1103:
The definition of a cup product is dual (and so is analogous) to the above definition of the intersection form on homology of a manifold, but is more abstract. However, the definition of a cup product generalizes to complexes and topological manifolds. This is an advantage for mathematicians who
362:
579:
1017:
1366:
simply-connected 4-manifold with positive definite intersection form has the diagonal (scalar 1) intersection form. So
Freedman's classification implies there are many non-smoothable 4-manifolds, for example the
227:
655:
180:
1098:
1217:
442:
733:
797:
460:
873:
1249:
The definition using cup product has a simpler analogue modulo 2 (which works for non-orientable manifolds). Of course one does not have this in de Rham cohomology.
1240:
1137:
911:
831:
1301:
1157:
219:
107:
927:
447:
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary).
1328:
used the intersection form to classify simply-connected topological 4-manifolds. Given any unimodular symmetric bilinear form over the integers,
584:
Using the notion of transversality, one can state the following results (which constitute an equivalent definition of the intersection form).
1450:
1314:
The signature of the intersection form is an important invariant. A 4-manifold bounds a 5-manifold if and only if it has zero signature.
357:{\displaystyle \cap _{M,2}:H_{2}(M;\mathbb {Z} /2\mathbb {Z} )\times H_{2}(M;\mathbb {Z} /2\mathbb {Z} )\to \mathbb {Z} /2\mathbb {Z} }
1430:
591:
116:
1353:
454:
is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2nd homology group
1352:
4-manifolds with the same intersection form are homeomorphic. In the odd case, the two manifolds are distinguished by their
1028:
1422:
1165:
373:
1348:
is odd, there are two, with at least one (possibly both) having no smooth structure. Thus two simply-connected closed
1473:
43:
of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a
668:
1311:
smooth 4-manifold (or more generally one with no 2-torsion residing in the first homology), the converse holds.
746:
72:
1315:
574:{\displaystyle Q_{M}=\cap _{M}=\cdot _{M}:H_{2}(M;\mathbb {Z} )\times H_{2}(M;\mathbb {Z} )\to \mathbb {Z} .}
36:
1359:
1308:
921:
be a closed oriented 4-manifold (PL or smooth). Define the intersection form on the 2nd cohomology group
1468:
1319:
1390:
914:
885:
1108:
836:
1104:
are interested in complexes and topological manifolds (not only in PL and smooth manifolds).
1446:
1426:
1225:
1122:
1012:{\displaystyle Q_{M}\colon H^{2}(M;\mathbb {Z} )\times H^{2}(M;\mathbb {Z} )\to \mathbb {Z} }
896:
810:
1325:
1271:
1243:
1142:
110:
44:
1440:
197:
85:
1258:
68:
60:
1265:
40:
1462:
1381:
1368:
891:
64:
20:
1408:
16:
A special symmetric bilinear form on the 2nd (co)homology group of a 4-manifold
1322:
implies that a smooth compact spin 4-manifold has signature a multiple of 16.
32:
1318:
implies that a spin 4-manifold has signature a multiple of eight. In fact,
28:
833:
has the sign +1 or −1 depending on the orientations, and
1159:, then the intersection form can be expressed by the integral
422:
799:
are represented by closed oriented surfaces (or 2-cycles)
657:
are represented by closed surfaces (or 2-cycles modulo 2)
650:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} /2\mathbb {Z} )}
175:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} /2\mathbb {Z} )}
1442:
