442:
1302:
1397:
299:
287:
1162:
1170:
803:
225:
588:
738:
1067:
955:
902:
869:
836:
131:
495:
1094:
615:
522:
165:
92:
644:
978:
1018:
998:
922:
688:
664:
562:
542:
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690:
to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at
1366:
1099:
However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true
437:{\displaystyle J(x,t)={\begin{cases}tf(x/t),&{\text{if }}0\leq \|x\|<t,\\x,&{\text{if }}t\leq \|x\|\leq 1.\end{cases}}}
1100:
230:
471:
calls this "combing all the tangles to one point". In the original 2-page paper, J. W. Alexander explains that for each
1455:
1384:
1350:
42:
1297:{\displaystyle F\colon D^{n}\to D^{n}{\text{ with }}F(rx)=rf(x){\text{ for all }}r\in {\text{ and }}x\in S^{n-1}}
741:
1109:
756:
1460:
184:
1332:
1406:
178:: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.
567:
329:
95:
693:
667:
65:
38:
1434:
1362:
1039:
927:
874:
841:
808:
103:
1424:
1414:
1354:
474:
468:
1376:
1072:
593:
500:
143:
70:
1372:
623:
1410:
960:
1429:
1388:
1003:
983:
907:
673:
649:
547:
527:
450:
17:
1449:
1320:
1316:
1309:
1033:
54:
1315:
The failure of smooth radial extension and the success of PL radial extension yield
447:
Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing'
27:
Two homeomorphisms of the n-ball which agree on the boundary sphere are isotopic
1398:
Proceedings of the
National Academy of Sciences of the United States of America
1358:
62:
1438:
1419:
134:
1349:. London Mathematical Society Student Texts. Vol. 18. Cambridge:
98:
670:
at the origin "jumps" from an infinitely stretched version of
430:
1069:
can be extended to a homeomorphism of the entire ball
1173:
1112:
1075:
1042:
1006:
986:
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844:
811:
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477:
453:
302:
233:
187:
146:
106:
73:
282:{\displaystyle f(x)=x{\text{ for all }}x\in S^{n-1}}
1296:
1156:
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992:
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609:
582:
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536:
516:
489:
459:
436:
281:
219:
159:
125:
86:
740:. This underlines that the Alexander trick is a
167:that are isotopic on the boundary are isotopic.
8:
544:at a different scale, on the disk of radius
418:
412:
377:
371:
1157:{\displaystyle f\colon S^{n-1}\to S^{n-1}}
1428:
1418:
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1267:
1241:
1203:
1197:
1184:
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1142:
1123:
1111:
1080:
1074:
1047:
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985:
962:
935:
929:
909:
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849:
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111:
105:
78:
72:
798:{\displaystyle f,g\colon D^{n}\to D^{n}}
750:: isotopic on boundary implies isotopic
220:{\displaystyle f\colon D^{n}\to D^{n}}
140:More generally, two homeomorphisms of
1347:Braids and coverings: selected topics
805:are two homeomorphisms that agree on
7:
1304:defines a homeomorphism of the ball.
25:
1345:Hansen, Vagn Lundsgaard (1989).
590:it is reasonable to expect that
1264:
1252:
1238:
1232:
1220:
1211:
1190:
1135:
782:
744:construction, but not smooth.
727:
715:
709:
697:
583:{\displaystyle t\rightarrow 0}
574:
352:
338:
318:
306:
243:
237:
204:
1:
1032:for the statement that every
289:, then an isotopy connecting
293:to the identity is given by
733:{\displaystyle (x,t)=(0,0)}
1477:
1389:"On the deformation of an
1351:Cambridge University Press
1028:Some authors use the term
1164:be a homeomorphism, then
1359:10.1017/CBO9780511613098
980:is then an isotopy from
904:, so we have an isotopy
620:The subtlety is that at
617:merges to the identity.
37:, is a basic result in
1062:{\displaystyle S^{n-1}}
950:{\displaystyle g^{-1}f}
897:{\displaystyle S^{n-1}}
864:{\displaystyle g^{-1}f}
831:{\displaystyle S^{n-1}}
126:{\displaystyle S^{n-1}}
1333:Clutching construction
1298:
1158:
1090:
1063:
1014:
994:
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518:
491:
490:{\displaystyle t>0}
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221:
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127:
88:
18:Alexander's Trick
1420:10.1073/pnas.9.12.406
1299:
1159:
1091:
1089:{\displaystyle D^{n}}
1064:
1015:
995:
975:
952:
924:from the identity to
919:
899:
866:
833:
800:
735:
685:
661:
641:
612:
610:{\displaystyle J_{t}}
585:
559:
539:
519:
517:{\displaystyle J_{t}}
492:
462:
439:
284:
222:
162:
160:{\displaystyle D^{n}}
128:
89:
87:{\displaystyle D^{n}}
1171:
1110:
1103:, but not smoothly.
1073:
1040:
1004:
984:
961:
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875:
842:
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694:
674:
650:
624:
594:
568:
548:
528:
501:
475:
467:down to the origin.
