2892:
under symplectic sums, proving various versions of a symplectic sum formula for Gromov–Witten invariants. Such a formula aids computation by allowing one to decompose a given manifold into simpler pieces, whose Gromov–Witten invariants should be easier to compute. Another approach is to use an
2850:
in algebraic geometry. In fact, symplectic treatments of Gromov–Witten theory often use the symplectic sum/cut for "rescaling the target" arguments, while algebro-geometric treatments use deformation to the normal cone for these same arguments.
2278:
1326:
1113:
1701:
2476:
593:
To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism
480:
3012:
400:
302:
2347:
2010:
2507:
2076:
839:
554:
2805:
1265:
2540:
1534:
2637:
2587:
2153:
1463:
43:, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to
2610:
1035:
2663:
1950:
196:
2690:
2424:
2377:
2203:
2180:
2106:
1915:
1895:
1868:
1821:
1794:
1736:
1581:
1495:
1396:
912:
688:
661:
611:
584:
250:
223:
118:
91:
1145:
1007:
984:
961:
938:
141:
3035:
2931:
2911:
2848:
2825:
2773:
2753:
2733:
2713:
2560:
2397:
2126:
1970:
1841:
1759:
1601:
1554:
1436:
1416:
1369:
1349:
1233:
1213:
1193:
1173:
885:
859:
796:
776:
756:
736:
708:
631:
161:
758:, gluing the manifold to itself along the two copies. The preceding description of the sum of two manifolds then corresponds to the special case where
2692:
is the symplectic cut of the generic fiber. So the symplectic sum and cut can be viewed together as a quadratic deformation of symplectic manifolds.
54:
The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the
3076:
3062:
3017:
A formula for the Gromov–Witten invariants of a symplectic sum then yields a recursive formula for the Gromov–Witten invariants of
2881:
Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the
2211:
1273:
1047:
1609:
1761:
for the symplectic sum. Such identity elements have been used both in establishing theory and in computations; see below.
3091:
2828:
861:
2432:
55:
419:
2939:
321:
258:
2289:
1981:
1769:
It is sometimes profitable to view the symplectic sum as a family of manifolds. In this framework, the given data
634:
410:
2867:
2484:
2021:
2889:
2875:
1739:
804:
495:
2866:
The symplectic sum was first clearly defined in 1995 by Robert Gompf. He used it to demonstrate that any
2778:
1238:
2512:
1500:
2882:
2615:
2565:
2185:
Loosely speaking, one constructs this family as follows. Choose a nonvanishing holomorphic section
2182:. (That is, the generic fibers are all members of the unique isotopy class of the symplectic sum.)
2131:
29:
21:
1441:
48:
2855:
2595:
3072:
3058:
2871:
633:(or rather their corresponding punctured unit disk bundles); then this composition is used to
1020:
867:
Other generalizations also exist. However, it is not possible to remove the requirement that
2642:
1920:
166:
2878:
of symplectic manifolds was shown to be much larger than the category of Kähler manifolds.
2668:
2402:
2355:
2188:
2158:
2084:
1900:
1873:
1846:
1799:
1772:
1709:
1559:
1468:
1374:
890:
718:
In greater generality, the symplectic sum can be performed on a single symplectic manifold
666:
639:
596:
562:
228:
201:
96:
69:
1121:
486:
613:
is composed with an orientation-reversing symplectic involution of the normal bundles of
989:
966:
943:
920:
123:
3020:
2916:
2896:
2833:
2810:
2758:
2738:
2718:
2698:
2545:
2382:
2111:
1955:
1826:
1744:
1586:
1539:
1421:
1401:
1354:
1334:
1218:
1198:
1178:
1158:
870:
844:
781:
761:
741:
721:
693:
616:
146:
40:
32:, which glues two given manifolds into a single new one. It is a symplectic version of
3085:
1014:
312:
33:
406:
2854:
However, the symplectic sum is not a complex operation in general. The sum of two
1147:. So the symplectic sum is possible only along a submanifold of codimension two.
308:
17:
1038:
44:
963:-dimensional annulus. If this involution exists, it can be used to patch two
2695:
An important example occurs when one of the summands is an identity element
1973:
1465:, which has opposite normal bundle. Therefore, one may symplectically sum
2888:
A number of researchers have subsequently investigated the behavior of
590:
operation whose result is a symplectic manifold, unique up to isotopy.
