1021:
1268:
1549:
948:
787:
242:
196:
844:
554:
522:
486:
454:
400:
325:
80:
45:
278:
1265:
888:
1166:
1045:
1242:
956:
1496:
139:
1614:
1193:
1119:
1076:
364:
1576:
1341:
1305:
1430:
1394:
1146:
639:
100:
1367:
666:
698:
596:
1271:, the direct sum of its free part and its torsion part. The rank of the free part is called the Novikov Betti number and is denoted by
1501:
403:
1643:
897:
710:
281:
201:
1635:
147:
1696:
799:
527:
495:
459:
408:
373:
298:
53:
22:
1087:
247:
576:
1247:
849:
1151:
1030:
1016:{\displaystyle \phi _{\xi }\colon \mathbb {Z} \to \operatorname {Nov} =\operatorname {Nov} (\mathbb {R} )}
677:
557:
328:
1706:
891:
1198:
1701:
1462:
105:
1582:
a formal variable, such that the multiplication is an extension of a multiplication in the integral
1588:
1171:
1093:
1050:
338:
1554:
1440:
1310:
1274:
289:
1639:
1399:
1372:
1124:
608:
85:
1649:
1346:
644:
1653:
1083:
701:
683:
581:
1690:
603:
599:
285:
1583:
569:
142:
950:. By the universal property, this map in turns gives a ring homomorphism,
1669:
The
Hamiltonian formalism and a multi-valued analogue of Morse theory.
288:
using a closed one-form instead of a function. The notion is used in
1662:
Multi-valued functions and functionals: An analogue of Morse theory.
1681:
796:
be a connected polyhedron with a base point. Each cohomology class
1307:. The number of cyclic modules in the torsion part is denoted by
846:
may be viewed as a linear functional on the first homology group
1544:{\displaystyle \sum _{\gamma \in \Gamma }n_{\gamma }t^{\gamma }}
1443:
holds for
Novikov numbers as well (cf. the reference for now.)
792:
In an analogy with this, one can define "Novikov numbers". Let
366:
consisting of those with leading term 1. Since the elements of
943:{\displaystyle \xi \colon \pi =\pi _{1}(X)\to \mathbb {R} }
782:{\displaystyle H^{*}(C_{*}(f))\cong H^{*}(M,\mathbb {Z} )}
1634:. Mathematical surveys and monographs. Vol. 108.
676:(called the Morse number). It computes the (integral)
1591:
1557:
1504:
1465:
1402:
1375:
1349:
1313:
1277:
1250:
1201:
1174:
1154:
1127:
1096:
1053:
1033:
959:
900:
852:
802:
713:
686:
647:
611:
584:
530:
498:
462:
411:
376:
341:
301:
250:
237:{\displaystyle \gamma _{1}>\gamma _{2}>\cdots }
204:
150:
108:
88:
56:
25:
284:in the papers that initiated the generalization of
191:{\displaystyle \sum n_{\gamma _{i}}t^{\gamma _{i}}}
1608:
1570:
1543:
1490:
1424:
1388:
1361:
1335:
1299:
1259:
1236:
1187:
1160:
1140:
1113:
1070:
1039:
1015:
942:
882:
838:
781:
692:
660:
633:
590:
548:
516:
480:
448:
394:
358:
319:
272:
236:
190:
133:
94:
74:
39:
1484:
1474:
127:
117:
1664:Soviet Mathematics - Doklady 24 (1981), 222–226.
1671:Russian Mathematical Surveys 35:5 (1982), 1–56.
1148:be a local coefficient system corresponding to
598:with nondegenerate critical points, the usual
839:{\displaystyle \xi \in H^{1}(X,\mathbb {R} )}
549:{\displaystyle \operatorname {Nov} (\Gamma )}
517:{\displaystyle \operatorname {Nov} (\Gamma )}
481:{\displaystyle \operatorname {Nov} (\Gamma )}
449:{\displaystyle \operatorname {Nov} (\Gamma )}
395:{\displaystyle \operatorname {Nov} (\Gamma )}
320:{\displaystyle \operatorname {Nov} (\Gamma )}
75:{\displaystyle \operatorname {Nov} (\Gamma )}
8:
894:, it can be viewed as a group homomorphism
40:{\displaystyle \Gamma \subset \mathbb {R} }
19:In mathematics, given an additive subgroup
1498:is the ring consisting of the formal sums
1593:
1592:
1590:
1562:
1556:
1535:
1525:
1509:
1503:
1467:
1466:
1464:
1407:
1401:
1380:
1374:
1348:
1318:
1312:
1282:
1276:
1249:
1225:
1206:
1200:
1179:
1173:
1153:
1132:
1126:
1098:
1097:
1095:
1055:
1054:
1052:
1032:
1006:
1005:
974:
973:
964:
958:
936:
935:
917:
899:
873:
872:
857:
851:
829:
828:
813:
801:
772:
771:
756:
731:
718:
712:
685:
652:
646:
616:
610:
583:
529:
497:
461:
434:
410:
375:
343:
342:
340:
300:
255:
249:
222:
209:
203:
180:
175:
163:
158:
149:
110:
109:
107:
87:
55:
33:
32:
24:
1452:
273:{\displaystyle \gamma _{i}\to -\infty }
1682:Different definitions of Novikov ring?
