1815:
1581:
258:
1363:
1715:
1684:
1623:
930:
1184:
1069:
1034:
1851:
1502:
412:
760:
502:
288:
168:
895:
1439:
1403:
712:
668:
594:
377:
328:
137:
86:
1704:
1490:
1459:
1299:
1244:
1134:
1107:
962:
621:
1907:
showed that the  genus is the index of the Atiyah–Singer operator, which is always integral, and is even in dimensions 4 mod 8. For a 4-dimensional manifold, the
536:
860:
840:
817:
452:
432:
1311:
2107:
140:
1109:, so its Rokhlin invariant is 1. This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in
2137:
1810:{\displaystyle \operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma )+8\operatorname {ks} (M){\bmod {1}}6}
509:
334:, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature.
2074:
1981:
1947:
2209:
728:
1896:
770:
766:
1632:
1854:
153:
93:
1940:
Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 85–97
2219:
2129:
1942:, Proceedings of Symposia in Pure Mathematics, vol. XXXII, Providence, Rhode Island: American Mathematics Society,
1908:
2199:
791:
Since
Rokhlin's theorem states that the signature of a spin smooth manifold is divisible by 16, the definition of the
1590:
1079:
627:. This manifold shows that Rokhlin's theorem fails for the set of merely topological (rather than smooth) manifolds.
2066:
2010:
50:
900:
1154:
1039:
999:
1576:{\displaystyle \operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma ){\bmod {1}}6}
2214:
2094:
Signature modulo 16, invariants de
Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle
2054:
1824:
984:
is divisible by 8, and an easy application of
Rokhlin's theorem shows that its value mod 16 depends only on
634:
is simply connected (or more generally if the first homology group has no 2-torsion), then the vanishing of
35:
1911:
shows that the signature is −8 times the Ă‚ genus, so in dimension 4 this implies
Rokhlin's theorem.
1369:
A characteristic sphere is an embedded 2-sphere whose homology class represents the
Stiefel–Whitney class
553:
1709:
A generalization of the
Freedman-Kirby theorem to topological (rather than smooth) manifolds states that
1921:
is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.
385:
2204:
1876:
557:
505:
253:{\displaystyle Q_{M}\colon H^{2}(M,\mathbb {Z} )\times H^{2}(M,\mathbb {Z} )\rightarrow \mathbb {Z} }
89:
291:
733:
457:
271:
2179:
379:
vanishes, and the signature is −16, so 16 is the best possible number in
Rokhlin's theorem.
265:
865:
2133:
2070:
1977:
1943:
1408:
1372:
715:
681:
637:
563:
346:
297:
106:
55:
1689:
1475:
1444:
1284:
2163:
2058:
2042:
1969:
1931:
1461:
to be any small sphere, which has self intersection number 0, so
Rokhlin's theorem follows.
1258:
671:
624:
545:
101:
2175:
2147:
2118:
2101:
2084:
2034:
2018:
1991:
1957:
1217:
1112:
1085:
935:
599:
2171:
2143:
2115:
2097:
2080:
2030:
2014:
1998:
1987:
1953:
2044:
An elementary proof of
Rochlin's signature theorem and its extension by Guillou and Marin
515:
932:
is defined to be the signature of any smooth compact spin 4-manifold with spin boundary
1904:
1900:
1207:
1148:
1137:
993:
973:
845:
825:
820:
802:
437:
417:
97:
46:
2193:
2183:
1884:
1626:
1935:
1872:
776:
670:
is equivalent to the intersection form being even. This is not true in general: an
17:
2002:
996:
so we can define the
Rokhlin invariant of a homology 3-sphere to be the element
549:
2167:
1888:
1358:{\displaystyle \operatorname {signature} (M)=\Sigma \cdot \Sigma {\bmod {1}}6}
539:
340:
27:
On the intersection form of a smooth, closed 4-manifold with a spin structure
2025:
Kervaire, Michel A.; Milnor, John W. (1961), "On 2-spheres in 4-manifolds",
331:
39:
2005:(1960), "Bernoulli numbers, homotopy groups, and a theorem of Rohlin",
1973:
1265:-valued lift of the Rokhlin invariant of integral homology 3-sphere.
