62:
439:
1242:
423:
1054:
17:
792:
526:
686:
883:
21:
913:
378:
18:
173:
1197:
1128:
20:
270:
226:
819:
590:
560:
697:
1383:
1357:
1324:
1223:
1291:
442:
The automorphism on the fundamental group of the torus induced by the self-homeomorphism of the Dehn twist along one of the generators of the torus.
1326:
explicit curves generate the mapping class group (this is called by the punning name "Lickorish twist theorem"); this number was later improved by
467:
19:
1434:
602:
1262:
1401:
831:
1049:{\displaystyle {T_{a}}_{\ast }:\pi _{1}\left(\mathbb {T} ^{2}\right)\to \pi _{1}\left(\mathbb {T} ^{2}\right):\mapsto \left}
304:
1388:
Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "
1500:
130:
1426:
1505:
1327:
1133:
1070:
593:
106:
794:
of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of
94:
1270:
451:
61:
1297:
later rediscovered this result with a simpler proof and in addition showed that Dehn twists along
787:{\displaystyle \left(e^{i\theta },t\right)\mapsto \left(e^{i\left(\theta +2\pi t\right)},t\right)}
438:
231:
201:
1461:
1294:
904:
797:
568:
538:
46:
30:
1430:
114:
1418:
1406:
1362:
1060:
565:
Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve
1485:
1472:
1456:
1482:
1469:
1453:
1449:
1389:
1333:
1300:
422:
1477:
W. B. R. Lickorish, "A finite set of generators for the homotopy group of a 2-manifold",
1202:
1241:
592:
will look like a band linked around a doughnut. This neighborhood is homeomorphic to an
1276:
903:
A homeomorphism between topological spaces induces a natural isomorphism between their
195:
1494:
118:
82:
42:
110:
34:
25:
A positive Dehn twist applied to the cylinder modifies the green curve as shown.
1269:
classes of orientation-preserving homeomorphisms of any closed, oriented
1266:
1258:
50:
521:{\displaystyle \mathbb {T} ^{2}\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2}.}
403:
1448:, Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722,
1440:
Stephen P. Humphries, "Generators for the mapping class group," in:
447:
437:
421:
78:
60:
15:
426:
An example of a Dehn twist on the torus, along the closed curve
1464:, "A representation of orientable combinatorial 3-manifolds."
434:
is an edge of the fundamental polygon representing the torus.
681:{\displaystyle a(0;0,1)=\{z\in \mathbb {C} :0<|z|<1\}}
398:
Dehn twists can also be defined on a non-orientable surface
878:{\displaystyle T_{a}:\mathbb {T} ^{2}\to \mathbb {T} ^{2}}
65:
General Dehn twist on a compact surface represented by a
888:
This self homeomorphism acts on the closed curve along
1365:
1336:
1303:
1279:
1205:
1136:
1073:
916:
834:
800:
700:
605:
571:
541:
470:
307:
234:
204:
133:
1423:
Automorphisms of
Surfaces After Nielsen and Thurston
892:. In the tubular neighborhood it takes the curve of
373:{\displaystyle f(s,t)=\left(se^{i2\pi t},t\right).}
1377:
1351:
1318:
1285:
1249:− 1 curves from the twist theorem, shown here for
1217:
1191:
1122:
1048:
877:
813:
786:
680:
584:
554:
520:
372:
264:
220:
167:
531:Let a closed curve be the line along the edge
168:{\displaystyle c\subset A\cong S^{1}\times I.}
8:
1385:, which he showed was the minimal number.
675:
633:
694:By extending to the torus the twisting map
290:to itself which is the identity outside of
1364:
1335:
1302:
1278:
1204:
1150:
1143:
1138:
1135:
1087:
1080:
1075:
1072:
1026:
992:
988:
987:
976:
959:
955:
954:
943:
930:
923:
918:
915:
869:
865:
864:
854:
850:
849:
839:
833:
805:
799:
742:
710:
699:
664:
656:
643:
642:
604:
576:
570:
546:
540:
509:
505:
504:
498:
492:
488:
487:
477:
473:
472:
469:
341:
306:
233:
209:
203:
150:
132:
1240:
821:, yields a Dehn twist of the torus by
1442:Topology of low-dimensional manifolds
7:
1261:that maps of this form generate the
907:. Therefore one has an automorphism
1192:{\displaystyle {T_{a}}_{\ast }()=}
1123:{\displaystyle {T_{a}}_{\ast }()=}
14:
1186:
1174:
1168:
1165:
1159:
1156:
1117:
1111:
1105:
1102:
1096:
1093:
1038:
1032:
1014:
1011:
1005:
969:
860:
730:
665:
657:
627:
609:
323:
311:
256:
241:
1:
1225:is the path travelled around
896:once along the curve of
402:, provided one starts with a
1479:Proc. Cambridge Philos. Soc.
