297:
340:
1540:, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish). There are 2 distinct non-degenerate homotopy classes. Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.
1151:
998:
817:
1250:
559:
662:
1003:
822:
699:
283:
203:
1304:
1436:
1507:
492:
389:
1347:
1656:
241:
87:
1474:
436:
165:
122:
47:, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to
1401:
1374:
694:
591:
463:
1156:
497:
320:
classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same
1516:
Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the
596:
1737:
1146:{\displaystyle \pi _{6}(SO(6))\cong \pi _{6}(\operatorname {Spin} (6))\cong \pi _{6}(SU(4))\cong \pi _{6}(U(4))\cong 0}
1742:
993:{\displaystyle \pi _{6}(SO(6))\to \pi _{6}(SO(7))\to \pi _{6}\left(S^{6}\right)\to \pi _{5}(SO(6))\to \pi _{5}(SO(7))}
812:{\displaystyle \pi _{2}(SO(3))\cong \pi _{2}\left(\mathbb {R} P^{3}\right)\cong \pi _{2}\left(S^{3}\right)\cong 0}
250:
170:
1255:
33:
51:
of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions
1631:
1699:
1406:
125:
1483:
468:
365:
1309:
1716:
1675:
1517:
208:
54:
296:
1626:
1441:
1708:
1665:
1603:
1570:
406:
396:
135:
92:
1379:
1352:
667:
564:
441:
1622:
1510:
392:
344:
325:
301:
1376:
in euclidean spaces of one more dimension are regular homotopic. In particular, spheres
89:
are homotopic if they represent points in the same path-components of the mapping space
1533:
339:
333:
321:
309:
44:
1731:
1687:
1644:
355:
348:
1691:
1648:
293:
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy.
1537:
1476:. A corollary of his work is that there is only one regular homotopy class of a
17:
1608:
1591:
1575:
1558:
329:
1245:{\displaystyle \pi _{5}(SO(6))\cong \mathbb {Z} ,\ \pi _{5}(SO(7))\cong 0}
48:
37:
29:
21:
1720:
1679:
554:{\displaystyle \pi _{k}\left(V_{k}\left(\mathbb {R} ^{n}\right)\right)}
40:
in another. The homotopy must be a 1-parameter family of immersions.
1712:
1670:
338:
1692:"The classification of immersions of spheres in Euclidean spaces"
657:{\displaystyle V_{n-1}\left(\mathbb {R} ^{n}\right)\cong SO(n)}
403:
partial derivatives not vanishing. More precisely, the set
343:
Smale's classification of immersions of spheres shows that
399:, which is a generalization of the Gauss map, with here
494:
is in one-to-one correspondence with elements of group
247:
if they represent points in the same path-component of
1592:"Third order nondegenerate homotopies of space curves"
1486:
1444:
1409:
1382:
1355:
1312:
1258:
1159:
1006:
825:
702:
670:
599:
567:
500:
471:
444:
409:
368:
253:
211:
173:
138:
95:
57:
1513:
exist, i.e. one can turn the 2-sphere "inside-out".