451:
300:
231:
185:
144:
104:
71:
33:, also known as the
1411:1923PNAS....9..406A
1243: for all
871:is the identity on
639:{\displaystyle t=0}
497:the transformation
254: for all
94:which agree on the
1456:Geometric topology
1294:
1154:
1101:piecewise-linearly
1086:
1059:
1010:
990:
973:{\displaystyle gJ}
970:
947:
914:
894:
861:
828:
795:
730:
680:
666:"disappears": the
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279:
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123:
84:
39:geometric topology
1270:
1244:
1206:
1013:{\displaystyle f}
993:{\displaystyle g}
917:{\displaystyle J}
683:{\displaystyle f}
659:{\displaystyle f}
557:{\displaystyle t}
537:{\displaystyle f}
460:{\displaystyle f}
404:
363:
255:
31:Alexander's trick
16:(Redirected from
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1442:
1432:
1422:
1385:Alexander, J. W.
1380:
1303:
1301:
1300:
1295:
1293:
1292:
1271:
1268:
1245:
1242:
1207:
1205: with
1204:
1202:
1201:
1189:
1188:
1163:
1161:
1160:
1155:
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1134:
1133:
1106:Concretely, let
1095:
1093:
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1084:
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1065:
1060:
1058:
1057:
1024:Radial extension
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736:
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689:
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665:
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541:
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494:
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469:William Thurston
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21:
1476:
1475:
1471:
1470:
1469:
1467:
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1465:
1446:
1445:
1405:(12): 406–407.
1383:
1369:
1344:
1341:
1329:
1321:twisted spheres
1313:
1278:
1269: and
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1180:
1169:
1168:
1138:
1119:
1108:
1107:
1076:
1071:
1070:
1043:
1038:
1037:
1030:Alexander trick
1026:
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1001:
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43:J. W. Alexander
35:Alexander trick
28:
23:
22:
15:
12:
11:
5:
1474:
1472:
1464:
1463:
1461:Homeomorphisms
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1447:
1444:
1443:
1381:
1367:
1340:
1337:
1336:
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1325:
1317:exotic spheres
1312:
1310:Exotic spheres
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55:homeomorphisms
50:
47:
41:, named after
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6:
4:
3:
2:
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1368:0-521-38757-4
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1120:
1116:
1113:
1104:
1102:
1097:
1081:
1077:
1054:
1051:
1048:
1044:
1035:
1034:homeomorphism
1031:
1023:
1021:
1007:
987:
967:
964:
944:
939:
936:
932:
911:
889:
886:
883:
879:
858:
853:
850:
846:
823:
820:
817:
813:
790:
786:
777:
773:
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766:
763:
760:
751:
749:
745:
743:
724:
721:
718:
712:
706:
703:
700:
677:
669:
653:
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618:
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598:
577:
571:
551:
531:
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484:
481:
478:
470:
454:
424:
421:
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406:
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368:
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234:
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97:
79:
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67:
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56:
48:
46:
44:
40:
36:
32:
19:
1402:
1396:
1390:
1346:
1314:
1105:
1098:
1029:
1027:
752:
748:General case
747:
746:
619:
446:
290:
180:
175:
174:
139:
58:
52:
34:
30:
29:
524:replicates
63:dimensional
1450:Categories
1339:References
957:. The map
564:, thus as
227:satisfies
1287:−
1276:∈
1250:∈
1191:→
1178::
1147:−
1136:→
1128:−
1117::
1052:−
937:−
887:−
851:−
821:−
783:→
770::
575:→
422:≤
419:‖
413:‖
410:≤
378:‖
372:‖
369:≤
272:−
261:∈
205:→
192::
176:Base case
116:−
49:Statement
1439:16586918
1387:(1923).
1327:See also
403:if
362:if
135:isotopic
96:boundary
1430:1085470
1407:Bibcode
1377:1247697
838:, then
57:of the
1437:
1427:
1393:-cell"
1375:
1365:
99:sphere
171:Proof
1435:PMID
1363:ISBN
1319:via
668:germ
482:>
381:<
133:are
66:ball
53:Two
1425:PMC
1415:doi
1355:doi
1036:of
1000:to
753:If
181:If
1452::
1433:.
1423:.
1413:.
1401:.
1395:.
1373:MR
1371:.
1361:.
1353:.
1323:.
1096:.
1020:.
742:PL
646:,
425:1.
137:.
45:.
1441:.
1417::
1409::
1403:9
1391:n
1379:.
1357::
1290:1
1284:n
1280:S
1273:x
1265:]
1262:1
1259:,
1256:0
1253:[
1247:r
1239:)
1236:x
1233:(
1230:f
1227:r
1224:=
1221:)
1218:x
1215:r
1212:(
1209:F
1199:n
1195:D
1186:n
1182:D
1175:F
1150:1
1144:n
1140:S
1131:1
1125:n
1121:S
1114:f
1082:n
1078:D
1055:1
1049:n
1045:S
1008:f
988:g
968:J
965:g
945:f
940:1
933:g
912:J
890:1
884:n
880:S
859:f
854:1
847:g
824:1
818:n
814:S
791:n
787:D
778:n
774:D
767:g
764:,
761:f
728:)
725:0
722:,
719:0
716:(
713:=
710:)
707:t
704:,
701:x
698:(
678:f
654:f
634:0
631:=
628:t
603:t
599:J
578:0
572:t
552:t
532:f
510:t
506:J
485:0
479:t
455:f
416:x
407:t
397:,
394:x
387:,
384:t
375:x
366:0
356:,
353:)
350:t
346:/
342:x
339:(
336:f
333:t
327:{
322:=
319:)
316:t
313:,
310:x
307:(
304:J
291:f
275:1
269:n
265:S
258:x
250:x
247:=
244:)
241:x
238:(
235:f
213:n
209:D
200:n
196:D
189:f
153:n
149:D
119:1
113:n
109:S
80:n
76:D
61:-
59:n
20:)
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