857:
proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.
696:
381:
825:
234:
39:
have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by
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768:
443:
599:
725:
469:
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143:
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1103:
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610:
1481:
44:
1471:
1320:
773:
182:
1476:
1445:
1380:
1131:
1425:
305:
1365:
32:
1486:
1440:
1400:
1395:
1355:
893:
309:
1450:
1059:
981:
606:
1435:
271:
1420:
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1410:
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1234:
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847:
146:
36:
28:
575:
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1287:
1252:
1166:(2016) . "Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture".
1163:
1135:
1099:
854:
173:
1057:(1952). "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung".
704:
448:
242:
76:
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874:
846:
which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a
835:
165:
102:
827:. In particular there are only a finite number of essentially distinct PL structures on
1274:
1124:
1119:
1054:
474:
408:
386:
154:
128:
40:
1465:
1024:
962:
614:
266:
161:
150:
1266:
1199:
732:
602:
1302:
1278:
976:
843:
60:
1344:
880:
979:(1961). "Two complexes which are homeomorphic but combinatorially distinct".
492:
71:
838:
found examples with an infinite number of inequivalent PL structures, and
939:"Ăśber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten"
404:
56:
1312:
1218:
1080:
1002:
954:
770:, and if this obstruction is 0, the PL structures are parametrized by
601:
is now seen as a relative version of the triangulation obstruction of
1239:
1191:
1072:
994:
938:
1182:
1248:
691:{\displaystyle \kappa (M)\in H^{4}(M;\mathbb {Z} /2\mathbb {Z} )}
376:{\displaystyle \kappa (f)\in H^{3}(M;\mathbb {Z} /2\mathbb {Z} )}
1316:
735:(i.e., it can be triangulated by a PL manifold) if and only if
1029:
Acta
Scientarum Mathematicarum Universitatis Szegediensis
160:
An obstruction to the manifold version was formulated by
834:
For compact simply-connected manifolds of dimension 4,
776:
741:
707:
632:
578:
501:
477:
451:
416:
389:
317:
274:
245:
185:
131:
105:
79:
1230:
Piecewise Linear
Structures on Topological Manifolds
1027:(1925). "Über den Begriff der Riemannschen Fläche".
918:
Steinitz, E. (1908). "Beiträge zur
Analysis situs".
820:{\displaystyle H^{3}(M;\mathbb {Z} /2\mathbb {Z} )}
229:{\displaystyle H^{3}(M;\mathbb {Z} /2\mathbb {Z} )}
31:is a now refuted conjecture asking whether any two
1126:Casson's invariant for oriented homology 3-spheres
1123:
819:
762:
719:
690:
593:
561:
483:
463:
437:
395:
375:
292:
257:
228:
137:
117:
91:
887:die Hauptvermutung der kombinatorischen Topologie
1223:Additional material, including original sources
1328:
8:
1169:Journal of the American Mathematical Society
1335:
1321:
1313:
1303:"High-dimensional manifolds then and now"
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1181:
810:
809:
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797:
796:
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668:
667:
652:
631:
577:
547:
546:
541:
500:
476:
450:
415:
407:to a piecewise linear (PL) homeomorphism
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219:
218:
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206:
205:
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130:
104:
78:
1096:Geometric Topology in Dimensions 2 and 3
445:. In the simply-connected case and with
18:Refuted conjecture of geometric topology
910:
867:
701:again using the Rochlin invariant. For
1219:"Triangulation and the Hauptvermutung"
157:in the 1920s and 1950s, respectively.
495:to a PL homeomorphism if and only if
7:
47:, but it is now known to be false.
620:-dimensional topological manifold
551:
548:
14:
70:The manifold version is true in
45:Heinrich Franz Friedrich Tietze
814:
787:
751:
745:
685:
658:
642:
636:
588:
582:
556:
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517:
514:
508:
502:
426:
420:
370:
343:
327:
321:
293:{\displaystyle f\colon N\to M}
284:
223:
196:
168:in 1967–69 (originally in the
1:
1122:; McCarthy, John D. (1990).