Algebraic
Topology From Geometric Viewpoint (in Russian)
1401:
The topology of 4-manifolds, Lecture Notes in Math. 1374
807:
meeting transversely, then every intersection point in
221:, respectively. Define the intersection form modulo 2
1257:
Poincare duality states that the intersection form is
1093:{\displaystyle Q_{M}(a,b)=\langle a\smile b,\rangle .}
1274:
1228:
1168:
1145:
1125:
1031:
930:
899:
839:
813:
749:
671:
594:
463:
376:
230:
200:
119:
88:
1212:{\displaystyle Q(a,b)=\int _{M}\alpha \wedge \beta }
1268:4-manifold must have even intersection form, i.e.,
917:(and so an equivalent) definition as follows. Let
437:{\displaystyle a\cap _{M,2}b=|A\cap B|{\bmod {2}}.}
1295:
1234:
1211:
1151:
1131:
1092:
1011:
905:
867:
825:
791:
727:
649:
573:
436:
356:
213:
174:
101:
1332:, there is a simply-connected closed 4-manifold
1344:is even, there is only one such manifold. If
728:{\displaystyle a\cap _{M,2}b=|A\cap B|\mod 2.}
8:
1084:
1060:
792:{\displaystyle a,b\in H_{2}(M;\mathbb {Z} )}
190:modulo 2 viewed as unions of 2-simplices of
1273:
1227:
1194:
1167:
1144:
1124:
1036:
1030:
1005:
1004:
994:
993:
978:
964:
963:
948:
935:
929:
898:
844:
838:
812:
782:
781:
766:
748:
721:
720:
711:
697:
679:
670:
640:
639:
631:
627:
626:
611:
593:
564:
563:
553:
552:
537:
523:
522:
507:
494:
481:
468:
462:
425:
421:
416:
402:
384:
375:
350:
349:
341:
337:
336:
326:
325:
317:
313:
312:
297:
283:
282:
274:
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269:
254:
235:
229:
205:
199:
165:
164:
156:
152:
151:
136:
118:
93:
87:
1107:When the 4-manifold is smooth, then in
7:
14:
1392:Intersection_number_of_immersions
716:
1290:
1278:
1184:
1172:
1081:
1075:
1054:
1042:
1001:
998:
984:
968:
954:
862:
850:
786:
772:
712:
698:
644:
617:
560:
557:
543:
527:
513:
417:
403:
333:
330:
303:
287:
260:
169:
142:
1:
1423:American Mathematical Society
1419:The wild world of 4-manifolds
665:meeting transversely, then
51:Definition using intersection
880:Definition using cup product
1439:Skopenkov, Arkadiy (2015),
1417:Scorpan, Alexandru (2005),
1253:Properties and applications
1119:are represented by 2-forms
1490:
1354:Kirby–Siebenmann invariant
883:
875:is the sum of these signs.
868:{\displaystyle Q_{M}(a,b)}
890:Using the notion of the
743:is oriented and classes
1336:with intersection form
1235:{\displaystyle \wedge }
1132:{\displaystyle \alpha }
906:{\displaystyle \smile }
826:{\displaystyle A\cap B}
37:symmetric bilinear form
1399:Kirby, Robion (1989),
1297:
1296:{\displaystyle Q(x,x)}
1236:
1213:
1153:
1152:{\displaystyle \beta }
1133:
1094:
1013:
907:
869:
827:
793:
729:
651:
575:
438:
358:
215:
176:
103:
1298:
1237:
1214:
1154:
1134:
1095:
1014:
908:
884:Further information:
870:
828:
794:
730:
652:
576:
439:
359:
216:
214:{\displaystyle T^{*}}
177:
113:. Represent classes
111:dual cell subdivision
104:
102:{\displaystyle T^{*}}
1316:Van der Blij's lemma
1272:
1226:
1166:
1143:
1123:
1029:
928:
897:
837:
811:
747:
669:
592:
461:
374:
228:
198:
117:
86:
1360:Donaldson's theorem
1264:By Wu's formula, a
886:Intersection theory
1474:Geometric topology
1303:is even for every
1293:
1232:
1209:
1149:
1129:
1109:de Rham cohomology
1090:
1009:
903:
865:
823:
789:
725:
647:
571:
434:
354:
211:
172:
99:
1452:978-5-4439-0293-7
1403:, Springer-Verlag
1383:Intersection form
1320:Rokhlin's theorem
1261:(up to torsion).