623:
of signature 8. Rokhlin's theorem implies that this manifold has no
2007:
Proceedings of the International Congress of Mathematicians, 1958
1968:, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag,
674:
is a compact smooth 4 manifold and has even intersection form II
1706:
is a sphere, so the Kervaire–Milnor theorem is a special case.
1194:
is well defined mod 16, and is called the Rokhlin invariant of
727:
Rokhlin's theorem can be deduced from the fact that the third
1795:
1561:
1343:
1186:
homology sphere), then the signature of any spin 4-manifold
762:
is cyclic of order 24; this is Rokhlin's original approach.
330:
implies that the intersection form is even. By a theorem of
1492:
is a characteristic surface in a smooth compact 4-manifold
1214:, and which evaluates to the Rokhlin invariant of the pair
678:
of signature −8 (not divisible by 16), but the class
1301:
is a characteristic sphere in a smooth compact 4-manifold
1679:{\displaystyle H_{1}(\Sigma ,\mathbb {Z} /2\mathbb {Z} )}
2112:
New results in the theory of four-dimensional manifolds
2114:, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224.
1827:
1718:
1692:
1635:
1593:
1505:
1478:
1447:
1411:
1375:
1314:
1287:
1220:
1157:
1115:
1088:
1042:
1002:
938:
903:
868:
848:
828:
805:
736:
684:
640:
602:
566:
518:
460:
440:
420:
388:
349:
300:
274:
171:
109:
58:
30:
In 4-dimensional topology, a branch of mathematics,
1938:(1978), "A geometric proof of Rochlin's theorem",
1845:
1809:
1698:
1678:
1617:
1575:
1484:
1453:
1433:
1397:
1357:
1293:
1238:
1178:
1128:
1101:
1075:any spin 4-manifold bounding the homology sphere.
1063:
1028:
956:
924:
889:
854:
834:
811:
754:
706:
662:
615:
588:
530:
496:
446:
426:
406:
371:
322:
282:
252:
131:
80:
2154:Szűcs, András (2003), "Two Theorems of Rokhlin",
1257:The Rokhlin invariant of M is equal to half the
1082:bounds a spin 4-manifold with intersection form
2027:Proceedings of the National Academy of Sciences
1891:is an integer, and is even if the dimension of
1618:{\displaystyle \operatorname {Arf} (M,\Sigma )}
139:, is divisible by 16. The theorem is named for
1278:
1469:
1261:mod 2. The Casson invariant is viewed as the
8:
976:3-manifold then it bounds a spin 4-manifold
1206:refers to the function whose domain is the
925:{\displaystyle \mathbb {Z} /16\mathbb {Z} }
1179:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
1064:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
1029:{\displaystyle \operatorname {sign} (M)/8}
2096:, MĂ©m. Soc. Math. France 1980/81, no. 5,
1895:is 4 mod 8. This can be deduced from the
1826:
1798:
1794:
1717:
1691:
1669:
1668:
1660:
1656:
1655:
1640:
1634:
1592:
1564:
1560:
1504:
1477:
1446:
1416:
1410:
1380:
1374:
1346:
1342:
1313:
1286:
1219:
1172:
1171:
1163:
1159:
1158:
1156:
1120:
1114:
1093:
1087:
1057:
1056:
1048:
1044:
1043:
1041:
1018:
1001:
937:
918:
917:
909:
905:
904:
902:
867:
847:
827:
804:
746:
741:
735:
689:
683:
645:
639:
607:
601:
571:
565:
517:
486:
474:
459:
439:
419:
398:
394:
391:
390:
387:
354:
348:
305:
299:
276:
275:
273:
246:
245:
235:
234:
219:
205:
204:
189:
176:
170:
114:
108:
63:
57:
1914:
714:does not vanish and is represented by a
1686:. This Arf invariant is obviously 0 if
1846:{\displaystyle \operatorname {ks} (M)}
1887:of dimension divisible by 4 then the
780:
7:
2029:, vol. 47, pp. 1651–1657,
1861:. The Kirby–Siebenmann invariant of
992:. Homology 3-spheres have a unique
1767:
1743:
1737:
1693:
1649:
1609:
1554:
1530:
1524:
1479:
1448:
1339:
1333:
1288:
25:
1879:proved the following theorem: If
538:gives back the last example of a
407:{\displaystyle \mathbb {CP} ^{3}}
2156:Journal of Mathematical Sciences
765:It can also be deduced from the
729:stable homotopy group of spheres
1629:of a certain quadratic form on
718:in the second cohomology group.