884:. It is an abbreviation for
763:{\displaystyle \kappa (M)=0}
611:Kirby–Siebenmann obstruction
438:{\displaystyle \kappa (f)=0}
1277:, ed. (30 September 1996).
943:Monatsh. FĂĽr Math. Und Phys
1503:
1221:. University of Edinburgh.
1132:Princeton University Press
920:Sitz-Ber. Berlin Math. Ges
594:{\displaystyle \kappa (f)}
306:piecewise linear manifolds
1351:
59:version was disproved by
1094:Moise, Edwin E. (1977).
609:, obtained in 1970. The
1482:Structures on manifolds
1280:The Hauptvermutung Book
892:the main conjecture of
720:{\displaystyle m\geq 5}
464:{\displaystyle m\geq 5}
258:{\displaystyle m\geq 5}
92:{\displaystyle m\leq 3}
894:combinatorial topology
890:, which translates as
821:
764:
721:
692:
595:
563:
562:{\displaystyle =0\in }
485:
465:
439:
397:
377:
294:
259:
230:
139:
119:
93:
1472:Disproved conjectures
1371:Euler's sum of powers
1227:Rudyak, Yuli (2016).
1060:Annals of Mathematics
982:Annals of Mathematics
822:
765:
722:
693:
607:Laurent C. Siebenmann
596:
564:
486:
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295:
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120:
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739:
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315:
272:
243:
183:
129:
103:
77:
65:Reidemeister torsion
937:Tietze, H. (1908).
613:is defined for any
118:{\displaystyle m=2}
1477:Geometric topology
1361:Chinese hypothesis
1164:Manolescu, Ciprian
955:10.1007/BF01736688
848:simplicial complex
817:
760:
717:
688:
591:
559:
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115:
89:
37:triangulable space
29:geometric topology
1459:
1458:
1301:Ranicki, Andrew.
1258:978-981-4733-78-6
1217:Ranicki, Andrew.
1105:978-0-387-90220-3
855:Ciprian Manolescu
484:{\displaystyle f}
396:{\displaystyle f}
174:Rochlin invariant
172:case), using the
138:{\displaystyle 3}
1494:
1411:Ono's inequality
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840:Michael Freedman
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178:cohomology group
170:simply-connected
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98:
96:
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90:
1502:
1501:
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1460:
1455:
1347:
1341:
1305:
1300:
1294:
1283:
1275:Ranicki, Andrew
1273:
1259:
1226:
1216:
1213:
1208:
1207:
1192:10.1090/jams829
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1161:
1157:
1142:
1120:Akbulut, Selman
1118:
1117:
1113:
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1093:
1092:
1088:
1073:10.2307/1969769
1055:Moise, Edwin E.
1053:
1052:
1048:
1023:
1022:
1018:
995:10.2307/1970299
977:Milnor, John W.
975:
974:
970:
936:
935:
931:
917:
916:
912:
907:
902:
901:
873:
869:
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836:Simon Donaldson
777:
772:
771:
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736:
727:, the manifold
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702:
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411:
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166:Dennis Sullivan
127:
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101:
100:
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53:
19:
12:
11:
5:
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1489:
1487:Surgery theory
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1386:Hauptvermutung
1383:
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1211:External links
1209:
1206:
1205:
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1140:
1111:
1104:
1086:
1067:(2): 101–121.
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1016:
989:(2): 575–590.
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572:This quantity
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409:if and only if
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155:Edwin E. Moise
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63:in 1961 using
52:
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41:Ernst Steinitz
33:triangulations
24:Hauptvermutung
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1293:0-7923-4174-0
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1035:(1): 96–114.
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845:
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837:
832:
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802:
793:
790:
782:
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757:
754:
748:
742:
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673:
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304:-dimensional
303:
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267:homeomorphism
252:
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239:In dimension
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171:
167:
163:
162:Andrew Casson
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30:
26:
25:
16:
1385:
1381:Hedetniemi's
1286:. Springer.