438:
of regular homotopy classes of embeddings of sphere
1649:"A classification of immersions of the two-sphere"
1501:
1468:
1430:
1395:
1368:
1341:
1298:
1244:
1145:
992:
811:
688:
656:
585:
553:
486:
457:
430:
383:
277:
235:
197:
159:
116:
81:
1657:Transactions of the American Mathematical Society
358:classified the regular homotopy classes of a
8:
1000:and due to Bott periodicity theorem we have
1669:
1607:
1574:
1493:
1489:
1488:
1485:
1443:
1416:
1412:
1411:
1408:
1387:
1381:
1360:
1354:
1333:
1317:
1311:
1263:
1257:
1209:
1195:
1194:
1164:
1158:
1113:
1079:
1045:
1011:
1005:
963:
929:
912:
898:
864:
830:
824:
793:
779:
761:
753:
752:
741:
707:
701:
669:
626:
622:
621:
604:
598:
566:
536:
532:
531:
520:
505:
499:
478:
474:
473:
470:
449:
443:
408:
375:
371:
370:
367:
278:{\displaystyle \operatorname {Imm} (M,N)}
252:
210:
198:{\displaystyle \operatorname {Imm} (M,N)}
172:
137:
94:
56:
295:
1627:"On regular closed curves in the plane"
1549:
1306:. Therefore all immersions of spheres
1299:{\displaystyle \pi _{6}(SO(7))\cong 0}
347:exist, which can be realized via this
1559:"Deformations of closed space curves"
328:; equivalently, if and only if their
167:consisting of immersions, denoted by
7:
1509:. In particular, this means that
1431:{\displaystyle \mathbb {R} ^{n+1}}
14:
1596:Journal of Differential Geometry
1563:Journal of Differential Geometry
1502:{\displaystyle \mathbb {R} ^{3}}
487:{\displaystyle \mathbb {R} ^{n}}
384:{\displaystyle \mathbb {R} ^{n}}
1287:
1284:
1278:
1269:
1233:
1230:
1224:
1215:
1188:
1185:
1179:
1170:
1134:
1131:
1125:
1119:
1103:
1100:
1094:
1085:
1069:
1066:
1060:
1051:
1035:
1032:
1026:
1017:
987:
984:
978:
969:
956:
953:
950:
944:
935:
922:
891:
888:
885:
879:
870:
857:
854:
851:
845:
836:
731:
728:
722:
713:
683:
677:
651:
645:
425:
413:
272:
260:
227:
192:
180:
154:
142:
111:
99:
73:
1:
1342:{\displaystyle S^{0},\ S^{2}}
28:refers to a special kind of
1759:
236:{\displaystyle f,g:M\to N}
82:{\displaystyle f,g:M\to N}
391:– they are classified by
318:Whitney–Graustein theorem
1590:Little, John A. (1971).
1557:Feldman, E. A. (1968).
1528:Non-degenerate homotopy
1469:{\displaystyle n=0,2,6}
1632:Compositio Mathematica
1609:10.4310/jdg/1214430012
1576:10.4310/jdg/1214501138
1524:-principle) approach.
1503:
1470:
1432:
1397:
1370:
1343:
1300:
1246:
1147:
994:
813:
690:
658:
587:
555:
488:
459:
432:
431:{\displaystyle I(n,k)}
385:
352:
313:
279:
237:
199:
161:
160:{\displaystyle C(M,N)}
118:
117:{\displaystyle C(M,N)}
83:
1738:Differential topology
1700:Annals of Mathematics
1504:
1471:
1433:
1398:
1396:{\displaystyle S^{n}}
1371:
1369:{\displaystyle S^{6}}
1344:
1301:
1247:
1148:
995:
814:
691:
689:{\displaystyle SO(1)}
659:
588:
586:{\displaystyle k=n-1}
556:
489:
460:
458:{\displaystyle S^{k}}
433:
386:
362:-sphere immersed in
342:
332:have the same degree/
299:
280:
238:
200:
162:
126:compact-open topology
119:
84:
1484:
1480:-sphere immersed in
1442:
1407:
1380:
1353:
1310:
1256:
1157:
1004:
823:
700:
668:
597:
565:
498:
469:
442:
407:
366:
251:
209:
171:
136:
93:
55:
696:is path connected,
245:regularly homotopic
132:is the subspace of
130:space of immersions
1743:Algebraic topology
1518:homotopy principle
1499:
1466:
1438:admit eversion if
1428:
1393:
1366:
1339:
1296:
1242:
1143:
990:
809:
686:
654:
583:
551:
484:
455:
428:
381:
353:
314:
275:
233:
195:
157:
114:
79:
1647:(February 1959).
