482:
441:
1336:
812:
1331:{\displaystyle {\begin{aligned}\partial _{1}c&=\partial _{1}(t_{1}+t_{2}+t_{3})\\&=\partial _{1}(t_{1})+\partial _{1}(t_{2})+\partial _{1}(t_{3})\\&=\partial _{1}()+\partial _{1}()+\partial _{1}()\\&=(-)+(-)+(-)\\&=-.\end{aligned}}}
305:
817:
497:−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.
569:
424:
Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients (which are typically integers). The set of all
334:
805:
745:
685:
414:
161:
625:
597:
208:
378:
356:
201:
181:
125:
1343:
The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.
17:
1371:
The plane punctured at the origin has nontrivial 1-homology group since the unit circle is a cycle, but not a boundary.
1446:
1522:
1487:
1421:
489:
The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a
514:
1517:
1413:
58:
312:
1359:
300:{\displaystyle C_{n}(X)=\left\{\sum \limits _{i}m_{i}\sigma _{i}|m_{i}\in \mathbb {Z} \right\}}
93:
750:
690:
630:
383:
130:
1375:
602:
574:
1497:
62:
39:
8:
1379:
1383:
363:
341:
186:
166:
110:
105:
73:
46:
31:
1407:
1350:
when its boundary is zero. A chain that is the boundary of another chain is called a
1483:
1452:
1442:
1417:
337:
1475:
456:. Assuming the segments all are oriented left-to-right (in increasing order from A
1493:
508:
504:
445:
77:
448:
is a linear combination of its nodes; in this case, some linear combination of A
1511:
1456:
1403:
1355:
485:
A closed polygonal curve, assuming consistent orientation, has null boundary.
429:
1474:. Applied Mathematical Sciences. Vol. 157. New York: Springer-Verlag.
35:
1358:, whose homology groups (cycles modulo boundaries) are called simplicial
481:
69:
96:; the elements of a homology group are equivalence classes of chains.
1479:
428:-chains forms a group and the sequence of these groups is called a
1470:
Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004).
1378:, the duality between the boundary operator on chains and the
440:
92:-cubes), but not necessarily connected. Chains are used in
1469:
815:
753:
693:
633:
605:
577:
517:
386:
366:
344:
315:
211:
189:
169:
133:
113:
416:
not necessary to be a connected simplicial complex.
507:is the formal difference of its endpoints: it is a
1330:
799:
739:
679:
619:
591:
563:
408:
372:
350:
328:
299:
195:
175:
155:
119:
1509:
435:
1354:. Boundaries are cycles, so chains form a
796:
736:
676:
616:
588:
560:
288:
480:
439:
419:
1402:
14:
1510:
807:are its constituent 1-simplices, then
1439:Introduction to topological manifolds
564:{\displaystyle c=t_{1}+t_{2}+t_{3}\,}
1441:(2nd ed.). New York: Springer.
1436:
240:
24:
1093:
1045:
997:
961:
932:
903:
841:
821:
25:
1534:
511:. To illustrate, if the 1-chain
1463:
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1396:
1318:
1305:
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1105:
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1086:
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1054:
1038:
1035:
1009:
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983:
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954:
941:
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793:
767:
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707:
673:
647:
403:
397:
270:
228:
222:
150:
144:
13:
1:
1389:
99:
1382:is expressed by the general
84:-chains are combinations of
7:
436:Boundary operator on chains
329:{\displaystyle \sigma _{i}}
27:A formal linear combination
10:
1539:
1414:Cambridge University Press
88:-simplices (respectively,
29:
59:formal linear combination
1523:Integration on manifolds
800:{\displaystyle t_{3}=\,}
740:{\displaystyle t_{2}=\,}
680:{\displaystyle t_{1}=\,}
409:{\displaystyle C_{n}(X)}
156:{\displaystyle C_{n}(X)}
34:. For the term chain in
18:Boundary (chain complex)
620:{\displaystyle v_{4}\,}
592:{\displaystyle v_{1}\,}
1472:Computational homology
1332:
801:
741:
681:
621:
593:
565:
486:
478:
410:
380:. that any element in
374:
352:
330:
301:
197:
177:
157:
121:
30:This article is about
1437:Lee, John M. (2011).