343:is compact, 4 dimensional, and
1840:
1834:
1791:
1785:
1770:
1758:
1731:
1725:
1673:
1646:
1612:
1600:
1557:
1545:
1518:
1512:
1428:
1422:
1392:
1386:
1327:
1321:
1233:
1221:
1198:. On a topological 3-manifold
1015:
1009:
951:
939:
884:
872:
701:
695:
657:
651:
583:
577:
480:
461:
366:
360:
317:
311:
242:
239:
225:
209:
195:
126:
120:
75:
69:
49:(or, equivalently, the second
1:
2130:American Mathematical Society
2126:The wild world of 4-manifolds
2041:Matsumoto, Yoichirou (1986),
1204:generalized Rokhlin invariant
1151:3-manifold (for example, any
771:Ă‚ genus and Rochlin's theorem
1909:Hirzebruch signature theorem
755:{\displaystyle \pi _{3}^{S}}
497:{\displaystyle (4-d^{2})d/3}
283:{\displaystyle \mathbb {Z} }
2124:Scorpan, Alexandru (2005),
1966:The Topology of 4-Manifolds
1897:Atiyah–Singer index theorem
783:) gives a geometric proof.
767:Atiyah–Singer index theorem
2236:
2067:Princeton University Press
2011:Cambridge University Press
1855:Kirby–Siebenmann invariant
1279:Kervaire & Milnor 1960
454:is even. It has signature
2065:, Princeton, New Jersey:
1470:Freedman & Kirby 1978
988:and not on the choice of
890:{\displaystyle \mu (N,s)}
504:, which can be seen from
143:, who proved it in 1952.
2055:Michelsohn, Marie-Louise
1434:{\displaystyle w_{2}(M)}
1398:{\displaystyle w_{2}(M)}
1080:Poincaré homology sphere
862:, the Rokhlin invariant
707:{\displaystyle w_{2}(M)}
663:{\displaystyle w_{2}(M)}
589:{\displaystyle w_{2}(M)}
372:{\displaystyle w_{2}(M)}
323:{\displaystyle w_{2}(M)}
132:{\displaystyle H^{2}(M)}
81:{\displaystyle w_{2}(M)}
2210:Differential structures
2168:10.1023/A:1021208007146
1699:{\displaystyle \Sigma }
1485:{\displaystyle \Sigma }
1454:{\displaystyle \Sigma }
1294:{\displaystyle \Sigma }
1275:Kervaire–Milnor theorem
1250:is a spin structure on
795:is deduced as follows:
434:is spin if and only if
294:, and the vanishing of
38:, orientable, closed 4-
1964:Kirby, Robion (1989),
1847:
1811:
1700:
1680:
1619:
1577:
1486:
1466:Freedman–Kirby theorem
1455:
1441:vanishes, we can take
1435:
1399:
1359:
1295:
1240:
1180:
1136:, nor does it bound a
1130:
1103:
1065:
1030:
958:
926:
891:
856:
836:
813:
756:
708:
664:
617:
596:and intersection form
590:
532:
498:
448:
428:
408:
373:
324:
284:
254:
133:
82:
2088:(especially page 280)
1848:
1812:
1701:
1681:
1620:
1578:
1487:
1456:
1436:
1400:
1360:
1296:
1241:
1239:{\displaystyle (N,s)}
1181:
1131:
1129:{\displaystyle S^{4}}
1104:
1102:{\displaystyle E_{8}}
1066:
1031:
959:
957:{\displaystyle (N,s)}
927:
892:
857:
837:
814:
787:The Rokhlin invariant
757:
709:
665:
618:
616:{\displaystyle E_{8}}
591:
533:
499:
449:
429:
409:
382:A complex surface in
374:
325:
285:
255:
134:
83:
51:Stiefel–Whitney class
2220:Theorems in topology
2108:Rokhlin, Vladimir A.