265:{\displaystyle \theta \in ,}
221:{\displaystyle e^{i\theta }}
1481:60 (1964), 769–778.
1402:Fenchel–Nielsen coordinates
814:{\displaystyle \gamma _{a}}
585:{\displaystyle \gamma _{a}}
555:{\displaystyle \gamma _{a}}
1522:
1427:Cambridge University Press
1446:Proc. Second Sussex Conf.
1379:
1378:{\displaystyle g>1}
1353:
1320:
1287:
1254:
1219:
1193:
1124:
1050:
879:
815:
788:
691:in the complex plane.
682:
586:
556:
522:
443:
435:
374:
266:
222:
169:
70:
26:
1468:(2) 76 1962 531—540.
1380:
1354:
1321:
1288:
1244:
1220:
1194:
1125:
1067:in the torus. Notice
1051:
880:
816:
789:
683:
587:
557:
523:
441:
425:
375:
267:
223:
170:
64:
41:is a certain type of
24:
1421:, Steven A Bleiler,
1363:
1352:{\displaystyle 2g+1}
1334:
1328:Stephen P. Humphries
1319:{\displaystyle 3g-1}
1301:
1277:
1203:
1134:
1071:
1063:of the closed curve
914:
832:
798:
698:
603:
569:
539:
468:
406:simple closed curve
305:
232:
202:
131:
95:tubular neighborhood
1263:mapping class group
1257:It is a theorem of
1237:Mapping class group
1218:{\displaystyle b*a}
452:fundamental polygon
79:simple closed curve
1501:Geometric topology
1462:W. B. R. Lickorish
1375:
1349:
1316:
1295:W. B. R. Lickorish
1283:
1255:
1215:
1189:
1120:
1046:
905:fundamental groups
875:
811:
784:
678:
582:
552:
518:
444:
436:
370:
262:
218:
165:
117:of a circle and a
71:
43:self-homeomorphism
31:geometric topology
27:
1286:{\displaystyle g}
450:represented by a
430:, in blue, where
115:Cartesian product
49:(two-dimensional
22:
1513:
1452:, Berlin, 1979.
1419:Andrew J. Casson
1407:Lantern relation
1390:Y-homeomorphisms
1384:
1382:
1381:
1376:
1358:
1356:
1355:
1350:
1325:
1323:
1322:
1317:
1292:
1290:
1289:
1284:
1224:
1222:
1221:
1216:
1198:
1196:
1195:
1190:
1155:
1154:
1149:
1148:
1147:
1129:
1127:
1126:
1121:
1092:
1091:
1086:
1085:
1084:
1061:homotopy classes
1055:
1053:
1052:
1047:
1045:
1041:
1031:
1030:
1001:
997:
996:
991:
981:
980:
968:
964:
963:
958:
948:
947:
935:
934:
929:
928:
927:
884:
882:
881:
876:
874:
873:
868:
859:
858:
853:
844:
843:
820:
818:
817:
812:
810:
809:
793:
791:
790:
785:
783:
779:
772:
771:
770:
766:
729:
725:
718:
717:
687:
685:
684:
679:
668:
660:
646:
591:
589:
588:
583:
581:
580:
561:
559:
558:
553:
551:
550:
527:
525:
524:
519:
514:
513:
508:
502:
497:
496:
491:
482:
481:
476:
391:about the curve
379:
377:
376:
371:
366:
362:
355:
354:
286:be the map from
278:
271:
269:
268:
263:
227:
225:
224:
219:
217:
216:
174:
172:
171:
166:
155:
154:
23:
1521:
1520:
1516:
1515:
1514:
1512:
1511:
1510:
1491:
1490:
1415:
1398:
1361:
1360:
1332:
1331:
1299:
1298:
1275:
1274:
1239:
1201:
1200:
1139:
1137:
1132:
1131:
1076:
1074:
1069:
1068:
1059:where are the
1022:
1021:
1017:
986:
982:
972:
953:
949:
939:
919:
917:
912:
911:
863:
848:
835:
830:
829:
801:
796:
795:
750:
746:
738:
737:
733:
706:
705:
701:
696:
695:
601:
600:
572:
567:
566:
542:
537:
536:
503:
486:
471:
466:
465:
420:
337:
333:
329:
303:
302:
273:
230:
229:
205:
200:
199:
146:
129:
128:
59:
16:
12:
11:
5:
1519:
1517:
1509:
1508:
1506:Homeomorphisms
1503:
1493:
1492:
1489:
1488:
1475:
1459:
1438:
1414:
1411:
1410:
1409:
1404:
1397:
1394:
1374:
1371:
1368:
1348:
1345:
1342:
1339:
1315:
1312:
1309:
1306:
1282:
1238:
1235:
1214:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1153:
1146:
1142:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1090:
1083:
1079:
1057:
1056:
1044:
1040:
1037:
1034:
1029:
1025:
1020:
1016:
1013:
1010:
1007:
1004:
1000:
995:
990:
985:
979:
975:
971:
967:
962:
957:
952:
946:
942:
938:
933:
926:
922:
886:
885:
872:
867:
862:
857:
852:
847:
842:
838:
808:
804:
782:
778:
775:
769:
765:
762:
759:
756:
753:
749:
745:
741:
736:
732:
728:
724:
721:
716:
713:
709:
704:
689:
688:
677:
674:
671:
667:
663:
659:
655:
652:
649:
645:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
579:
575:
549:
545:
529:
528:
517:
512:
507:
501:
495:
490:
485:
480:
475:
419:
416:
381:
380:
369:
365:
361:
358:
353:
350:
347:
344:
340:
336:
332:
328:
325:
322:
319:
316:
313:
310:
261:
258:
255:
252:
249:
246:
243:
240:
237:
215:
212:
208:
196:complex number
176:
175:
164:
161:
158:
153:
149:
145:
142:
139:
136:
58:
55:
33:, a branch of
13:
10:
9:
6:
4:
3:
2:
1518:
1507:
1504:
1502:
1499:
1498:
1496:
1487:
1484:
1480:
1476:
1474:
1471:
1467:
1466:Ann. of Math.
1463:
1460:
1458:
1455:
1451:
1447:
1443:
1439:
1436:
1435:0-521-34985-0
1432:
1428:
1424:
1420:
1417:
1416:
1412:
1408:
1405:
1403:
1400:
1399:
1395:
1393:
1391:
1386:
1372:
1369:
1366:
1346:
1343:
1340:
1337:
1329:
1313:
1310:
1307:
1304:
1296:
1280:
1272:
1268:
1264:
1260:
1252:
1248:
1243:
1236:
1234:
1232:
1228:
1212:
1209:
1206:
1183:
1180:
1177:
1171:
1162:
1151:
1144:
1140:
1114:
1108:
1099:
1088:
1081:
1077:
1066:
1062:
1042:
1035:
1027:
1023:
1018:
1008:
1002:
998:
993:
983:
977:
973:
965:
960:
950:
944:
940:
936:
931:
924:
920:
910:
909:
908:
906:
901:
899:
895:
891:
870:
855:
845:
840:
836:
828:
827:
826:
824:
806:
802:
780:
776:
773:
767:
763:
760:
757:
754:
751:
747:
743:
739:
734:
726:
722:
719:
714:
711:
707:
702:
692:
672:
669:
661:
653:
650:
647:
639:
636:
630:
624:
621:
618:
615:
612:
606:
599:
598:
597:
595:
577:
573:
563:
547:
543:
534:
515:
510:
499:
493:
483:
478:
464:
463:
462:
461:
457:
453:
449:
446:Consider the
440:
433:
429:
424:
417:
415:
413:
409:
405:
401:
396:
394:
390:
386:
367:
363:
359:
356:
351:
348:
345:
342:
338:
334:
330:
326:
320:
317:
314:
308:
301:
300:
299:
297:
293:
289:
285:
280:
276:
259:
253:
250:
247:
244:
238:
235:
213:
210:
206:
197:
193:
189:
185:
182:coordinates (
181:
162:
159:
156:
151:
147:
143:
140:
137:
134:
127:
126:
125:
123:
120:
119:unit interval
116:
112:
108:
104:
100:
96:
92:
88:
84:
81:in a closed,
80:
76:
73:Suppose that
68:
63:
56:
54:
52:
48:
44:
40:
36:
32:
1478:
1465:
1445:
1441:
1422:
1387:
1256:
1250:
1246:
1230:
1226:
1064:
1058:
902:
897:
893:
889:
887:
822:
693:
690:
564:
532:
530:
459:
455:
445:
431:
427:
411:
407:
399:
397:
392:
388:
384:
382:
295:
291:
287:
283:
281:
274:
198:of the form
191:
187:
183:
179:
177:
121:
111:homeomorphic
102:
98:
90:
86:
74:
72:
66:
38:
28:
454:with edges
294:and inside
35:mathematics
1495:Categories
1413:References
389:Dehn twist
83:orientable
57:Definition
39:Dehn twist
1311:−
1293:surface.