1328:
1204:
397:Stiefel manifolds
205:. Two immersions
1750:
1724:
1696:
1683:
1673:
1653:
1640:
1623:Whitney, Hassler
1614:
1613:
1611:
1587:
1581:
1580:
1578:
1554:
1511:sphere eversions
1508:
1506:
1505:
1500:
1498:
1497:
1492:
1475:
1473:
1472:
1467:
1437:
1435:
1434:
1429:
1427:
1426:
1415:
1402:
1400:
1399:
1394:
1392:
1391:
1375:
1373:
1372:
1367:
1365:
1364:
1348:
1346:
1345:
1340:
1338:
1337:
1326:
1322:
1321:
1305:
1303:
1302:
1297:
1268:
1267:
1251:
1249:
1248:
1243:
1214:
1213:
1202:
1198:
1169:
1168:
1152:
1150:
1149:
1144:
1118:
1117:
1084:
1083:
1050:
1049:
1016:
1015:
999:
997:
996:
991:
968:
967:
934:
933:
921:
917:
916:
903:
902:
869:
868:
835:
834:
818:
816:
815:
810:
802:
798:
797:
784:
783:
771:
767:
766:
765:
756:
746:
745:
712:
711:
695:
693:
692:
687:
663:
661:
660:
655:
635:
631:
630:
625:
615:
614:
592:
590:
589:
584:
560:
558:
557:
552:
550:
546:
545:
541:
540:
535:
525:
524:
510:
509:
493:
491:
490:
485:
483:
482:
477:
464:
462:
461:
456:
454:
453:
437:
435:
434:
429:
390:
388:
387:
382:
380:
379:
374:
345:sphere eversions
324:– equivalently,
284:
282:
281:
276:
242:
240:
239:
234:
204:
202:
201:
196:
166:
164:
163:
158:
123:
121:
120:
115:
88:
86:
85:
80:
45:homotopy classes
26:regular homotopy
1758:
1757:
1753:
1752:
1751:
1749:
1748:
1747:
1728:
1727:
1713:10.2307/1970186
1694:
1686:
1671:10.2307/1993205
1651:
1643:
1621:
1618:
1617:
1589:
1588:
1584:
1556:
1555:
1551:
1546:
1530:
1487:
1482:
1481:
1440:
1439:
1410:
1405:
1404:
1383:
1378:
1377:
1356:
1351:
1350:
1329:
1313:
1308:
1307:
1259:
1254:
1253:
1205:
1160:
1155:
1154:
1109:
1075:
1041:
1007:
1002:
1001:
959:
925:
908:
904:
894:
860:
826:
821:
820:
789:
785:
775:
757:
751:
747:
737:
703:
698:
697:
666:
665:
620:
616:
600:
595:
594:
563:
562:
530:
526:
516:
515:
511:
501:
496:
495:
472:
467:
466:
445:
440:
439:
405:
404:
393:homotopy groups
369:
364:
363:
326:total curvature
302:total curvature
300:This curve has
291:
249:
248:
207:
206:
169:
168:
134:
133:
91:
90:
53:
52:
12:
11:
5:
1756:
1754:
1746:
1745:
1740:
1730:
1729:
1726:
1725:
1707:(2): 327–344.
1690:(March 1959).
1688:Smale, Stephen
1684:
1664:(2): 281–290.
1645:Smale, Stephen
1641:
1616:
1615:
1602:(3): 503–515.