2013:, pp. 454–458,
1883:is a smooth compact
1877:Friedrich Hirzebruch
1825:
1716:
1690:
1633:
1591:
1503:
1476:
1445:
1409:
1373:
1312:
1285:
1218:
1155:
1113:
1086:
1040:
1000:
936:
901:
866:
846:
826:
803:
734:
682:
638:
600:
564:
558:topological manifold
516:
506:Friedrich Hirzebruch
458:
438:
418:
386:
347:
298:
272:
169:
107:
88:vanishes), then the
56:
1999:Kervaire, Michel A.
1143:More generally, if
980:. The signature of
751:
531:{\displaystyle d=4}
2200:Geometric topology
1974:10.1007/BFb0089031
1843:
1807:
1696:
1676:
1615:
1573:
1482:
1451:
1431:
1395:
1355:
1291:
1236:
1176:
1126:
1099:
1061:
1026:
954:
922:
887:
852:
832:
809:
752:
737:
704:
660:
613:
586:
528:
494:
444:
424:
404:
369:
320:
280:
250:
129:
78:
2139:978-0-8218-3749-8
2092:Ochanine, Serge,
2059:Lawson, H. Blaine
1932:Freedman, Michael
1472:) states that if
1281:) states that if
1078:For example, the
855:{\displaystyle N}
835:{\displaystyle s}
812:{\displaystyle N}
793:Rokhlin invariant
510:signature theorem
447:{\displaystyle d}
427:{\displaystyle d}
154:intersection form
94:intersection form
34:states that if a
32:Rokhlin's theorem
18:Rochlin invariant
16:(Redirected from
2227:
2186:
2150:
2104:
2087:
2050:
2049:
2037:
2021:
1994:
1960:
1852:
1850:
1849:
1844:
1816:
1814:
1813:
1808:
1803:
1802:
1705:
1703:
1702:
1697:
1685:
1683:
1682:
1677:
1672:
1664:
1659:
1645:
1644:
1624:
1622:
1621:
1616:
1582:
1580:
1579:
1574:
1569:
1568:
1491:
1489:
1488:
1483:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1421:
1420:
1404:
1402:
1401:
1396:
1385:
1384:
1364:
1362:
1361:
1356:
1351:
1350:
1300:
1298:
1297:
1292:
1259:Casson invariant
1245:
1243:
1242:
1237:
1185:
1183:
1182:
1177:
1175:
1167:
1162:
1135:
1133:
1132:
1127:
1125:
1124:
1108:
1106:
1105:
1100:
1098:
1097:
1070:
1068:
1067:
1062:
1060:
1052:
1047:
1035:
1033:
1032:
1027:
1022:
963:
961:
960:
955:
931:
929:
928:
923:
921:
913:
908:
896:
894:
893:
888:
861:
859:
858:
853:
841:
839:
838:
833:
818:
816:
815:
810:
777:Robion Kirby
761:
759:
758:
753:
750:
745:
713:
711:
710:
705:
694:
693:
672:Enriques surface
669:
667:
666:
661:
650:
649:
630:If the manifold
625:smooth structure
622:
620:
619:
614:
612:
611:
595:
593:
592:
587:
576:
575:
554:simply connected
546:Michael Freedman
537:
535:
534:
529:
503:
501:
500:
495:
490:
479:
478:
453:
451:
450:
445:
433:
431:
430:
425:
413:
411:
410:
405:
403:
402:
397:
378:
376:
375:
370:
359:
358:
329:
327:
326:
321:
310:
309:
292:Poincaré duality
289:
287:
286:
281:
279:
259:
257:
256:
251:
249:
238:
224:
223:
208:
194:
193:
181:
180:
141:Vladimir Rokhlin
138:
136:
135:
130:
119:
118:
102:cohomology group
87:
85:
84:
79:
68:
67:
21:
2235:
2234:
2230:
2229:
2228:
2226:
2225:
2224:
2190:
2189:
2153:
2140:
2123:
2091:
2077:
2053:
2047:
2040:
2024:
2003:Milnor, John W.