801:
Additionally, the sum can be performed simultaneously on submanifolds
2827:. This is analogous to deformation to the normal cone along a smooth
1010:
778:
consists of two connected components, each containing a copy of
3071:(2004) American Mathematical Society Colloquium Publications,
559:
meeting several conditions of compatibility with the summands
2273:{\displaystyle N_{M_{1}}V\otimes _{\mathbb {C} }N_{M_{2}}V.}
2562:
deleted) onto this locus; the result is the symplectic sum
1321:{\displaystyle P:=\mathbb {P} (N_{M}V\oplus \mathbb {C} ).}
3046:
Robert Gompf: A new construction of symplectic manifolds,
1108:{\displaystyle \in H^{2}(\mathbb {S} ^{2k},\mathbb {R} ).}
2715:. For then the generic fiber is a symplectic manifold
3023:
2942:
2919:
2899:
2836:
2813:
2781:
2761:
2741:
2721:
2701:
2671:
2645:
2618:
2598:
2568:
2548:
2515:
2487:
2435:
2405:
2385:
2358:
2292:
2214:
2191:
2161:
2134:
2114:
2087:
2024:
1984:
1958:
1923:
1903:
1876:
1849:
1829:
1802:
1775:
1747:
1712:
1696:{\displaystyle (M,V)=((M,V)\#(P,V_{\infty }),V_{0}).}
1612:
1589:
1562:
1542:
1503:
1471:
1444:
1424:
1404:
1377:
1357:
1337:
1276:
1241:
1221:
1201:
1195:, one may projectively complete the normal bundle of
1181:
1161:
1124:
1050:
1023:
992:
969:
946:
923:
893:
873:
847:
807:
784:
764:
744:
724:
696:
669:
642:
619:
599:
565:
498:
422:
324:
261:
231:
204:
169:
149:
126:
99:
72:
917:
A symplectic sum along a submanifold of codimension
489:
class of symplectic structures on the connected sum
405:In the 1995 paper that defined the symplectic sum,
3029:
3006:
2925:
2905:
2842:
2819:
2799:
2767:
2747:
2727:
2707:
2684:
2657:
2631:
2604:
2581:
2554:
2534:
2501:
2470:
2418:
2391:
2371:
2341:
2272:
2197:
2174:
2147:
2120:
2100:
2070:
2004:
1964:
1944:
1909:
1889:
1862:
1835:
1815:
1788:
1753:
1730:
1695:
1595:
1575:
1548:
1528:
1489:
1457:
1430:
1410:
1390:
1363:
1343:
1320:
1259:
1227:
1207:
1187:
1167:
1139:
1107:
1029:
1001:
978:
955:
932:
906:
879:
853:
833:
790:
770:
750:
730:
702:
682:
655:
625:
605:
578:
548:
474:
394:
296:
244:
217:
190:
155:
135:
112:
85:
2471:{\displaystyle v_{1}\otimes v_{2}=\epsilon \eta }
2015:in which the central fiber is the singular space
986:-dimensional balls together to form a symplectic
1118:But this second cohomology group is zero unless
2426:, consider the locus of the quadratic equation
198:-manifold, embedded as a submanifold into both
36:along a submanifold, often called a fiber sum.
2885:and other operations on symplectic manifolds.
475:{\displaystyle \psi :N_{M_{1}}V\to N_{M_{2}}V}
3007:{\displaystyle (M,V)=(M,V)\#(P,V_{\infty }).}
395:{\displaystyle e(N_{M_{1}}V)=-e(N_{M_{2}}V).}
297:{\displaystyle j_{i}:V\hookrightarrow M_{i},}
8:
3069:J-Holomorphic Curves and Symplectic Topology
2342:{\displaystyle N_{M_{1}}V\oplus N_{M_{2}}V,}
2005:{\displaystyle Z\to D\subseteq \mathbb {C} }
1175:with codimension-two symplectic submanifold
1398:, which has normal bundle equal to that of
3022:
2992:
2941:
2918:
2898:
2835:
2812:
2791:
2787:
2784:
2783:
2780:
2760:
2740:
2720:
2700:
2676:
2670:
2644:
2623:
2617:
2597:
2573:
2567:
2547:
2520:
2514:
2502:{\displaystyle \epsilon \in \mathbb {C} }
2495:
2494:
2486:
2453:
2440:
2434:
2410:
2404:
2384:
2363:
2357:
2325:
2320:
2302:
2297:
2291:
2256:
2251:
2241:
2240:
2239:
2224:
2219:
2213:
2190:
2166:
2160:
2139:
2133:
2113:
2092:
2086:
2071:{\displaystyle Z_{0}=M_{1}\cup _{V}M_{2}}
2062:
2052:
2042:
2029:
2023:
1998:
1997:
1983:
1957:
1922:
1902:
1881:
1875:
1854:
1848:
1828:
1807:
1801:
1780:
1774:
1746:
1711:
1681:
1665:
1611:
1588:
1567:
1561:
1541:
1517:
1502:
1470:
1449:
1443:
1423:
1403:
1382:
1376:
1356:
1336:
1308:
1307:
1295:
1284:
1283:
1275:
1251:
1247:
1244:
1243:
1240:
1220:
1200:
1180:
1160:
1123:
1095:
1094:
1082:
1078:
1077:
1067:
1049:
1022:
991:
968:
945:
922:
898:
892:
872:
846:
825:
812:
806:
783:
763:
743:
723:
695:
674:
668:
647:
641:
618:
598:
570:
564:
531:
506:
497:
461:
456:
438:
433:
421:
375:
370:
340:
335:
323:
285:
266:
260:
236:
230:
209:
203:
168:
148:
125:
104:
98:
77:
71:
39:The symplectic sum is the inverse of the
2874:of a symplectic four-manifold. Thus the
586:. In other words, the theorem defines a
3057:(1998) Oxford Mathematical Monographs,
2526:
2775:"pinched off at infinity" to form the
940:requires a symplectic involution of a
1765:Symplectic sum and cut as deformation
7:
834:{\displaystyle X_{i}\subseteq M_{i}}
549:{\displaystyle (M_{1},V)\#(M_{2},V)}
3055:Introduction to Symplectic Topology
2665:described above. The central fiber
2205:of the trivial complex line bundle
914:, as the following argument shows.