1376:differential geometry
1333:
802:
742:
682:
622:
594:
571:is a path from point
566:
484:
443:
420:Integration on chains
411:
375:
353:
331:
302:
198:
178:
158:
122:
1346:A chain is called a
813:
751:
691:
631:
603:
575:
515:
469:), the boundary is A
384:
364:
342:
313:
209:
187:
167:
131:
111:
74:simplicial complexes
40:chain (order theory)
1380:exterior derivative
1518:Algebraic topology
1409:Algebraic Topology
1328:
1326:
797:
737:
677:
617:
589:
561:
503:The boundary of a
487:
479:
444:The boundary of a
406:
370:
348:
326:
297:
248:
193:
173:
153:
117:
106:simplicial complex
47:algebraic topology
32:algebraic topology
373:{\displaystyle X}
351:{\displaystyle n}
239:
196:{\displaystyle X}
176:{\displaystyle n}
120:{\displaystyle X}
78:cubical complexes
16:(Redirected from
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1005:
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1397:
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1384:Stokes' theorem
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582:
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541:
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509:telescoping sum
476:
472:
468:
461:
455:
451:
446:polygonal curve
438:
422:
391:
387:
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382:
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365:
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343:
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168:
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109:
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91:
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76:(respectively,
65:
52:
43:
28:
23:
22:
15:
12:
11:
5:
1536:
1526:
1525:
1520:
1504:
1503:
1488:
1480:10.1007/b97315
1462:
1448:978-1441979391
1447:
1429:
1422:
1404:Hatcher, Allen
1394:
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1320:
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1267:
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1259:
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1234:
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1196:
1193:
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1133:
1128:
1124:
1120:
1115:
1111:
1107:
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1076:
1072:
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1024:
1019:
1015:
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1003:
999:
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992:
990:
988:
985:
980:
976:
972:
967:
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956:
951:
947:
943:
938:
934:
930:
927:
922:
918:
914:
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905:
901:
898:
896:
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891:
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878:
873:
869:
865:
860:
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839:
836:
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827:
823:
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773:
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713:
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675:
670:
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657:
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641:
637:
613:
609:
585:
581:
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553:
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540:
536:
531:
527:
523:
520:
474:
470:
463:
457:
453:
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437:
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421:
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369:
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98:
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85:
81:
63:
50:
26:
9:
6:
4:
3:
2:
1535:
1524:
1521:
1519:
1516:
1515:
1513:
1499:
1495:
1491:
1489:0-387-40853-3
1485:
1481:
1477:
1473:
1466:
1458:
1454:
1450:
1444:
1440:
1433:
1425:
1423:0-521-79540-0
1419:
1415:
1411:
1410:
1405:
1399:
1395:
1387:
1385:
1381:
1377:
1372:
1370:
1366:
1363:
1361:
1357:
1356:chain complex
1353:
1349:
1344:
1342:
1338:
1321:
1313:
1309:
1302:
1294:
1290:
1283:
1281:
1265:
1261:
1254:
1246:
1242:
1232:
1221:
1217:
1210:
1202:
1198:
1188:
1177:
1173:
1166:
1158:
1154:
1144:
1142:
1126:
1122:
1118:
1113:
1109:
1097:
1089:
1078:
1074:
1070:
1065:
1061:
1049:
1041:
1030:
1026:
1022:
1017:
1013:
1001:
993:
991:
978:
974:
965:
957:
949:
945:
936:
928:
920:
916:
907:
899:
897:
884:
880:
876:
871:
867:
863:
858:
854:
845:
837:
835:
830:
825:
808:
788:
784:
780:
775:
771:
764:
759:
755:
728:
724:
720:
715:
711:
704:
699:
695:
668:
664:
660:
655:
651:
644:
639:
635:
611:
607:
583:
579:
555:
551:
547:
542:
538:
534:
529:
525:
521:
518:
510:
506:
502:
498:
496:
493:-chain is a (
492:
483:
466:
460:
447:
442:
433:
431:
430:chain complex
427:
417:
400:
392:
388:
367:
359:
345:
321:
317:
307:
293:
284:
279:
275:
264:
260:
254:
250:
244:
235:
231:
225:
217:
213:
204:
203:is given by:
190:
170:
147:
139:
135:
114:
107:
97:
95:
79:
75:
71:
67:
60:
56:
48:
41:
37:
33:
19:
1471:
1465:
1438:
1432:
1408:
1398:
1373:
1368:
1367:
1364:
1351:
1347:
1345:
1340:
1339:
809:
500:
499:
494:
490:
488:
464:
458:
425:
423:
308:
205:
127:, the group
103:
70:cell complex
54:
44:
36:order theory
183:-chains of
1512:Categories
1390:References
1369:Example 3:
1341:Example 2:
501:Example 1:
358:-simplices
100:Definition
1457:697506452
1303:−
1255:−
1211:−
1167:−
1094:∂
1046:∂
998:∂
962:∂
933:∂
904:∂
842:∂
822:∂
627:, where
599:to point
473:− A
452:through A
338:singular
318:σ
285:∈
261:σ
241:∑
1406:(2002).
