a
35:, an
1895:ISBN
1857:OCLC
1847:ISBN
1822:ISBN
1680:and
1631:and
1276:and
1213:− 2)
1186:and
1052:and
932:and
363:loop
263:) =
228:and
209:and
205:Let
1703:If
1558:in
1474:of
1427:Grp
1360:Top
1264:).
1249:to
1167:of
716:in
708:in
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368:in
245:X→Y
235:in
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213:be
125:or
66:of
27:In
1911::
1870:^
1855:.
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957:,
951:→
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908:,
881:,
875:→
866:,
801:,
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755:.
538:,
526:→
517:,
440::=
404::
391:,
352:,
331:,
310:,
176:.
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1901:.
1863:.
1830:.
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1771:X
1768:(
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1522:F
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1510:(
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1377::
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1239:X
1231:A
1224:R
1215:-
1211:n
1209:(
1205:R
1201:R
1197:R
1193:R
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1184:R
1173:R
1169:R
1138:R
1112:)
1109:h
1106:(
1093:h
1073:)
1070:h
1067:(
1054:Y
1050:X
1042:h
1025:.
1022:)
1019:h
1016:(
1007:)
1004:k
1001:(
995:=
992:)
989:h
983:k
980:(
964:)
962:0
959:z
955:Z
953:(
949:)
947:0
944:y
940:Y
938:(
934:k
930:)
928:0
925:y
921:Y
919:(
915:)
913:0
910:x
906:X
904:(
900:h
888:)
886:0
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879:Y
877:(
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871:0
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864:X
862:(
858:h
834:h
830:=
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824:h
821:(
808:)
806:0
803:x
799:X
797:(
793:)
791:0
788:x
784:X
782:(
780:1
777:π
768:π
764:h
762:(
760:π
753:h
729:h
718:Y
714:+
710:X
706:+
702:+
685:)
682:]
679:g
676:[
673:(
664:h
660:+
657:)
654:]
651:f
648:[
645:(
636:h
632:=
629:)
626:]
623:g
620:+
617:f
614:[
611:(
602:h
572:h
557:h
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549:X
545:)
543:0
540:y
536:Y
534:(
532:1
529:π
524:)
522:0
519:x
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513:(
511:1
508:π
487:h
476:f
455:]
452:f
446:h
443:[
437:)
434:]
431:f
428:[
425:(
416:h
402:h
398:)
396:0
393:y
389:Y
387:(
385:1
382:π
377:0
374:x
370:X
366:f
359:)
357:0
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350:X
348:(
346:1
343:π
338:)
336:0
333:y
329:Y
327:(
325:1
322:π
317:)
315:0
312:x
308:X
306:(
304:1
301:π
280:h
268:0
265:y
261:0
258:x
256:(
254:h
241:h
237:Y
233:0
230:y
226:X
222:0
219:x
211:Y
207:X
158:h
137:h
72:Y
68:X
56:Y
52:X
20:)
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