1582:
1548:
1547:
1545:
1542:
1534:locally convex
1529:
1526:
1496:
1491:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1425:
1422:
1419:
1414:
1390:
1386:
1363:
1359:
1336:
1332:
1325:
1320:
1316:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1266:
1262:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1212:
1208:
1201:
1197:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1167:
1163:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1116:
1112:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1082:
1078:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1048:
1044:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1014:
1010:
989:
986:
983:
980:
977:
974:
971:
966:
962:
958:
955:
952:
949:
946:
943:
940:
937:
932:
928:
924:
920:
915:
911:
907:
901:
897:
893:
890:
887:
884:
881:
878:
875:
872:
867:
863:
859:
856:
853:
850:
847:
844:
841:
838:
833:
829:
808:
805:
801:
796:
792:
788:
782:
778:
774:
770:
764:
760:
755:
750:
744:
740:
736:
733:
730:
727:
724:
721:
718:
715:
710:
706:
685:
682:
679:
676:
673:
653:
650:
647:
644:
641:
638:
634:
629:
624:
619:
613:
610:
607:
603:
582:
579:
576:
573:
570:
549:
544:
539:
534:
529:
523:
519:
514:
508:
504:
481:
476:
452:
448:
427:
424:
421:
418:
415:
412:
378:
373:
334:winding number
322:turning number
310:turning number
290:
287:
274:
271:
268:
265:
262:
259:
256:
232:
229:
226:
223:
220:
217:
214:
194:
191:
188:
185:
182:
179:
176:
156:
153:
150:
147:
144:
141:
113:
110:
107:
104:
101:
98:
78:
75:
72:
69:
66:
63:
60:
13:
10:
9:
6:
4:
3:
2:
1755:
1744:
1741:
1739:
1736:
1735:
1733:
1722:
1718:
1714:
1710:
1706:
1702:
1701:
1693:
1689:
1685:
1681:
1677:
1672:
1667:
1663:
1659:
1658:
1650:
1646:
1642:
1638:
1634:
1633:
1628:
1624:
1620:
1619:
1610:
1605:
1601:
1597:
1593:
1586:
1583:
1577:
1572:
1568:
1564:
1560:
1553:
1550:
1543:
1541:
1539:
1535:
1527:
1525:
1523:
1519:
1514:
1512:
1494:
1479:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1423:
1420:
1417:
1388:
1384:
1361:
1357:
1334:
1330:
1323:
1318:
1314:
1293:
1290:
1281:
1275:
1272:
1264:
1260:
1252:then we have
1239:
1236:
1227:
1221:
1218:
1210:
1206:
1199:
1191:
1182:
1176:
1173:
1165:
1161:
1140:
1137:
1128:
1122:
1114:
1110:
1106:
1097:
1091:
1088:
1080:
1076:
1072:
1063:
1057:
1054:
1046:
1042:
1038:
1029:
1023:
1020:
1012:
1008:
981:
975:
972:
964:
960:
947:
941:
938:
930:
926:
918:
913:
909:
905:
899:
895:
882:
876:
873:
865:
861:
848:
842:
839:
831:
827:
806:
803:
799:
794:
790:
786:
780:
776:
772:
768:
762:
758:
748:
742:
738:
734:
725:
719:
716:
708:
704:
680:
674:
671:
648:
642:
639:
636:
632:
627:
617:
611:
608:
605:
601:
580:
577:
574:
571:
568:
547:
542:
537:
527:
521:
517:
512:
506:
502:
479:
450:
446:
422:
419:
416:
410:
402:
398:
394:
376:
361:
357:
356:Stephen Smale
350:
349:Morin surface
346:
341:
337:
335:
331:
327:
323:
319:
311:
307:
303:
298:
294:
288:
286:
269:
266:
263:
257:
254:
246:
230:
224:
221:
218:
215:
212:
189:
186:
183:
177:
174:
151:
148:
145:
139:
131:
127:
108:
105:
102:
96:
76:
70:
67:
64:
61:
58:
50:
46:
41:
39:
35:
31:
27:
23:
19:
1704:
1698:
1661:
1655:
1636:
1630:
1599:
1595:
1585:
1569:(1): 67–75.
1566:
1562:
1552:
1538:space curves
1531:
1521:
1515:
1477:
1403:embedded in
400:
359:
354:
317:
315:
305:
292:
244:
129:
124:, given the
42:
25:
18:mathematical
15:
43:Similar to
1732:Categories
1639:: 276–284.