1279:
1249:10.1142/9887
1240:math/0105047
1229:
1173:
1167:
1158:
1125:
1114:
1098:. Springer.
1095:
1089:
1064:
1058:
1049:
1032:
1028:
1019:
986:
980:
971:
946:
942:
932:
923:
919:
913:
891:
886:
885:
878:
870:
852:
833:
828:
733:PL structure
728:
700:
621:
617:
603:Robion Kirby
571:
301:
238:
159:
99:. The cases
69:
54:
23:
22:
20:
15:
1441:Von Neumann
1345:conjectures
1176:: 147–176.
1025:RadĂł, Tibor
844:E8 manifold
61:John Milnor
1466:Categories
1451:Williamson
1446:Weyl–Berry
1426:Schoen–Yau
1343:Disproved
1041:51.0273.01
905:References
881:conjecture
842:found the
383:such that
151:Tibor RadĂł
72:dimensions
1183:1303.2354
963:120998023
949:: 1–118.
853:In 2013,
743:κ
712:≥
646:∈
634:κ
580:κ
527:∈
506:κ
493:homotopic
456:≥
418:κ
331:∈
319:κ
310:invariant
285:→
279::
250:≥
84:≤
1421:Ragsdale
1401:Keller's
1396:Kalman's
1356:Borsuk's
1267:16750789
1200:16403004
926:: 29–49.
405:isotopic
176:and the
57:manifold
55:The non-
1431:Seifert
1406:Mertens
1150:1030042
1081:1969769
1011:0133127
1003:1970299
615:compact
308:has an
51:History
1436:Tait's
1391:Hirsch
1366:Connes
1290:
1265:
1255:
1198:
1148:
1138:
1102:
1079:
1039:
1009:
1001:
961:
875:German
731:has a
147:proved
1416:PĂłlya
1376:Ganea
1306:(PDF)
1284:(PDF)
1263:S2CID
1235:arXiv
1196:S2CID
1178:arXiv
1077:JSTOR
999:JSTOR
959:S2CID
879:main
862:Notes
145:were
35:of a
1288:ISBN
1253:ISBN
1136:ISBN
1100:ISBN
877:for
605:and
265:, a
164:and
153:and
125:and
43:and
21:The
1245:doi
1188:doi
1069:doi
1037:JFM
991:doi
951:doi
491:is
403:is
300:of
149:by
27:of
1468::
1261:.
1251:.
1243:.
1233:.
1194:.
1186:.
1174:29
1172:.
1146:MR
1144:.
1134:.
1130:.
1075:.
1065:56
1063:.
1031:.
1007:MR
1005:.
997:.
987:74
985:.
957:.
947:19
945:.
941:.
922:.
850:.
831:.
569:.
471:,
236:.
67:.
1336:e
1329:t
1322:v
1308:.
1296:.
1269:.
1247::
1237::
1202:.
1190::
1180::
1152:.
1108:.
1083:.
1071::
1043:.
1033:2
1013:.
993::
965:.
953::
924:7
897:.
829:M
815:)
811:Z
807:2
803:/
798:Z
794:;
791:M
788:(
783:3
779:H
758:0
755:=
752:)
749:M
746:(
729:M
715:5
709:m
686:)
682:Z
678:2
674:/
669:Z
665:;
662:M
659:(
654:4
650:H
643:)
640:M
637:(
622:M
618:m
589:)
586:f
583:(
557:]
552:L
549:P
543:/
539:G
536:,
533:M
530:[
524:0
521:=
518:]
515:)
512:f
509:(
503:[
479:f
459:5
453:m
433:0
430:=
427:)
424:f
421:(
391:f
371:)
367:Z
363:2
359:/
354:Z
350:;
347:M
344:(
339:3
335:H
328:)
325:f
322:(
302:m
288:M
282:N
276:f
253:5
247:m
224:)
220:Z
216:2
212:/
207:Z
203:;
200:M
197:(
192:3
188:H
133:3
113:2
110:=
107:m
87:3
81:m
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