1210:∗
1181:∗
1152:∗
1089:∗
1015:↦
974:π
970:→
941:π
932:∗
861:→
803:γ
761:π
752:θ
731:↦
715:θ
640:∈
574:γ
544:γ
484:≅
349:π
254:π
239:∈
236:θ
214:θ
157:×
144:≅
138:⊂
1450:Springer
1429:, 1988.
1396:See also
1259:Max Dehn
1199:, where
298:we have
277:∈
190:) where
101:. Then
85:surface
51:manifold
1486:0171269
1473:0151948
1457:0547453
1267:isotopy
596:, say
594:annulus
535:called
418:Example
404:2-sided
113:to the
107:annulus
89:. Let
47:surface
1433:
1359:, for
105:is an
1271:genus
1245:The 3
1229:then
448:torus
387:is a
383:Then
228:with
194:is a
178:Give
93:be a
77:is a
69:-gon.
45:of a
1431:ISBN
1370:>
1253:= 3.
1130:and
670:<
654:<
458:and
395:.
282:Let
279:.
272:and
37:, a
1392:."
1330:to
1265:of
410:on
124::
97:of
53:).
29:In
1497::
1483:MR
1470:MR
1454:MR
1425:,
1233:.
900:.
825:.
562:.
414:.
186:,
109:,
1444:(
1437:.
1373:1
1367:g
1347:1
1344:+
1341:g
1338:2
1314:1
1308:g
1305:3
1281:g
1273:-
1251:g
1247:g
1231:a
1227:b
1213:a
1207:b
1187:]
1184:a
1178:b
1175:[
1172:=
1169:)
1166:]
1163:b
1160:[
1157:(
1145:a
1141:T
1118:]
1115:a
1112:[
1109:=
1106:)
1103:]
1100:a
1097:[
1094:(
1082:a
1078:T
1065:x
1043:]
1039:)
1036:x
1033:(
1028:a
1024:T
1019:[
1012:]
1009:x
1006:[
1003::
999:)
994:2
989:T
984:(
978:1
966:)
961:2
956:T
951:(
945:1
937::
925:a
921:T
898:a
894:b
890:b
871:2
866:T
856:2
851:T
846::
841:a
837:T
823:a
807:a
781:)
777:t
774:,
768:)
764:t
758:2
755:+
748:(
744:i
740:e
735:(
727:)
723:t
720:,
712:i
708:e
703:(
676:}
673:1
666:|
662:z
658:|
651:0
648::
644:C
637:z
634:{
631:=
628:)
625:1
622:,
619:0
616:;
613:0
610:(
607:a
578:a
548:a
533:a
516:.
511:2
506:Z
500:/
494:2
489:R
479:2
474:T
460:b
456:a
432:a
428:a
412:S
408:c
400:S
393:c
385:f
368:.
364:)
360:t
357:,
352:t
346:2
343:i
339:e
335:s
331:(
327:=
324:)
321:t
318:,
315:s
312:(
309:f
296:A
292:A
288:S
284:f
275:t
260:,
257:]
251:2
248:,
245:0
242:[
211:i
207:e
192:s
188:t
184:s
180:A
163:.
160:I
152:1
148:S
141:A
135:c
122:I
103:A
99:c
91:A
87:S
75:c
67:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