1997:
1984:
1963:
1950:
1930:
1927:
1917:proved that if
1915:Ochanine (1980)
1823:
1822:
1714:
1713:
1688:
1687:
1636:
1631:
1630:
1589:
1588:
1501:
1500:
1474:
1473:
1443:
1442:
1412:
1407:
1406:
1376:
1371:
1370:
1310:
1309:
1283:
1282:
1271:
1269:Generalizations
1216:
1215:
1208:spin structures
1153:
1152:
1116:
1111:
1110:
1089:
1084:
1083:
1038:
1037:
998:
997:
934:
933:
899:
898:
864:
863:
844:
843:
824:
823:
801:
800:
799:For 3-manifold
789:
732:
731:
725:
716:torsion element
685:
680:
679:
677:
641:
636:
635:
603:
598:
597:
567:
562:
561:
560:with vanishing
514:
513:
470:
456:
455:
436:
435:
416:
415:
389:
384:
383:
350:
345:
344:
301:
296:
295:
270:
269:
215:
185:
172:
167:
166:
149:
110:
105:
104:
59:
54:
53:
28:
23:
22:
15:
12:
11:
5:
2233:
2231:
2223:
2222:
2217:
2215:Surgery theory
2212:
2207:
2202:
2192:
2191:
2188:
2187:
2162:(6): 888–892,
2151:
2138:
2121:
2105:
2089:
2075:
2051:
2038:
2022:
1995:
1982:
1961:
1948:
1926:
1923:
1905:Isadore Singer
1901:Michael Atiyah
1842:
1839:
1836:
1833:
1830:
1819:
1818:
1806:
1801:
1797:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1695:
1675:
1671:
1667:
1663:
1658:
1654:
1651:
1648:
1643:
1639:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1585:
1584:
1572:
1567:
1563:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1481:
1450:
1430:
1427:
1424:
1419:
1415:
1394:
1391:
1388:
1383:
1379:
1367:
1366:
1354:
1349:
1345:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1290:
1270:
1267:
1235:
1232:
1229:
1226:
1223:
1190:with boundary
1174:
1170:
1166:
1161:
1138:Mazur manifold
1123:
1119:
1096:
1092:
1059:
1055:
1051:
1046:
1025:
1021:
1017:
1014:
1011:
1008:
1005:
994:spin structure
966:
965:
953:
950:
947:
944:
941:
920:
916:
912:
907:
886:
883:
880:
877:
874:
871:
851:
831:
821:spin structure
808:
788:
785:
749:
744:
740:
724:
721:
720:
719:
703:
700:
697:
692:
688:
675:
659:
656:
653:
648:
644:
628:
610:
606:
585:
582:
579:
574:
570:
543:
527:
524:
521:
493:
489:
485:
482:
477:
473:
469:
466:
463:
443:
423:
401:
396:
393:
380:
368:
365:
362:
357:
353:
336:
335:
319:
316:
313:
308:
304:
278:
262:
261:
260:
248:
244:
241:
237:
233:
230:
227:
222:
218:
214:
211:
207:
203:
200:
197:
192:
188:
184:
179:
175:
161:
160:
148:
145:
128:
125:
122:
117:
113:
100:on the second
98:quadratic form
77:
74:
71:
66:
62:
47:spin structure
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2232:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2197:
2195:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2152:
2149:
2145:
2141:
2135:
2131:
2127:
2122:
2120:
2117:
2113:
2109:
2106:
2103:
2099:
2095:
2090:
2086:
2082:
2078:
2076:0-691-08542-0
2072:
2068:
2064:
2063:Spin geometry
2060:
2056:
2052:
2046:
2045:
2039:
2036:
2032:
2028:
2023:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1993:
1989:
1985:
1983:0-387-51148-2
1979:
1975:
1971:
1967:
1962:
1959:
1955:
1951:
1949:0-8218-1432-X
1945:
1941:
1937:
1936:Kirby, Robion
1933:
1929:
1928:
1924:
1922:
1920:
1916:
1912:
1910:
1906:
1902:
1898:
1894:
1890:
1886:
1885:spin manifold
1882:
1878:
1874:
1870:
1868:
1864:
1860:
1856:
1837:
1831:
1828:
1804:
1799:
1788:
1782:
1779:
1776:
1773:
1764:
1761:
1755:
1752:
1749:
1746:
1740:
1734:
1728:
1722:
1719:
1712:
1711:
1710:
1707:
1665:
1661:
1652:
1641:
1637:
1628:
1627:Arf invariant
1606:
1603:
1597:
1594:
1570:
1565:
1551:
1548:
1542:
1539:
1536:
1533:
1527:
1521:
1515:
1509:
1506:
1499:
1498:
1497:
1495:
1471:
1467:
1462:
1425:
1417:
1413:
1389:
1381:
1377:
1352:
1347:
1336:
1330:
1324:
1318:
1315:
1308:
1307:
1306:
1304:
1280:
1276:
1268:
1266:
1264:
1260:
1255:
1253:
1249:
1230:
1227:
1224:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1168:
1164:
1150:
1146:
1141:
1139:
1121:
1117:
1094:
1090:
1081:
1076:
1074:
1053:
1049:
1023:
1019:
1012:
1006:
1003:
995:
991:
987:
983:
979:
975:
971:
948:
945:
942:
914:
910:
881:
878:
875:
869:
849:
829:
822:
806:
798:
797:
796:
794:
786:
784:
782:
778:
774:
772:
768:
763:
747:
742:
738:
730:
722:
717:
698:
690:
686:
673:
654:
646:
642:
633:
629:
626:
608:
604:
580:
572:
568:
559:
555:
551:
547:
544:
541:
525:
522:
519:
511:
507:
491:
487:
483:
475:
471:
467:
464:
441:
421:
399:
381:
363:
355:
351:
342:
338:
337:
333:
314:
306:
302:
293:
267:
263:
231:
228:
220:
216:
212:
201:
198:
190:
186:
182:
177:
173:
165:
164:
163:
162:
159:
155:
151:
150:
146:
144:
142:
123:
115:
111:
103:
99:
95:
91:
72:
64:
60:
52:
48:
44:
41:
37:
33:
19:
2159:
2155:
2125:
2111:
2093:
2062:
2043:
2026:
2009:, New York:
2006:
1965:
1939:
1918:
1913:
1892:
1880:
1873:Armand Borel
1871:
1866:
1862:
1858:
1820:
1708:
1586:
1493:
1465:
1463:
1368:
1302:
1274:
1272:
1262:
1256:
1251:
1247:
1211:
1203:
1199:
1195:
1191:
1187:
1144:
1142:
1077:
1072:
989:
985:
981:
977:
969:
967:
792:
790:
775:
764:
726:
631:
157:
42:
31:
29:
2205:4-manifolds
1869:is smooth.