1260:{\displaystyle \operatorname {Nov} ,}
883:{\displaystyle H_{1}(X,\mathbb {R} )}
7:
1244:is a finitely generated module over
1161:{\displaystyle \operatorname {Nov} }
1090:over it corresponds one-to-one to a
1040:{\displaystyle \operatorname {Nov} }
668:is the number of critical points of
1600:
1516:
1478:
540:
508:
472:
421:
386:
350:
311:
267:
121:
89:
66:
26:
14:
1237:{\displaystyle H_{p}(X,L_{\xi })}
641:such that the (integral) rank of
1491:{\displaystyle \mathbb {Z} \!]}
1168:with module structure given by
280:. The notion was introduced by
134:{\displaystyle \mathbb {Z} \!]}
1603:
1597:
1485:
1481:
1475:
1471:
1419:
1413:
1330:
1324:
1294:
1288:
1231:
1212:
1108:
1102:
1065:
1059:
1010:
1002:
987:
984:
978:
932:
929:
923:
877:
863:
833:
819:
776:
762:
746:
743:
737:
724:
628:
622:
543:
537:
524:called the "rational part" of
511:
505:
475:
469:
443:
427:
424:
418:
389:
383:
353:
347:
314:
308:
261:
128:
124:
118:
114:
69:
63:
1:
1636:American Mathematical Society
1432:is the usual Betti number of
1632:Topology of closed one-forms
1609:{\displaystyle \mathbb {Z} }
1188:{\displaystyle \phi _{\xi }}
1114:{\displaystyle \mathbb {Z} }
1071:{\displaystyle \mathbb {Z} }
359:{\displaystyle \mathbb {Z} }
1571:{\displaystyle n_{\gamma }}
1336:{\displaystyle q_{p}(\xi )}
1300:{\displaystyle b_{p}(\xi )}
1723:
890:; when composed with the
1630:Farber, Michael (2004).
1425:{\displaystyle b_{p}(0)}
1389:{\displaystyle L_{\xi }}
1141:{\displaystyle L_{\xi }}
1088:local coefficient system
634:{\displaystyle C_{*}(f)}
95:{\displaystyle \Gamma }
1610:
1572:
1545:
1492:
1426:
1390:
1363:
1362:{\displaystyle \xi =0}
1337:
1301:
1261:
1238:
1189:
1162:
1142:
1115:
1072:
1041:
1017:
944:
884:
840:
783:
694:
662:
635:
592:
558:principal ideal domain
550:
518:
482:
450:
396:
360:
329:principal ideal domain
321:
274:
238:
192:
135:
96:
76:
41:
16:Mathematical construct
1611:
1573:
1546:
1493:
1427:
1391:
1364:
1338:
1302:
1262:
1239:
1195:. The homology group
1190:
1163:
1143:
1116:
1073:
1042:
1018:
945:
892:Hurewicz homomorphism
885:
841:
784:
695:
663:
661:{\displaystyle C_{p}}
636:
593:
551:
519:
483:
451:
397:
370:are unit elements of
361:
322:
275:
239:
193:
136:
97:
77:
42:
1589:
1555:
1502:
1463:
1400:
1373:
1347:
1311:
1275:
1248:
1199:
1172:
1152:
1125:
1094:
1051:
1031:
957:
898:
850:
800:
711:
684:
645:
609:
582:
528:
496:
460:
409:
374:
339:
299:
292:, among the others.
