117:. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem.
416:
1338:) subset of the set of mappings. To make such a statement precise, it is necessary to define the space of mappings under consideration, and what is the topology in it. There are several possibilities; see the book by Hirsch.
828:
606:
938:
1828:
2586:
2367:
1065:
875:
977:
313:
1887:
684:
160:
77:
725:
501:
2485:
2025:
1639:
1414:
1773:
2169:
1271:
1570:
655:
266:
2104:
1970:
103:
2714:
2631:
2290:
2252:
1545:
1468:
1336:
2435:
2078:
1924:
1168:
629:
2316:
2133:
1665:
321:
2393:
1716:
1311:
Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or, in some cases, only a dense
1214:
2658:
2532:
2216:
1601:
1441:
1241:
1498:
2678:
2505:
2189:
2045:
1944:
1685:
1518:
1291:
1188:
1145:
1125:
1105:
1085:
1017:
997:
745:
568:
544:
521:
460:
440:
240:
220:
200:
180:
1349:
transversality. See the books by Hirsch and by
Golubitsky and Guillemin. The original reference is Thom, Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59–71.
2887:
753:
1370:
The infinite-dimensional version of the transversality theorem takes into account that the manifolds may be modeled in Banach spaces.
2868:
2825:
2810:
2786:
2764:
2742:
1301:
The parametric transversality theorem above is sufficient for many elementary applications (see the book by
Guillemin and Pollack).
573:
2892:
880:
39:, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that
1782:
40:
1352:
2540:
2321:
1032:
842:
943:
271:
1833:
660:
133:
50:
689:
465:
20:
2440:
1975:
1606:
1381:
1721:
2141:
1246:
79:, may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold
1550:
634:
245:
2083:
1949:
1308:) that imply the parametric transversality theorem and are needed for more advanced applications.
82:
2800:
2683:
2591:
2259:
2221:
1523:
1446:
1314:
2861:
Nonlinear
Functional Analysis and Its Applications: Part 4: Applications to Mathematical Physics
2398:
2050:
1896:
1150:
611:
411:{\displaystyle \operatorname {im} \left(df_{x}\right)+T_{f\left(x\right)}Z=T_{f\left(x\right)}Y}
2864:
2806:
2782:
2760:
2738:
1776:
1357:
1346:
2295:
2109:
1644:
2834:
2774:
2752:
2731:
2372:
1695:
1193:
110:
44:
2636:
2510:
2194:
1579:
1419:
1219:
2726:
1477:
2663:
2490:
2174:
2030:
1929:
1670:
1503:
1276:
1173:
1130:
1110:
1090:
1070:
1002:
982:
730:
553:
529:
506:
445:
425:
225:
205:
185:
165:
114:
2881:
2796:
2820:
631:
is smooth, and it allows us to state an extension of the previous result: if both
36:
16:
Describes the transverse intersection properties of a smooth family of smooth maps
823:{\displaystyle \partial f^{-1}\left(Z\right)=f^{-1}\left(Z\right)\cap \partial X}
106:
2680:
such that there are at most finitely many solutions for each fixed parameter
547:
2838:
2823:(1954). "Quelques propriétés globales des variétés differentiables".
422:
An important result about transversality states that if a smooth map
32:
2846:
Thom, René (1956). "Un lemme sur les applications différentiables".