2993:
2976:
1666:
1649:
1518:
1450:
738:containing two disjoint copies of
521:
14:
3067:Dusa McDuff and Dietmar Salamon:
3053:Dusa McDuff and Dietmar Salamon:
2800:{\displaystyle \mathbb {CP} ^{1}}
2081:obtained by joining the summands
1952:-dimensional symplectic manifold
1351:contains two canonical copies of
1260:{\displaystyle \mathbb {CP} ^{1}}
2535:{\displaystyle M_{i}\setminus V}
2379:representing a normal vector to
1529:{\displaystyle (P,V_{\infty })}
841:of equal dimension and meeting
28:is a geometric modification on
2998:
2979:
2973:
2961:
2955:
2943:
2649:
1988:
1939:
1924:
1725:
1713:
1687:
1671:
1652:
1646:
1634:
1631:
1625:
1613:
1523:
1504:
1484:
1472:
1312:
1288:
1099:
1073:
1057:
1051:
543:
524:
518:
499:
449:
386:
363:
351:
328:
278:
185:
170:
45:deformation to the normal cone
1:
2632:{\displaystyle Z_{\epsilon }}
2582:{\displaystyle Z_{\epsilon }}
2148:{\displaystyle Z_{\epsilon }}
887:be of codimension two in the
1017:manifold, a symplectic form
2155:is a symplectic sum of the
1706:So for any particular pair
1458:{\displaystyle V_{\infty }}
1438:, and the infinity-section
3108:
2755:with the normal bundle of
2639:naturally form the family
1917:determine a unique smooth
1013:. Because the sphere is a
2735:and the central fiber is
2605:{\displaystyle \epsilon }
58:of symplectic manifolds.
2890:pseudoholomorphic curves
2868:finitely presented group
2862:History and applications
2283:Then, in the direct sum
2128:, and the generic fiber
1583:now playing the role of
1037:on it induces a nonzero
690:along the two copies of
56:Gromov–Witten invariants
1030:{\displaystyle \omega }
413:-reversing isomorphism
3031:
3008:
2927:
2913:to write the manifold
2907:
2844:
2821:
2801:
2769:
2749:
2729:
2709:
2686:
2659:
2658:{\displaystyle Z\to D}
2633:
2606:
2583:
2556:
2536:
2503:
2472:
2420:
2393:
2373:
2343:
2274:
2199:
2176:
2149:
2122:
2102:
2072:
2006:
1966:
1946:
1945:{\displaystyle (2n+2)}
1911:
1891:
1864:
1837:
1817:
1790:
1755:
1732:
1697:
1597:
1577:
1550:
1536:; the result is again
1530:
1491:
1459:
1432:
1412:
1392:
1365:
1345:
1322:
1261:
1229:
1209:
1189:
1169:
1141:
1109:
1031:
1003:
980:
957:
934:
908:
881:
855:
835:
792:
772:
752:
732:
704:
684:
657:
627:
607:
580:
550:
476:
396:
298:
246:
219:
192:
191:{\displaystyle (2n-2)}
157:
137:
114:
87:
3048:Annals of Mathematics
3032:
3009:
2928:
2908:
2845:
2822:
2802:
2770:
2750:
2730:
2710:
2687:
2685:{\displaystyle Z_{0}}
2660:
2634:
2607:
2584:
2557:
2537:
2504:
2473:
2421:
2419:{\displaystyle M_{i}}
2394:
2374:
2372:{\displaystyle v_{i}}
2344:
2275:
2200:
2198:{\displaystyle \eta }
2177:
2175:{\displaystyle M_{i}}
2150:
2123:
2103:
2101:{\displaystyle M_{i}}
2073:
2007:
1967:
1947:
1912:
1910:{\displaystyle \psi }
1892:
1890:{\displaystyle j_{2}}
1865:
1863:{\displaystyle j_{1}}
1838:
1818:
1816:{\displaystyle M_{2}}
1791:
1789:{\displaystyle M_{1}}
1756:
1733:
1731:{\displaystyle (M,V)}
1698:
1598:
1578:
1576:{\displaystyle V_{0}}
1551:
1531:
1492:
1490:{\displaystyle (M,V)}
1460:
1433:
1413:
1393:
1391:{\displaystyle V_{0}}
1366:
1346:
1323:
1262:
1230:
1210:
1190:
1170:
1142:
1110:
1032:
1004:
981:
958:
935:
909:
907:{\displaystyle M_{i}}
882:
856:
836:
793:
773:
753:
733:
705:
685:
683:{\displaystyle M_{2}}
658:
656:{\displaystyle M_{1}}
628:
608:
606:{\displaystyle \psi }
581:
579:{\displaystyle M_{i}}
551:
485:there is a canonical
477:
397:
299:
247:
245:{\displaystyle M_{2}}
220:
218:{\displaystyle M_{1}}
193:
158:
138:
115:
113:{\displaystyle M_{2}}
88:
86:{\displaystyle M_{1}}
3021:
2940:
2933:as a symplectic sum
2917:
2897:
2858:need not be Kähler.
2834:
2811:
2779:
2759:
2739:
2719:
2699:
2669:
2643:
2616:
2596:
2566:
2546:
2513:
2509:. One can glue both
2485:
2433:
2403:
2383:
2356:
2290:
2212:
2189:
2159:
2132:
2112:
2085:
2022:
1982:
1956:
1921:
1901:
1874:
1847:
1827:
1800:
1773:
1745:
1710:
1610:
1587:
1560:
1540:
1501:
1469:
1442:
1422:
1402:
1375:
1355:
1335:
1274:
1239:
1219:
1199:
1179:
1159:
1140:{\displaystyle 2k=2}
1122:
1048:
1021:
990:
967:
944:
921:
891:
871:
845:
805:
782:
762:
742:
722:
694:
667:
640:
617:
597:
563:
496:
420:
409:proved that for any
322:
259:
229:
202:
167:
147:
124:
97:
70:
30:symplectic manifolds
3092:Symplectic topology
3050:142 (1995), 527-595
2883:symplectic quotient
2542:(the summands with
2481:for a chosen small
1371:: the zero-section
34:connected summation
22:symplectic geometry
3027:
3004:
2923:
2903:
2840:
2817:
2797:
2765:
2745:
2725:
2705:
2682:
2655:
2629:
2602:
2579:
2552:
2532:
2499:
2468:
2416:
2389:
2369:
2339:
2270:
2195:
2172:
2145:
2118:
2098:
2068:
2002:
1962:
1942:
1907:
1887:
1860:
1833:
1813:
1786:
1751:
1728:
1693:
1593:
1573:
1546:
1526:
1487:
1455:
1428:
1408:
1388:
1361:
1341:
1318:
1257:
1225:
1205:
1185:
1165:
1137:
1105:
1027:
1002:{\displaystyle 2k}
999:
979:{\displaystyle 2k}
976:
956:{\displaystyle 2k}
953:
933:{\displaystyle 2k}
930:
904:
877:
851:
831:
788:
768:
748:
728:
700:
680:
653:
623:
603:
576:
546:
472:
392:
294:
242:
215:
188:
153:
136:{\displaystyle 2n}
133:
120:be two symplectic
110:
83:
49:algebraic geometry
20:, specifically in
3030:{\displaystyle M}
2926:{\displaystyle M}
2906:{\displaystyle P}
2893:identity element
2872:fundamental group
2843:{\displaystyle V}
2820:{\displaystyle P}
2768:{\displaystyle V}
2748:{\displaystyle M}
2728:{\displaystyle M}
2708:{\displaystyle P}
2612:varies, the sums
2555:{\displaystyle V}
2392:{\displaystyle V}
2121:{\displaystyle V}
1965:{\displaystyle Z}
1836:{\displaystyle V}
1754:{\displaystyle P}
1596:{\displaystyle V}
1549:{\displaystyle M}
1431:{\displaystyle M}
1411:{\displaystyle V}
1364:{\displaystyle V}
1344:{\displaystyle P}
1228:{\displaystyle M}
1208:{\displaystyle V}
1188:{\displaystyle V}
1168:{\displaystyle M}
880:{\displaystyle V}
854:{\displaystyle V}
791:{\displaystyle V}
771:{\displaystyle X}
751:{\displaystyle V}
731:{\displaystyle M}
703:{\displaystyle V}
626:{\displaystyle V}
156:{\displaystyle V}
3099:
3036:
3034:
3033:
3028:
3013:
3011:
3010:
3005:
2997:
2996:
2932:
2930:
2929:
2924:
2912:
2910:
2909:
2904:
2856:Kähler manifolds
2849:
2847:
2846:
2841:
2826:
2824:
2823:
2818:
2806:
2804:
2803:
2798:
2796:
2795:
2790:
2774:
2772:
2771:
2766:
2754:
2752:
2751:
2746:
2734:
2732:
2731:
2726:
2714:
2712:
2711:
2706:
2691:
2689:
2688:
2683:
2681:
2680:
2664:
2662:
2661:
2656:
2638:
2636:
2635:
2630:
2628:
2627:
2611:
2609:
2608:
2603:
2588:
2586:
2585:
2580:
2578:
2577:
2561:
2559:
2558:
2553:
2541:
2539:
2538:
2533:
2525:
2524:
2508:
2506:
2505:
2500:
2498:
2477:
2475:
2474:
2469:
2458:
2457:
2445:
2444:
2425:
2423:
2422:
2417:
2415:
2414:
2398:
2396:
2395:
2390:
2378:
2376:
2375:
2370:
2368:
2367:
2348:
2346:
2345:
2340:
2332:
2331:
2330:
2329:
2309:
2308:
2307:
2306:
2279:
2277:
2276:
2271:
2263:
2262:
2261:
2260:
2246:
2245:
2244:
2231:
2230:
2229:
2228:
2204:
2202:
2201:
2196:
2181:
2179:
2178:
2173:
2171:
2170:
2154:
2152:
2151:
2146:
2144:
2143:
2127:
2125:
2124:
2119:
2107:
2105:
2104:
2099:
2097:
2096:
2077:
2075:
2074:
2069:
2067:
2066:
2057:
2056:
2047:
2046:
2034:
2033:
2011:
2009:
2008:
2003:
2001:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1916:
1914:
1913:
1908:
1896:
1894:
1893:
1888:
1886:
1885:
1869:
1867:
1866:
1861:
1859:
1858:
1842:
1840:
1839:
1834:
1822:
1820:
1819:
1814:
1812:
1811:
1795:
1793:
1792:
1787:
1785:
1784:
1760:
1758:
1757:
1752:
1740:identity element
1738:there exists an
1737:
1735:
1734:
1729:
1702:
1700:
1699:
1694:
1686:
1685:
1670:
1669:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1572:
1571:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1522:
1521:
1496:
1494:
1493:
1488:
1464:
1462:
1461:
1456:
1454:
1453:
1437:
1435:
1434:
1429:
1417:
1415:
1414:
1409:
1397:
1395:
1394:
1389:
1387:
1386:
1370:
1368:
1367:
1362:
1350:
1348:
1347:
1342:
1327:
1325:
1324:
1319:
1311:
1300:
1299:
1287:
1266:
1264:
1263:
1258:
1256:
1255:
1250:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1194:
1192:
1191:
1186:
1174:
1172:
1171:
1166:
1151:Identity element
1146:
1144:
1143:
1138:
1114:
1112:
1111:
1106:
1098:
1090:
1089:
1081:
1072:
1071:
1036:
1034:
1033:
1028:
1008:
1006:
1005:
1000:
985:
983:
982:
977:
962:
960:
959:
954:
939:
937:
936:
931:
913:
911:
910:
905:
903:
902:
886:
884:
883:
878:
860:
858:
857:
852:
840:
838:
837:
832:
830:
829:
817:
816:
797:
795:
794:
789:
777:
775:
774:
769:
757:
755:
754:
749:
737:
735:
734:
729:
709:
707:
706:
701:
689:
687:
686:
681:
679:
678:
662:
660:
659:
654:
652:
651:
632:
630:
629:
624:
612:
610:
609:
604:
585:
583:
582:
577:
575:
574:
555:
553:
552:
547:
536:
535:
511:
510:
481:
479:
478:
473:
468:
467:
466:
465:
445:
444:
443:
442:
401:
399:
398:
393:
382:
381:
380:
379:
347:
346:
345:
344:
303:
301:
300:
295:
290:
289:
271:
270:
251:
249:
248:
243:
241:
240:
224:
222:
221:
216:
214:
213:
197:
195:
194:
189:
162:
160:
159:
154:
142:
140:
139:
134:
119:
117:
116:
111:
109:
108:
92:
90:
89:
84:
82:
81:
3107:
3106:
3102:
3101:
3100:
3098:
3097:
3096:
3082:
3081:
3043:
3019:
3018:
2988:
2938:
2937:
2915:
2914:
2895:
2894:
2870:appears as the
2864:
2832:
2831:
2809:
2808:
2782:
2777:
2776:
2757:
2756:
2737:
2736:
2717:
2716:
2697:
2696:
2672:
2667:
2666:
2641:
2640:
2619:
2614:
2613:
2594:
2593:
2569:
2564:
2563:
2544:
2543:
2516:
2511:
2510:
2483:
2482:
2449:
2436:
2431:
2430:
2406:
2401:
2400:
2381:
2380:
2359:
2354:
2353:
2321:
2316:
2298:
2293:
2288:
2287:
2252:
2247:
2235:
2220:
2215:
2210:
2209:
2187:
2186:
2162:
2157:
2156:
2135:
2130:
2129:
2110:
2109:
2088:
2083:
2082:
2058:
2048:
2038:
2025:
2020:
2019:
1980:
1979:
1954:
1953:
1919:
1918:
1899:
1898:
1877:
1872:
1871:
1850:
1845:
1844:
1825:
1824:
1803:
1798:
1797:
1776:
1771:
1770:
1767:
1743:
1742:
1708:
1707:
1677:
1661:
1608:
1607:
1585:
1584:
1563:
1558:
1557:
1538:
1537:
1513:
1499:
1498:
1467:
1466:
1445:
1440:
1439:
1420:
1419:
1400:
1399:
1378:
1373:
1372:
1353:
1352:
1333:
1332:
1291:
1272:
1271:
1242:
1237:
1236:
1217:
1216:
1197:
1196:
1177:
1176:
1157:
1156:
1153:
1120:
1119:
1076:
1063:
1046:
1045:
1019:
1018:
988:
987:
965:
964:
942:
941:
919:
918:
894:
889:
888:
869:
868:
843:
842:
821:
808:
803:
802:
780:
779:
760:
759:
740:
739:
720:
719:
716:
714:Generalizations
692:
691:
670:
665:
664:
643:
638:
637:
615:
614:
595:
594:
566:
561:
560:
527:
502:
494:
493:
457:
452:
434:
429:
418:
417:
371:
366:
336:
331:
320:
319:
281:
262:
257:
256:
232:
227:
226:
205:
200:
199:
165:
164:
145:
144:
143:-manifolds and
122:
121:
100:
95:
94:
73:
68:
67:
64:
12:
11:
5:
3105:
3103:
3095:
3094:
3084:
3083:
3080:
3079:
3065:
3051:
3042:
3039:
3026:
3015:
3014:
3003:
3000:
2995:
2991:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2922:
2902:
2863:
2860:
2839:
2816:
2794:
2789:
2786:
2764:
2744:
2724:
2704:
2679:
2675:
2654:
2651:
2648:
2626:
2622:
2601:
2576:
2572:
2551:
2531:
2528:
2523:
2519:
2497:
2493:
2490:
2479:
2478:
2467:
2464:
2461:
2456:
2452:
2448:
2443:
2439:
2413:
2409:
2388:
2366:
2362:
2350:
2349:
2338:
2335:
2328:
2324:
2319:
2315:
2312:
2305:
2301:
2296:
2281:
2280:
2269:
2266:
2259:
2255:
2250:
2243:
2238:
2234:
2227:
2223:
2218:
2194:
2169:
2165:
2142:
2138:
2117:
2095:
2091:
2079:
2078:
2065:
2061:
2055:
2051:
2045:
2041:
2037:
2032:
2028:
2013:
2012:
2000:
1996:
1993:
1990:
1987:
1961:
1941:
1938:
1935:
1932:
1929:
1926:
1906:
1884:
1880:
1857:
1853:
1832:
1810:
1806:
1783:
1779:
1766:
1763:
1750:
1727:
1724:
1721:
1718:
1715:
1704:
1703:
1692:
1689:
1684:
1680:
1676:
1673:
1668:
1664:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1592:
1570:
1566:
1545:
1525:
1520:
1516:
1512:
1509:
1506:
1486:
1483:
1480:
1477:
1474:
1452:
1448:
1427:
1407:
1385:
1381:
1360:
1340:
1329:
1328:
1317:
1314:
1310:
1306:
1303:
1298:
1294:
1290:
1286:
1282:
1279:
1254:
1249:
1246:
1224:
1204:
1184:
1164:
1152:
1149:
1136:
1133:
1130:
1127:
1116:
1115:
1104:
1101:
1097:
1093:
1088:
1085:
1080:
1075:
1070:
1066:
1062:
1059:
1056:
1053:
1026:
998:
995:
975:
972:
952:
949:
929:
926:
901:
897:
876:
850:
828:
824:
820:
815:
811:
787:
767:
747:
727:
715:
712:
699:
677:
673:
650:
646:
622:
602:
588:symplectic sum
573:
569:
557:
556:
545:
542:
539:
534:
530:
526:
523:
520:
517:
514:
509:
505:
501:
483:
482:
471:
464:
460:
455:
451:
448:
441:
437:
432:
428:
425:
403:
402:
391:
388:
385:
378:
374:
369:
365:
362:
359:
356:
353:
350:
343:
339:
334:
330:
327:
315:are opposite:
313:normal bundles
307:such that the
305:
304:
293:
288:
284:
280:
277:
274:
269:
265:
239:
235:
212:
208:
187:
184:
181:
178:
175:
172:
152:
132:
129:
107:
103:
80:
76:
63:
60:
41:symplectic cut
26:symplectic sum
13:
10:
9:
6:
4:
3:
2:
3104:
3093:
3090:
3089:
3087:
3078:
3077:0-8218-3485-1
3074:
3070:
3066:
3064:
3063:0-19-850451-9
3060:
3056:
3052:
3049:
3045:
3044:
3040:
3038:
3024:
3001:
2989:
2985:
2982:
2970:
2967:
2964:
2958:
2952:
2949:
2946:
2936:
2935:
2934:
2920:
2900:
2891:
2886:
2884:
2879:
2877:
2873:
2869:
2861:
2859:
2857:
2852:
2837:
2830:
2814:
2792:
2762:
2742:
2722:
2702:
2693:
2677:
2673:
2652:
2646:
2624:
2620:
2599:
2590:
2574:
2570:
2549:
2529:
2521:
2517:
2491:
2488:
2465:
2462:
2459:
2454:
2450:
2446:
2441:
2437:
2429:
2428:
2427:
2411:
2407:
2386:
2364:
2360:
2336:
2333:
2326:
2322:
2317:
2313:
2310:
2303:
2299:
2294:
2286:
2285:
2284:
2267:
2264:
2257:
2253:
2248:
2236:
2232:
2225:
2221:
2216:
2208:
2207:
2206:
2192:
2183:
2167:
2163:
2140:
2136:
2115:
2093:
2089:
2063:
2059:
2053:
2049:
2043:
2039:
2035:
2030:
2026:
2018:
2017:
2016:
1994:
1991:
1985:
1978:
1977:
1976:
1975:
1959:
1936:
1933:
1930:
1927:
1904:
1882:
1878:
1855:
1851:
1830:
1808:
1804:
1781:
1777:
1764:
1762:
1748:
1741:
1722:
1719:
1716:
1690:
1682:
1678:
1674:
1662:
1658:
1655:
1643:
1640:
1637:
1628:
1622:
1619:
1616:
1606:
1605:
1604:
1590:
1568:
1564:
1543:
1514:
1510:
1507:
1481:
1478:
1475:
1446:
1425:
1405:
1383:
1379:
1358:
1338:
1315:
1304:
1301:
1296:
1292:
1280:
1277:
1270:
1269:
1268:
1252:
1222:
1202:
1182:
1162:
1150:
1148:
1134:
1131:
1128:
1125:
1102:
1091:
1086:
1083:
1068:
1064:
1060:
1054:
1044:
1043:
1042:
1040:
1024:
1016:
1012:
1009:-dimensional
996:
993:
973:
970:
950:
947:
927:
924:
915:
899:
895:
874:
865:
863:
862:transversally
848:
826:
822:
818:
813:
809:
799:
785:
765:
745:
725:
713:
711:
697:
675:
671:
648:
644:
636:
620:
600:
591:
589:
571:
567:
540:
537:
532:
528:
515:
512:
507:
503:
492:
491:
490:
488:
469:
462:
458:
453:
446:
439:
435:
430:
426:
423:
416:
415:
414:
412:
408:
389:
383:
376:
372:
367:
360:
357:
354:
348:
341:
337:
332:
325:
318:
317:
316:
314:
310:
309:Euler classes
291:
286:
282:
275:
272:
267:
263:
255:
254:
253:
237:
233:
210:
206:
182:
179:
176:
173:
163:a symplectic
150:
130:
127:
105:
101:
78:
74:
61:
59:
57:
52:
50:
46:
42:
37:
35:
31:
27:
23:
19:
3068:
3054:
3047:
3016:
2887:
2880:
2865:
2853:
2694:
2591:
2480:
2351:
2282:
2184:
2080:
2014:
1768:
1705:
1330:
1154:
1117:
916:
866:
800:
717:
592:
587:
558:
484:
407:Robert Gompf
404:
306:
65:
53:
38:
25:
15:
411:orientation
18:mathematics
3041:References
1039:cohomology
62:Definition
2994:∞
2977:#
2650:→
2625:ϵ
2600:ϵ
2575:ϵ
2527:∖
2492:∈
2489:ϵ
2466:η
2463:ϵ
2447:⊗
2314:⊕
2237:⊗
2193:η
2141:ϵ
2050:∪
1995:⊆
1989:→
1974:fibration
1905:ψ
1667:∞
1650:#
1519:∞
1451:∞
1305:⊕
1061:∈
1055:ω
1025:ω
819:⊆
601:ψ
522:#
450:→
424:ψ
358:−
279:↪
180:−
3086:Category
2876:category
2807:-bundle
1267:-bundle
2829:divisor
1556:, with
1235:to the
1015:compact
487:isotopy
311:of the
3075:
3061:
2108:along
1972:and a
1155:Given
1041:class
1011:sphere
24:, the
2352:with
1497:with
1331:This
3073:ISBN
3059:ISBN
635:glue
252:via
225:and
93:and
66:Let
2592:As
2399:in
1418:in
1215:in
663:to
47:in
16:In
3088::
3037:.