248:
202:
148:
106:
86:
54:
23:
1697:Commutative algebra
1606:
1568:
1541:
1520:
1488:
1441:Morse inequalities
1422:
1386:
1359:
1333:
1297:
1257:
1234:
1185:
1158:
1138:
1111:
1068:
1037:
1013:
940:
880:
836:
779:
690:
658:
631:
602:constructs a free
588:
546:
514:
478:
446:
392:
356:
317:
290:quantum cohomology
270:
234:
188:
131:
102:is the subring of
92:
72:
37:
1505:
1269:structure theorem
1267:which is, by the
693:{\displaystyle M}
591:{\displaystyle M}
335:be the subset of
295:The Novikov ring
1714:
1657:
1617:
1615:
1613:
1612:
1607:
1596:
1577:
1575:
1574:
1569:
1567:
1566:
1550:
1548:
1547:
1542:
1540:
1539:
1530:
1529:
1519:
1497:
1495:
1494:
1489:
1470:
1457:
1431:
1429:
1428:
1423:
1412:
1411:
1395:
1393:
1392:
1387:
1385:
1384:
1368:
1366:
1365:
1360:
1342:
1340:
1339:
1334:
1323:
1322:
1306:
1304:
1303:
1298:
1287:
1286:
1266:
1264:
1263:
1258:
1243:
1241:
1240:
1235:
1230:
1229:
1211:
1210:
1194:
1192:
1191:
1186:
1184:
1183:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1137:
1136:
1120:
1118:
1117:
1112:
1101:
1077:
1075:
1074:
1069:
1058:
1046:
1044:
1043:
1038:
1022:
1020:
1019:
1014:
1009:
977:
969:
968:
949:
947:
946:
941:
939:
922:
921:
889:
887:
886:
881:
876:
862:
861:
845:
843:
842:
837:
832:
818:
817:
788:
786:
785:
780:
775:
761:
760:
736:
735:
723:
722:
699:
697:
696:
691:
667:
665:
664:
659:
657:
656:
640:
638:
637:
632:
621:
620:
597:
595:
594:
589:
555:
553:
552:
547:
523:
521:
520:
515:
492:is a subring of
488:with respect to
487:
485:
484:
479:
455:
453:
452:
447:
442:
441:
401:
399:
398:
393:
365:
363:
362:
357:
346:
326:
324:
323:
318:
279:
277:
276:
271:
260:
259:
243:
241:
240:
235:
227:
226:
214:
213:
197:
195:
194:
189:
187:
186:
185:
184:
170:
169:
168:
167:
140:
138:
137:
132:
113:
101:
99:
98:
93:
81:
79:
78:
73:
46:
44:
43:
38:
36:
1722:
1721:
1717:
1716:
1715:
1713:
1712:
1711:
1687:
1686:
1678:
1667:S. P. Novikov:
1660:S. P. Novikov,
1646:
1629:
1626:
1621:
1620:
1587:
1586:
1558:
1553:
1552:
1531:
1521:
1500:
1499:
1461:
1460:
1458:
1454:
1449:
1403:
1398:
1397:
1396:is trivial and
1376:
1371:
1370:
1345:
1344:
1314:
1309:
1308:
1278:
1273:
1272:
1246:
1245:
1221:
1202:
1197:
1196:
1175:
1170:
1169:
1150:
1149:
1128:
1123:
1122:
1092:
1091:
1049:
1048:
1029:
1028:
960:
955:
954:
913:
896:
895:
853:
848:
847:
809:
798:
797:
752:
727:
714:
709:
708:
682:
681:
648:
643:
642:
612:
607:
606:
580:
579:
577:smooth manifold
570:smooth function
566:
564:Novikov numbers
556:; it is also a
526:
525:
494:
493:
458:
457:
430:
407:
406:
372:
371:
337:
336:
297:
296:
251:
246:
245:
218:
205:
200:
199:
176:
171:
159:
154:
146:
145:
104:
103:
84:
83:
52:
51:
21:
20:
17:
12:
11:
5:
1720:
1718:
1710:
1709:
1704:
1699:
1689:
1688:
1685:
1684:
1677:
1676:External links
1674:
1673:
1672:
1665:
1658:
1644:
1625:
1622:
1619:
1618:
1605:
1602:
1599:
1595:
1565:
1561:
1538:
1534:
1528:
1524:
1518:
1515:
1512:
1508:
1487:
1483:
1480:
1477:
1473:
1469:
1451:
1450:
1448:
1445:
1439:The analog of
1421:
1418:
1415:
1410:
1406:
1383:
1379:
1358:
1355:
1352:
1332:
1329:
1326:
1321:
1317:
1296:
1293:
1290:
1285:
1281:
1256:
1253:
1233:
1228:
1224:
1220:
1217:
1214:
1209:
1205:
1182:
1178:
1157:
1135:
1131:
1110:
1107:
1104:
1100:
1086:polyhedron, a
1067:
1064:
1061:
1057:
1047:a module over
1036:
1025:
1024:
1012:
1008:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
976:
972:
967:
963:
938:
934:
931:
928:
925:
920:
916:
912:
909:
906:
903:
879:
875:
871:
868:
865:
860:
856:
835:
831:
827:
824:
821:
816:
812:
808:
805:
790:
789:
778:
774:
770:
767:
764:
759:
755:
751:
748:
745:
742:
739:
734:
730:
726:
721:
717:
702:Morse homology
689:
655:
651:
630:
627:
624:
619:
615:
587:
565:
562:
545:
542:
539:
536:
533:
513:
510:
507:
504:
501:
477:
474:
471:
468:
465:
445:
440:
437:
433:
429:
426:
423:
420:
417:
414:
391:
388:
385:
382:
379:
355:
352:
349:
345:
316:
313:
310:
307:
304:
282:Sergei Novikov
269:
266:
263:
258:
254:
233:
230:
225:
221:
217:
212:
208:
183:
179:
174:
166:
162:
157:
153:
141:consisting of
130:
126:
123:
120:
116:
112:
91:
71:
68:
65:
62:
59:
35:
31:
28:
15:
13:
10:
9:
6:
4:
3:
2:
1719:
1708:
1705:
1703:
1700:
1698:
1695:
1694:
1692:
1683:
1680:
1679:
1675:
1670:
1666:
1663:
1659:
1655:
1651:
1647:
1645:0-8218-3531-9
1641:
1637:
1633:
1628:
1627:
1623:
1585:
1581:
1578:integers and
1563:
1559:
1536:
1532:
1526:
1522:
1513:
1510:
1506:
1456:
1453:
1446:
1444:
1442:
1437:
1435:
1416:
1408:
1404:
1381:
1377:
1356:
1353:
1350:
1327:
1319:
1315:
1291:
1283:
1279:
1270:
1254:
1251:
1226:
1222:
1218:
1215:
1207:
1203:
1180:
1176:
1155:
1133:
1129:
1121:-module. Let
1105:
1089:
1085:
1081:
1062:
1034:
999:
996:
993:
990:
981:
970:
965:
961:
953:
952:
951:
926:
918:
914:
910:
907:
904:
901:
893:
869:
866:
858:
854:
825:
822:
814:
810:
806:
803:
795:
768:
765:
757:
753:
749:
740:
732:
728:
719:
715:
707:
706:
705:
703:
687:
679:
675:
671:
653:
649:
625:
617:
613:
605:
604:chain complex
601:
585:
578:
574:
571:
563:
561:
559:
534:
531:
502:
499:
491:
466:
463:
438:
435:
431:
415:
412:
405:
380:
377:
369:
334:
330:
305:
302:
293:
291:
287:
283:
264:
256:
252:
231:
228:
223:
219:
215:
210:
206:
181:
177:
172:
164:
160:
155:
151:
144:
60:
57:
50:
29:
1707:Morse theory
1668:
1661:
1631:
1579:
1455:
1438:
1433:
1079:
1026:
793:
791:
673:
669:
600:Morse theory
572:
567:
489:
404:localization
367:
332:
294:
286:Morse theory
49:Novikov ring
48:
18:
1702:Ring theory
143:formal sums
1691:Categories
1654:1052.58016
1624:References
1584:group ring
198:such that
1601:Γ
1564:γ
1537:γ
1527:γ
1517:Γ
1514:∈
1511:γ
1507:∑
1479:Γ
1382:ξ
1351:ξ
1328:ξ
1292:ξ
1227:ξ
1181:ξ
1177:ϕ
1134:ξ
1106:π
1084:connected
1063:π
1000:
988:→
982:π
971::
966:ξ
962:ϕ
933:→
915:π
908:π
905::
902:ξ
807:∈
804:ξ
758:∗
750:≅
733:∗
720:∗
672:of index
618:∗
541:Γ
535:
509:Γ
503:
473:Γ
467:
436:−
422:Γ
416:
387:Γ
381:
351:Γ
312:Γ
306:
268:∞
265:−
262:→
253:γ
232:⋯
220:γ
207:γ
178:γ
161:γ
152:∑
122:Γ
90:Γ
67:Γ
61:
30:⊂
27:Γ
1078:. Since
678:homology
568:Given a
1027:making
1652:
1642:
1459:Here,
402:, the
331:. Let
47:, the
1447:Notes
1343:. If
1082:is a
700:(cf.
575:on a
327:is a
1640:ISBN
704:):
244:and
229:>
216:>
1650:Zbl
1252:Nov
1156:Nov
1035:Nov
997:Nov
991:Nov
680:of
532:Nov
500:Nov
464:Nov
456:of
413:Nov
378:Nov
303:Nov
82:of
58:Nov
1693::
1648:.
1638:.
1551:,
1436:.
1369:,
560:.
1656:.
1616:.
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