237:
991:
1053:
532:
306:
136:
369:
1150:
881:
724:
1101:
394:
465:
808:
764:
690:
653:
590:
563:
148:
424:
due to
Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact
891:. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a
1292:
1250:
1211:
1111:(1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its
1165:
931:
996:
1206:
1181:
47:
1340:
888:
614:
has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in
472:
1112:
1345:
482:
397:
264:
94:
25:
330:
1278:
1121:
596:
is not connected, its signature is defined to be the sum of the signatures of its connected components.
1216:
1189:
1153:
892:
43:
827:
699:
1283:
1177:
900:
321:
1180:
is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the
1071:
377:
908:
819:
693:
1056:
441:
1288:
1246:
777:
767:
733:
36:
662:
625:
1115:
317:
86:
568:
541:
1309:
896:
59:
1193:
421:
79:
1334:
904:
232:{\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}
1108:
1237:
884:
75:
1169:
425:
1192:
says that the signature of a 4-dimensional simply connected manifold with a
255:
615:
605:
396:. Therefore the cup product, under these hypotheses, does give rise to a
63:
29:
17:
42:
This invariant of a manifold has been studied in detail, starting with
1157:
1156:(1954) found an explicit expression for this linear combination as the
692:
and these invariants do not always vanish for other dimensions. The
1245:(Repr. ed.). Cambridge: Cambridge Univ. Pr. p. 250.
1310:"Quelques proprietes globales des varietes differentiables"
986:{\displaystyle \sigma (M\sqcup N)=\sigma (M)+\sigma (N)}
1048:{\displaystyle \sigma (M\times N)=\sigma (M)\sigma (N)}
1315:(in French). Comm. Math. Helvetici 28 (1954), S. 17–86
907:. The resulting invariant of a manifold is called the
1124:
1074:
999:
934:
830:
780:
736:
702:
665:
628:
571:
544:
485:
444:
380:
333:
267:
151:
97:
903:need not be equivalent, being distinguished by the
1277:. Annals of Mathematics Studies 246. p. 224.
1144:
1118:. For example, in four dimensions, it is given by
1095:
1047:
985:
875:
802:
758:
718:
684:
647:
622:-dimensional (simply connected) symmetric L-group
584:
557:
526:
459:
388:
363:
300:
231:
130:
770:is a mod 2 invariant of manifolds of dimension 4
1168:(1962) proved that a simply connected compact
8:
618:: the signature can be interpreted as the 4
887:), the same construction gives rise to an
810:); the other dimensional L-groups vanish.
1282:
1131:
1125:
1123:
1073:
998:
933:
829:
785:
779:
741:
735:
708:
703:
701:
670:
664:
633:
627:
576:
570:
549:
543:
518:
505:
484:
443:
381:
379:
353:
338:
332:
290:
272:
266:
220:
207:
191:
169:
156:
150:
120:
102:
96:
1228:
895:of the form, which occurs if one has a
142:The basic identity for the cup product
1273:Milnor, John; Stasheff, James (1962).
527:{\displaystyle \sigma (M)=n_{+}-n_{-}}
301:{\displaystyle H^{4k}(M,\mathbf {R} )}
131:{\displaystyle H^{2k}(M,\mathbf {R} )}
726:) for framed manifolds of dimension 4
412:); and therefore to a quadratic form
364:{\displaystyle H^{0}(M,\mathbf {R} )}
7:
534:where any diagonal matrix defining
1212:Genus of a multiplicative sequence
1145:{\displaystyle {\frac {p_{1}}{3}}}
14:
28:which is defined for an oriented
704:
696:is a mod 2 (i.e., an element of
382:
354:
291:
121:
659:-dimensional quadratic L-group
432:-dimensional Poincaré duality.
1084:
1078:
1068:is an oriented boundary, then
1042:
1036:
1030:
1024:
1015:
1003:
980:
974:
965:
959:
950:
938:
876:{\displaystyle d=4k+2=2(2k+1)}
870:
855:
719:{\displaystyle \mathbf {Z} /2}
495:
489:
454:
448:
358:
344:
295:
281:
226:
200:
188:
178:
125:
111:
1:
374:which can be identified with
1207:Hirzebruch signature theorem
1182:Hirzebruch signature theorem
1096:{\displaystyle \sigma (M)=0}
389:{\displaystyle \mathbf {R} }
48:Hirzebruch signature theorem
993:by definition, and satisfy
920:Compact oriented manifolds
889:antisymmetric bilinear form
1362:
817:
774:+1 (the symmetric L-group
730:+2 (the quadratic L-group
603:
460:{\displaystyle \sigma (M)}
883:is twice an odd integer (
803:{\displaystyle L^{4k+1}}
759:{\displaystyle L_{4k+2}}
1236:Hatcher, Allen (2003).