2589:.
1897:,
1870:,
1843:,
1823:,
1796:,
1603::
1281::=
864:.
798:.
710:.
51:.
3025:M
3002:.
2999:)
2990:V
2986:,
2983:P
2980:(
2974:)
2971:V
2968:,
2965:M
2962:(
2959:=
2956:)
2953:V
2950:,
2947:M
2944:(
2921:M
2901:P
2838:V
2815:P
2793:1
2788:P
2785:C
2763:V
2743:M
2723:M
2703:P
2678:0
2674:Z
2653:D
2647:Z
2621:Z
2571:Z
2550:V
2530:V
2522:i
2518:M
2496:C
2460:=
2455:2
2451:v
2442:1
2438:v
2412:i
2408:M
2387:V
2365:i
2361:v
2337:,
2334:V
2327:2
2323:M
2318:N
2311:V
2304:1
2300:M
2295:N
2268:.
2265:V
2258:2
2254:M
2249:N
2242:C
2233:V
2226:1
2222:M
2217:N
2168:i
2164:M
2137:Z
2116:V
2094:i
2090:M
2064:2
2060:M
2054:V
2044:1
2040:M
2036:=
2031:0
2027:Z
1999:C
1992:D
1986:Z
1960:Z
1940:)
1937:2
1934:+
1931:n
1928:2
1925:(
1883:2
1879:j
1856:1
1852:j
1831:V
1809:2
1805:M
1782:1
1778:M
1749:P
1726:)
1723:V
1720:,
1717:M
1714:(
1691:.
1688:)
1683:0
1679:V
1675:,
1672:)
1663:V
1659:,
1656:P
1653:(
1647:)
1644:V
1641:,
1638:M
1635:(
1632:(
1629:=
1626:)
1623:V
1620:,
1617:M
1614:(
1591:V
1569:0
1565:V
1544:M
1524:)
1515:V
1511:,
1508:P
1505:(
1485:)
1482:V
1479:,
1476:M
1473:(
1447:V
1426:M
1406:V
1384:0
1380:V
1359:V
1339:P
1316:.
1313:)
1309:C
1302:V
1297:M
1293:N
1289:(
1285:P
1278:P
1253:1
1248:P
1245:C
1223:M
1203:V
1183:V
1163:M
1135:2
1132:=
1129:k
1126:2
1103:.
1100:)
1096:R
1092:,
1087:k
1084:2
1079:S
1074:(
1069:2
1065:H
1058:]
1052:[
997:k
994:2
974:k
971:2
951:k
948:2
928:k
925:2
900:i
896:M
875:V
849:V
827:i
823:M
814:i
810:X
786:V
766:X
746:V
726:M
698:V
676:2
672:M
649:1
645:M
621:V
572:i
568:M
544:)
541:V
538:,
533:2
529:M
525:(
519:)
516:V
513:,
508:1
504:M
500:(
470:V
463:2
459:M
454:N
447:V
440:1
436:M
431:N
427::
390:.
387:)
384:V
377:2
373:M
368:N
364:(
361:e
355:=
352:)
349:V
342:1
338:M
333:N
329:(
326:e
292:,
287:i
283:M
276:V
273::
268:i
264:j
238:2
234:M
211:1
207:M
186:)
183:2
177:n
174:2
171:(
151:V
131:n
128:2
106:2
102:M
79:1
75:M
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