550:E8 manifold
512:. The case
2194:Categories
1925:References
540:K3 surface
414:of degree
341:K3 surface
266:unimodular
2184:117175810
1832:
1783:
1768:Σ
1756:
1744:Σ
1741:⋅
1738:Σ
1723:
1720:signature
1694:Σ
1650:Σ
1610:Σ
1598:
1555:Σ
1543:
1531:Σ
1528:⋅
1525:Σ
1510:
1507:signature
1480:Σ
1449:Σ
1340:Σ
1337:⋅
1334:Σ
1319:
1316:signature
1289:Σ
1007:
870:μ
739:π
468:−
332:Cahit Arf
243:→
213:×
183::
90:signature
2061:(1989),
1865:is 0 if
1496:, then
1305:, then
1071:, where
556:compact
147:Examples
40:manifold
2176:1809832
2148:2136212
2119:0052101
2102:1809832
2085:1031992
2035:0133134
2019:0121801
1992:1001966
1958:0520525
1889:Ă‚ genus
1853:is the
1625:is the
779: (
92:of its
2182:
2174:
2146:
2136:
2100:
2083:
2073:
2033:
2017:
1990:
1980:
1956:
1946:
1821:where
1587:where
1246:where
1202:, the
819:and a
769:. See
723:Proofs
45:has a
36:smooth
2180:S2CID
2048:(PDF)
1405:. If
1147:is a
972:is a
552:is a
2134:ISBN
2071:ISBN
1978:ISBN
1944:ISBN
1903:and
1875:and
1464:The
1273:The
1149:spin
1004:sign
974:spin
781:1989
152:The
96:, a
2164:doi
2160:113
1970:doi
1857:of
1796:mod
1753:Arf
1595:Arf
1562:mod
1540:Arf
1344:mod
1210:on
1036:of
968:If
897:in
842:on
676:1,9
548:'s
508:'s
290:by
268:on
264:is
156:on
2196::
2178:,
2172:MR
2170:,
2158:,
2144:MR
2142:,
2132:,
2128:,
2116:MR
2110:,
2098:MR
2081:MR
2079:,
2069:,
2057:;
2031:MR
2015:MR
2001:;
1988:MR
1986:,
1976:,
1954:MR
1952:,
1934:;
1899::
1829:ks
1780:ks
1254:.
1140:.
915:16
773:.
339:A
2166::
1972::
1919:X
1893:X
1881:X
1867:M
1863:M
1859:M
1841:)
1838:M
1835:(
1817:,
1805:6
1800:1
1792:)
1789:M
1786:(
1777:8
1774:+
1771:)
1765:,
1762:M
1759:(
1750:8
1747:+
1735:=
1732:)
1729:M
1726:(
1674:)
1670:Z
1666:2
1662:/
1657:Z
1653:,
1647:(
1642:1
1638:H
1613:)
1607:,
1604:M
1601:(
1583:.
1571:6
1566:1
1558:)
1552:,
1549:M
1546:(
1537:8
1534:+
1522:=
1519:)
1516:M
1513:(
1494:M
1468:(
1429:)
1426:M
1423:(
1418:2
1414:w
1393:)
1390:M
1387:(
1382:2
1378:w
1365:.
1353:6
1348:1
1331:=
1328:)
1325:M
1322:(
1303:M
1277:(
1263:Z
1252:N
1248:s
1234:)
1231:s
1228:,
1225:N
1222:(
1212:N
1200:N
1196:N
1192:N
1188:M
1173:Z
1169:2
1165:/
1160:Z
1145:N
1122:4
1118:S
1095:8
1091:E
1073:M
1058:Z
1054:2
1050:/
1045:Z
1024:8
1020:/
1016:)
1013:M
1010:(
990:M
986:N
982:M
978:M
970:N
964:.
952:)
949:s
946:,
943:N
940:(
919:Z
911:/
906:Z
885:)
882:s
879:,
876:N
873:(
850:N
830:s
807:N
748:S
743:3
702:)
699:M
696:(
691:2
687:w
658:)
655:M
652:(
647:2
643:w
632:M
609:8
605:E
584:)
581:M
578:(
573:2
569:w
542:.
526:4
523:=
520:d
492:3
488:/
484:d
481:)
476:2
472:d
465:4
462:(
442:d
422:d
400:3
395:P
392:C
367:)
364:M
361:(
356:2
352:w
318:)
315:M
312:(
307:2
303:w
277:Z
247:Z
240:)
236:Z
232:,
229:M
226:(
221:2
217:H
210:)
206:Z
202:,
199:M
196:(
191:2
187:H
178:M
174:Q
158:M
127:)
124:M
121:(
116:2
112:H
76:)
73:M
70:(
65:2
61:w
43:M
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.