1362:groups.
1360:homology
1352:boundary
94:homology
1498:2028588
61:of the
1496:
1486:
1455:
1445:
1420:
309:where
104:For a
66:-cells
38:, see
1348:cycle
72:. In
68:in a
57:is a
55:chain
1484:ISBN
1453:OCLC
1443:ISBN
1418:ISBN
747:and
505:path
462:to A
336:are
49:, a
1476:doi
1374:In
360:of
163:of
80:),
45:In
1514::
1494:MR
1492:.
1482:.
1451:.
1416:.
1412:.
1386:.
687:,
467:+1
432:.
1500:.
1478::
1459:.
1426:.
1322:.
1319:]
1314:1
1310:v
1306:[
1300:]
1295:4
1291:v
1287:[
1284:=
1274:)
1271:]
1266:3
1262:v
1258:[
1252:]
1247:4
1243:v
1239:[
1236:(
1233:+
1230:)
1227:]
1222:2
1218:v
1214:[
1208:]
1203:3
1199:v
1195:[
1192:(
1189:+
1186:)
1183:]
1178:1
1174:v
1170:[
1164:]
1159:2
1155:v
1151:[
1148:(
1145:=
1135:)
1132:]
1127:4
1123:v
1119:,
1114:3
1110:v
1106:[
1103:(
1098:1
1090:+
1087:)
1084:]
1079:3
1075:v
1071:,
1066:2
1062:v
1058:[
1055:(
1050:1
1042:+
1039:)
1036:]
1031:2
1027:v
1023:,
1018:1
1014:v
1010:[
1007:(
1002:1
994:=
984:)
979:3
975:t
971:(
966:1
958:+
955:)
950:2
946:t
942:(
937:1
929:+
926:)
921:1
917:t
913:(
908:1
900:=
890:)
885:3
881:t
877:+
872:2
868:t
864:+
859:1
855:t
851:(
846:1
838:=
831:c
826:1
794:]
789:4
785:v
781:,
776:3
772:v
768:[
765:=
760:3
756:t
734:]
729:3
725:v
721:,
716:2
712:v
708:[
705:=
700:2
696:t
674:]
669:2
665:v
661:,
656:1
652:v
648:[
645:=
640:1
636:t
612:4
608:v
584:1
580:v
556:3
552:t
548:+
543:2
539:t
535:+
530:1
526:t
522:=
519:c
495:k
491:k
477:.
475:1
471:6
465:k
459:k
454:6
450:1
426:k
404:)
401:X
398:(
393:n
389:C
368:X
346:n
322:i
294:}
289:Z
280:i
276:m
271:|
265:i
255:i
251:m
245:i
236:{
232:=
229:)
226:X
223:(
218:n
214:C
191:X
171:n
151:)
148:X
145:(
140:n
136:C
115:X
90:k
86:k
82:k
64:k
53:-
51:k
42:.
20:)
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