685:{\displaystyle L_{4k},}
648:{\displaystyle L^{4k},}
398:symmetric bilinear form
312:If we assume also that
1275:Characteristic classes
1146:
1097:
1049:
987:
877:
804:
760:
720:
686:
649:
586:
559:
528:
461:
390:
365:
302:
233:
132:
1147:
1098:
1050:
988:
899:, then the resulting
878:
805:
761:
721:
687:
650:
604:Further information:
592:negative entries. If
587:
585:{\displaystyle n_{-}}
565:positive entries and
560:
558:{\displaystyle n_{+}}
529:
471:is by definition the
462:
391:
366:
324:identifies this with
303:
258:. It takes values in
234:
133:
85:on the 'middle' real
46:for 4-manifolds, and
1154:Friedrich Hirzebruch
1122:
1072:
997:
932:
893:quadratic refinement
828:
778:
734:
700:
663:
626:
569:
542:
483:
442:
378:
331:
265:
149:
95:
1196:is divisible by 16.
1341:Geometric topology
1239:Algebraic topology
1142:
1093:
1045:
983:
909:Kervaire invariant
873:
820:Kervaire invariant
814:Kervaire invariant
800:
756:
716:
694:Kervaire invariant
682:
645:
582:
555:
524:
457:
386:
361:
298:
229:
128:
1217:Rokhlin's theorem
1190:Rokhlin's theorem
1140:
901:ε-quadratic forms
768:de Rham invariant
44:Rokhlin's theorem
37:divisible by four
1353:
1325:
1324:
1322:
1320:
1314:
1305:
1299:
1298:
1286:
1270:
1264:
1263:
1261:
1259:
1244:
1233:
1178:Poincaré duality
1160:of the manifold.
1151:
1149:
1148:
1143:
1141:
1136:
1135:
1126:
1102:
1100:
1099:
1094:
1054:
1052:
1051:
1046:
992:
990:
989:
984:
882:
880:
879:
874:
809:
807:
806:
801:
799:
798:
765:
763:
762:
757:
755:
754:
725:
723:
722:
717:
712:
707:
691:
689:
688:
683:
678:
677:
654:
652:
651:
646:
641:
640:
600:Other dimensions
591:
589:
588:
583:
581:
580:
564:
562:
561:
556:
554:
553:
533:
531:
530:
525:
523:
522:
510:
509:
466:
464:
463:
458:
395:
393:
392:
387:
385:
370:
368:
367:
362:
357:
343:
342:
322:Poincaré duality
307:
305:
304:
299:
294:
280:
279:
242:shows that with
238:
236:
235:
230:
225:
224:
212:
211:
199:
198:
174:
173:
161:
160:
137:
135:
134:
129:
124:
110:
109:
87:cohomology group
78:gives rise to a
16:In the field of
1361:
1360:
1356:
1355:
1354:
1352:
1351:
1350:
1346:Quadratic forms
1331:
1330:
1329:
1328:
1318:
1316:
1312:
1307:
1306:
1302:
1295:
1272:
1271:
1267:
1257:
1255:
1253:
1242:
1235:
1234:
1230:
1225:
1203:
1166:William Browder
1127:
1120:
1119:
1070:
1069:
1057:KĂĽnneth formula
995:
994:
930:
929:
917:
897:framed manifold
826:
825:
822:
816:
781:
776:
775:
737:
732:
731:
698:
697:
666:
661:
660:
629:
624:
623:
608:
602:
572:
567:
566:
545:
540:
539:
514:
501:
481:
480:
440:
439:
376:
375:
334:
329:
328:
268:
263:
262:
254:the product is
216:
203:
187:
165:
152:
147:
146:
98:
93:
92:
56:
12:
11:
5:
1359:
1357:
1349:
1348:
1343:
1333:
1332:
1327:
1326:
1300:
1294:978-0691081229
1293:
1284:10.1.1.448.869
1265:
1252:978-0521795401
1251:
1227:
1226:
1224:
1221:
1220:
1219:
1214:
1209:
1202:
1199:
1198:
1197:
1194:spin structure
1186:
1185:
1162:
1161:
1139:
1134:
1130:
1105:
1104:
1092:
1089:
1086:
1083:
1080:
1077:
1061:
1060:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
916:
913:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
818:Main article:
815:
812:
797:
794:
791:
788:
784:
753:
750:
747:
744:
740:
715:
711:
706:
681:
676:
673:
669:
644:
639:
636:
632:
601:
598:
579:
575:
552:
548:
521:
517:
513:
508:
504:
500:
497:
494:
491:
488:
456:
453:
450:
447:
422:non-degenerate
384:
372:
371:
360:
356:
352:
349:
346:
341:
337:
310:
309:
297:
293:
289:
286:
283:
278:
275:
271:
240:
239:
228:
223:
219:
215:
210:
206:
202:
197:
194:
190:
186:
183:
180:
177:
172:
168:
164:
159:
155:
140:
139:
127:
123:
119:
116:
113:
108:
105:
101:
80:quadratic form
70:of dimension 4
55:
52:
24:is an integer
13:
10:
9:
6:
4:
3:
2:
1358:
1347:
1344:
1342:
1339:
1338:
1336:
1311:
1304:
1301:
1296:
1290:
1285:
1280:
1276:
1269:
1266:
1254:
1248:
1241:
1240:
1232:
1229:
1222:
1218:
1215:
1213:
1210:
1208:
1205:
1204:
1200:
1195:
1191:
1188:
1187:
1183:
1179:
1176:-dimensional
1175:
1171:
1167:
1164:
1163:
1159:
1155:
1137:
1132:
1128:
1117:
1114:
1110:
1107:
1106:
1090:
1087:
1081:
1075:
1067:
1063:
1062:
1058:
1039:
1033:
1027:
1021:
1018:
1012:
1009:
1006:
1000:
977:
971:
968:
962:
956:
953:
947:
944:
941:
935:
927:
923:
919:
918:
914:
912:
910:
906:
905:Arf invariant
902:
898:
894:
890:
886:
867:
864:
861:
858:
852:
849:
846:
843:
840:
837:
834:
831:
821:
813:
811:
795:
792:
789:
786:
782:
773:
769:
766:), while the
751:
748:
745:
742:
738:
729:
713:
709:
695:
679:
674:
671:
667:
658:
642:
637:
634:
630:
621:
617:
613:
607:
599:
597:
595:
577:
573:
550:
546:
537:
519:
515:
511:
506:
502:
498:
492:
486:
478:
474:
470:
451:
445:
438:
433:
431:
427:
423:
419:
415:
411:
407:
403:
399:
350:
347:
339:
335:
327:
326:
325:
323:
319:
315:
287:
284:
276:
273:
269:
261:
260:
259:
257:
253:
249:
245:
221:
217:
213:
208:
204:
195:
192:
184:
181:
175:
170:
166:
162:
157:
153:
145:
144:
143:
117:
114:
106:
103:
99:
91:
90:
89:
88:
84:
81:
77:
73:
69:
65:
61:
53:
51:
49:
45:
40:
38:
35:of dimension
34:
31:
27:
23:
19:
1317:. Retrieved
1308:Thom, René.
1303:
1274:
1268:
1256:. Retrieved
1238:
1231:
1173:
1065:
925:
921:
823:
771:
727:
656:
619:
611:
609:
593:
535:
476:
468:
436:
434:
429:
417:
413:
409:
405:
401:
373:
313:
311:
251:
247:
243:
241:
141:
82:
71:
67:
57:
41:
32:
21:
15:
885:singly even
655:or as the 4
479:, that is,
416:. The form
76:cup product
1335:Categories
1319:26 October
1223:References
1170:polyhedron
1113:Pontryagin
915:Properties
426:polyhedron
54:Definition
1279:CiteSeerX
1258:8 January
1109:René Thom
1076:σ
1034:σ
1022:σ
1010:×
1001:σ
972:σ
957:σ
945:⊔
936:σ
578:−
520:−
512:−
487:σ
473:signature
446:σ
437:signature
256:symmetric
218:α
214:⌣
205:β
182:−
167:β
163:⌣
154:α
66:manifold
60:connected
26:invariant
22:signature
1201:See also
928:satisfy
616:L-theory
606:L-theory
64:oriented
58:Given a
30:manifold
18:topology
1158:L genus
1116:numbers
318:compact
1291:
1281:
1249:
1172:with 4
74:, the
20:, the
1313:(PDF)
1243:(PDF)
1055:by a
824:When
428:with
1321:2019
1289:ISBN
1260:2017
1247:ISBN
924:and
538:has
435:The
62:and
1064:If
610:If
475:of
467:of
420:is
400:on
316:is
250:= 2
1337::
1287:.
1152:.
911:.
430:4n
320:,
246:=
50:.
39:.
1323:.
1297:.
1262:.
1184:.
1174:n
1138:3
1133:1
1129:p
1103:.
1091:0
1088:=
1085:)
1082:M
1079:(
1066:M
1059:.
1043:)
1040:N
1037:(
1031:)
1028:M
1025:(
1019:=
1016:)
1013:N
1007:M
1004:(
981:)
978:N
975:(
969:+
966:)
963:M
960:(
954:=
951:)
948:N
942:M
939:(
926:N
922:M
871:)
868:1
865:+
862:k
859:2
856:(
853:2
850:=
847:2
844:+
841:k
838:4
835:=
832:d
796:1
793:+
790:k
787:4
783:L
772:k
752:2
749:+
746:k
743:4
739:L
728:k
714:2
710:/
705:Z
680:,
675:k
672:4
668:L
657:k
643:,
638:k
635:4
631:L
620:k
612:M
594:M
574:n
551:+
547:n
536:Q
516:n
507:+
503:n
499:=
496:)
493:M
490:(
477:Q
469:M
455:)
452:M
449:(
418:Q
414:Q
410:R
408:,
406:M
404:(
402:H
383:R
359:)
355:R
351:,
348:M
345:(
340:0
336:H
314:M
308:.
296:)
292:R
288:,
285:M
282:(
277:k
274:4
270:H
252:k
248:q
244:p
227:)
222:p
209:q
201:(
196:q
193:p
189:)
185:1
179:(
176:=
171:q
158:p
138:.
126:)
122:R
118:,
115:M
112:(
107:k
104:2
100:H
83:Q
72:k
68:M
33:M
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.