913:, one can give a
25:intersection form
1481:
1455:
1435:
1413:
1404:
1395:
1386:
1326:Michael Freedman
1309:simply-connected
1302:
1300:
1299:
1294:
1241:
1239:
1238:
1233:
1218:
1216:
1215:
1210:
1199:
1198:
1158:
1156:
1155:
1150:
1138:
1136:
1135:
1130:
1099:
1097:
1096:
1091:
1041:
1040:
1018:
1016:
1015:
1010:
1008:
997:
983:
982:
967:
953:
952:
940:
939:
912:
910:
909:
904:
874:
872:
871:
866:
849:
848:
832:
830:
829:
824:
798:
796:
795:
790:
785:
771:
770:
734:
732:
731:
726:
715:
701:
690:
689:
656:
654:
653:
648:
643:
635:
630:
616:
615:
580:
578:
577:
572:
567:
556:
542:
541:
526:
512:
511:
499:
498:
486:
485:
473:
472:
443:
441:
440:
435:
430:
429:
420:
406:
395:
394:
363:
361:
360:
355:
353:
345:
340:
329:
321:
316:
302:
301:
286:
278:
273:
259:
258:
246:
245:
220:
218:
217:
212:
210:
209:
181:
179:
178:
173:
168:
160:
155:
141:
140:
108:
106:
105:
100:
98:
97:
45:smooth structure
1489:
1488:
1484:
1483:
1482:
1480:
1479:
1478:
1459:
1458:
1453:
1438:
1433:
1416:
1407:
1398:
1389:
1380:
1377:
1270:
1269:
1255:
1224:
1223:
1190:
1164:
1163:
1141:
1140:
1121:
1120:
1032:
1027:
1026:
1022:by the formula
974:
944:
931:
926:
925:
895:
894:
888:
882:
840:
835:
834:
809:
808:
762:
745:
744:
675:
667:
666:
607:
590:
589:
533:
503:
490:
477:
464:
459:
458:
380:
372:
371:
367:by the formula
293:
250:
231:
226:
225:
201:
196:
195:
132:
115:
114:
89:
84:
83:
53:
39:on the 2nd (co)
17:
12:
11:
5:
1487:
1485:
1477:
1476:
1471:
1461:
1460:
1457:
1456:
1451:
1436:
1431:
1414:
1405:
1396:
1387:
1376:
1373:
1292:
1289:
1286:
1283:
1280:
1277:
1254:
1251:
1231:
1220:
1219:
1208:
1205:
1202:
1197:
1193:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1148:
1128:
1101:
1100:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1039:
1035:
1020:
1019:
1007:
1003:
1000:
996:
992:
989:
986:
981:
977:
973:
970:
966:
962:
959:
956:
951:
947:
943:
938:
934:
902:
881:
878:
877:
876:
864:
861:
858:
855:
852:
847:
843:
822:
819:
816:
788:
784:
780:
777:
774:
769:
765:
761:
758:
755:
752:
736:
735:
724:
719:
714:
710:
707:
704:
700:
696:
693:
688:
685:
682:
678:
674:
646:
642:
638:
634:
629:
625:
622:
619:
614:
610:
606:
603:
600:
597:
582:
581:
570:
566:
562:
559:
555:
551:
548:
545:
540:
536:
532:
529:
525:
521:
518:
515:
510:
506:
502:
497:
493:
489:
484:
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476:
471:
467:
445:
444:
433:
428:
424:
419:
415:
412:
409:
405:
401:
398:
393:
390:
387:
383:
379:
365:
364:
352:
348:
344:
339:
335:
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328:
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315:
311:
308:
305:
300:
296:
292:
289:
285:
281:
277:
272:
268:
265:
262:
257:
253:
249:
244:
241:
238:
234:
208:
204:
171:
167:
163:
159:
154:
150:
147:
144:
139:
135:
131:
128:
125:
122:
96:
92:
52:
49:
41:homology group
15:
13:
10:
9:
6:
4:
3:
2:
1486:
1475:
1472:
1470:
1467:
1466:
1464:
1454:
1448:
1444:
1443:
1437:
1434:
1432:0-8218-3749-4
1428:
1424:
1420:
1415:
1412:
1411:
1406:
1402:
1397:
1394:
1393:
1388:
1385:
1384:
1379:
1378:
1374:
1372:
1370:
1365:
1361:
1357:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1321:
1317:
1312:
1310:
1306:
1287:
1284:
1281:
1275:
1267:
1262:
1260:
1252:
1250:
1247:
1245:
1244:wedge product
1229:
1206:
1203:
1200:
1195:
1191:
1187:
1181:
1178:
1175:
1169:
1162:
1161:
1160:
1146:
1126:
1118:
1114:
1110:
1105:
1087:
1078:
1072:
1069:
1066:
1063:
1057:
1051:
1048:
1045:
1037:
1033:
1025:
1024:
1023:
990:
987:
979:
975:
971:
960:
957:
949:
945:
941:
936:
932:
924:
923:
922:
920:
916:
900:
893:
887:
879:
859:
856:
853:
845:
841:
820:
817:
814:
806:
802:
778:
775:
767:
763:
759:
756:
753:
750:
742:
738:
737:
722:
717:
708:
705:
702:
694:
691:
686:
683:
680:
676:
672:
664:
660:
636:
632:
623:
620:
612:
608:
604:
601:
598:
595:
587:
586:
585:
568:
549:
546:
538:
534:
530:
519:
516:
508:
504:
500:
495:
491:
487:
482:
478:
474:
469:
465:
457:
456:
455:
453:
448:
431:
426:
413:
410:
407:
399:
396:
391:
388:
385:
381:
377:
370:
369:
368:
346:
342:
322:
318:
309:
306:
298:
294:
290:
279:
275:
266:
263:
255:
251:
247:
242:
239:
236:
232:
224:
223:
222:
206:
202:
193:
189:
185:
161:
157:
148:
145:
137:
133:
129:
126:
123:
120:
112:
94:
90:
82:. Denote by
81:
77:
74:
73:triangulation
70:
66:
62:
58:
50:
48:
46:
42:
38:
35:is a special
34:
30:
26:
22:
1441:
1418:
1410:Linking_form
1409:
1400:
1391:
1382:
1363:
1358:
1349:
1345:
1341:
1337:
1333:
1329:
1324:
1313:
1304:
1263:
1256:
1248:
1221:
1116:
1112:
1106:
1102:
1021:
918:
889:
804:
800:
740:
662:
658:
583:
451:
449:
446:
366:
191:
187:
183:
182:by 2-cycles
79:
75:
63:4-manifold (
56:
54:
24:
18:
1469:4-manifolds
1369:E8 manifold
892:cup product
588:If classes
71:). Take a
21:mathematics
1463:Categories
1375:References
1259:unimodular
33:4-manifold
1445:, MCCME,
1362:states a
1307:. For a
1230:∧
1207:β
1204:∧
1201:α
1192:∫
1147:β
1127:α
1085:⟩
1067:⌣
1061:⟨
1002:→
972:×
942::
901:⌣
818:∩
760:∈
706:∩
677:∩
605:∈
561:→
531:×
492:⋅
479:∩
411:∩
382:∩
334:→
291:×
233:∩
207:∗
130:∈
95:∗
31:compact
29:oriented
1242:is the
194:and of
1449:
1429:
1364:smooth
1350:smooth
1340:. If
1222:where
69:smooth
61:closed
27:of an
23:, the
1111:, if
59:be a
1447:ISBN
1427:ISBN
1266:spin
1139:and
1115:and
915:dual
803:and
661:and
186:and
109:the
55:Let
739:If
718:mod
450:If
423:mod
78:of
67:or
19:In
1465::
1425:,
1421:,
1371:.
1356:.
1246:.
723:2.
65:PL
47:.
1346:Q
1342:Q
1338:Q
1334:M
1330:Q
1305:x
1291:)
1288:x
1285:,
1282:x
1279:(
1276:Q
1196:M
1188:=
1185:)
1182:b
1179:,
1176:a
1173:(
1170:Q
1117:b
1113:a
1088:.
1082:]
1079:M
1076:[
1073:,
1070:b
1064:a
1058:=
1055:)
1052:b
1049:,
1046:a
1043:(
1038:M
1034:Q
1006:Z
999:)
995:Z
991:;
988:M
985:(
980:2
976:H
969:)
965:Z
961:;
958:M
955:(
950:2
946:H
937:M
933:Q
919:M
863:)
860:b
857:,
854:a
851:(
846:M
842:Q
821:B
815:A
805:B
801:A
787:)
783:Z
779:;
776:M
773:(
768:2
764:H
757:b
754:,
751:a
741:M
713:|
709:B
703:A
699:|
695:=
692:b
687:2
684:,
681:M
673:a
663:B
659:A
645:)
641:Z
637:2
633:/
628:Z
624:;
621:M
618:(
613:2
609:H
602:b
599:,
596:a
569:.
565:Z
558:)
554:Z
550:;
547:M
544:(
539:2
535:H
528:)
524:Z
520:;
517:M
514:(
509:2
505:H
501::
496:M
488:=
483:M
475:=
470:M
466:Q
452:M
432:.
427:2
418:|
414:B
408:A
404:|
400:=
397:b
392:2
389:,
386:M
378:a
351:Z
347:2
343:/
338:Z
331:)
327:Z
323:2
319:/
314:Z
310:;
307:M
304:(
299:2
295:H
288:)
284:Z
280:2
276:/
271:Z
267:;
264:M
261:(
256:2
252:H
248::
243:2
240:,
237:M
203:T
192:T
188:B
184:A
170:)
166:Z
162:2
158:/
153:Z
149:;
146:M
143:(
138:2
134:H
127:b
124:,
121:a
91:T
80:M
76:T
57:M
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