1544:References
1153:and since
561:. In case
330:Gauss maps
34:immersions
1536:, closed
1291:≅
1261:π
1237:≅
1207:π
1192:≅
1162:π
1138:≅
1111:π
1107:≅
1077:π
1073:≅
1058:
1043:π
1039:≅
1009:π
961:π
957:→
927:π
923:→
896:π
892:→
862:π
858:→
828:π
804:≅
777:π
773:≅
739:π
735:≅
705:π
637:≅
609:−
578:−
503:π
258:
228:→
178:
74:→
20:field of
1625:(1937).
664:. Since
593:we have
289:Examples
38:manifold
32:between
30:homotopy
22:topology
1721:1970186
1680:1993205
49:isotopy
36:of one
16:In the
1719:
1678:
1327:
1203:
308:, and
128:. The
1717:JSTOR
1695:(PDF)
1676:JSTOR
1652:(PDF)
1532:For
1520:(or
1349:and
1055:Spin
819:and
316:The
243:are
24:, a
1709:doi
1666:doi
1604:doi
1571:doi
465:in
395:of
255:Imm
175:Imm
1734::
1715:.
1705:69
1703:.
1697:.
1674:.
1662:90
1660:.
1654:.
1635:.
1629:.
1598:.
1594:.
1565:.
1561:.
336:.
312:3.
285:.
1723:.
1711::
1682:.
1668::
1637:4
1612:.
1606::
1600:5
1579:.
1573::
1567:2
1522:h
1495:3
1490:R
1478:2
1464:6
1461:,
1458:2
1455:,
1452:0
1449:=
1446:n
1424:1
1421:+
1418:n
1413:R
1389:n
1385:S
1362:6
1358:S
1335:2
1331:S
1324:,
1319:0
1315:S
1294:0
1288:)
1285:)
1282:7
1279:(
1276:O
1273:S
1270:(
1265:6
1240:0
1234:)
1231:)
1228:7
1225:(
1222:O
1219:S
1216:(
1211:5
1200:,
1196:Z
1189:)
1186:)
1183:6
1180:(
1177:O
1174:S
1171:(
1166:5
1141:0
1135:)
1132:)
1129:4
1126:(
1123:U
1120:(
1115:6
1104:)
1101:)
1098:4
1095:(
1092:U
1089:S
1086:(
1081:6
1070:)
1067:)
1064:6
1061:(
1052:(
1047:6
1036:)
1033:)
1030:6
1027:(
1024:O
1021:S
1018:(
1013:6
988:)
985:)
982:7
979:(
976:O
973:S
970:(
965:5
954:)
951:)
948:6
945:(
942:O
939:S
936:(
931:5
919:)
914:6
910:S
906:(
900:6
889:)
886:)
883:7
880:(
877:O
874:S
871:(
866:6
855:)
852:)
849:6
846:(
843:O
840:S
837:(
832:6
807:0
800:)
795:3
791:S
787:(
781:2
769:)
763:3
759:P
754:R
749:(
743:2
732:)
729:)
726:3
723:(
720:O
717:S
714:(
709:2
684:)
681:1
678:(
675:O
672:S
652:)
649:n
646:(
643:O
640:S
633:)
628:n
623:R
618:(
612:1
606:n
602:V
581:1
575:n
572:=
569:k
548:)
543:)
538:n
533:R
528:(
522:k
518:V
513:(
507:k
480:n
475:R
451:k
447:S
426:)
423:k
420:,
417:n
414:(
411:I
401:k
377:n
372:R
360:k
351:.
306:π
304:6
273:)
270:N
267:,
264:M
261:(
231:N
225:M
222::
219:g
216:,
213:f
193:)
190:N
187:,
184:M
181:(
155:)
152:N
149:,
146:M
143:(
140:C
112:)
109:N
106:,
103:M
100:(
97:C
77:N
71:M
68::
65:g
62:,
59:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