2733:
Geometrical
Methods in the Theory of Ordinary Differential Equations
1355:
proved in the 1970s an even more general result called the
2138:
If (i)-(iv) hold, then there exists an open, dense subset
1304:
There are more powerful statements (collectively known as
601:{\displaystyle \partial f\colon \partial X\rightarrow Y}
979:. We require that the family vary smoothly by assuming
2686:
2666:
2639:
2594:
2543:
2513:
2493:
2443:
2401:
2375:
2324:
2298:
2262:
2224:
2197:
2177:
2144:
2112:
2086:
2053:
2033:
1978:
1952:
1932:
1899:
1836:
1785:
1724:
1698:
1673:
1647:
1609:
1582:
1553:
1526:
1506:
1480:
1449:
1422:
1384:
1317:
1279:
1249:
1222:
1196:
1176:
1153:
1133:
1113:
1093:
1073:
1035:
1005:
985:
946:
933:{\displaystyle f_{s}\left(x\right)=F\left(x,s\right)}
883:
845:
756:
733:
692:
663:
637:
614:
576:
556:
532:
509:
468:
448:
428:
324:
274:
248:
228:
208:
188:
168:
136:
85:
53:
2730:
2708:
2672:
2652:
2625:
2580:
2526:
2499:
2479:
2429:
2387:
2361:
2310:
2284:
2246:
2210:
2183:
2163:
2127:
2098:
2072:
2047:implies the existence of a convergent subsequence
2039:
2019:
1964:
1938:
1918:
1881:
1823:{\displaystyle \operatorname {ind} Df_{s}(x)<k}
1822:
1767:
1710:
1679:
1659:
1633:
1595:
1564:
1539:
1512:
1492:
1462:
1435:
1408:
1330:
1285:
1265:
1235:
1208:
1182:
1162:
1139:
1119:
1099:
1079:
1059:
1011:
991:
971:
932:
869:
822:
739:
719:
678:
649:
623:
600:
562:
538:
515:
495:
454:
434:
410:
307:
260:
234:
214:
194:
174:
162:be a smooth map between smooth manifolds, and let
154:
97:
71:
1547:-Banach manifolds with chart spaces over a field
550:, then we can define the restriction of the map
2581:{\displaystyle \operatorname {ind} Df_{s}(x)=0}
2534:-Banach manifold or the solution set is empty.
2362:{\displaystyle \operatorname {ind} Df_{s}(x)=n}
2716:In addition, all these solutions are regular.
1060:{\displaystyle F\colon X\times S\rightarrow Y}
870:{\displaystyle F\colon X\times S\rightarrow Y}
8:
2471:
2465:
1870:
1864:
1362:. See the book by Golubitsky and Guillemin.
972:{\displaystyle f_{s}\colon X\rightarrow Y}
2697:
2685:
2665:
2644:
2638:
2599:
2593:
2557:
2542:
2518:
2512:
2492:
2453:
2448:
2442:
2406:
2400:
2374:
2338:
2323:
2297:
2273:
2261:
2235:
2223:
2202:
2196:
2176:
2149:
2143:
2111:
2085:
2058:
2052:
2032:
2002:
1989:
1977:
1951:
1931:
1904:
1898:
1852:
1847:
1835:
1799:
1784:
1729:
1723:
1697:
1672:
1646:
1608:
1587:
1581:
1555:
1554:
1552:
1531:
1525:
1505:
1479:
1454:
1448:
1427:
1421:
1383:
1322:
1316:
1278:
1257:
1248:
1227:
1221:
1195:
1175:
1152:
1132:
1112:
1092:
1072:
1067:is a smooth map of manifolds, where only
1034:
1004:
984:
951:
945:
888:
882:
844:
791:
764:
755:
732:
697:
691:
662:
636:
613:
575:
555:
531:
508:
473:
467:
447:
427:
388:
361:
343:
323:
308:{\displaystyle x\in f^{-1}\left(Z\right)}
285:
273:
247:
227:
207:
187:
167:
135:
84:
52:
2757:Stable Mappings and Their Singularities
2633:then there exists an open dense subset
1882:{\displaystyle x\in f_{s}^{-1}(\{y\}).}
940:. This generates a family of mappings
679:{\displaystyle \partial f\pitchfork Z}
155:{\displaystyle f\colon X\rightarrow Y}
72:{\displaystyle f\colon X\rightarrow Y}
7:
1297:More general transversality theorems
720:{\displaystyle f^{-1}\left(Z\right)}
496:{\displaystyle f^{-1}\left(Z\right)}
1345:is a more powerful statement about
2093:
1959:
1532:
1455:
1250:
1154:
814:
757:
664:
615:
586:
577:
14:
2888:Theorems in differential topology
2826:Commentarii Mathematici Helvetici
2480:{\displaystyle f_{s}^{-1}(\{y\})}
1024:parametric transversality theorem
835:Parametric transversality theorem
2020:{\displaystyle F(x_{n},s_{n})=y}
1634:{\displaystyle F:X\times S\to Y}
1409:{\displaystyle F:X\times S\to Y}
1768:{\displaystyle f_{s}(x)=F(x,s)}
109:, it is the technical heart of
2611:
2605:
2569:
2563:
2474:
2462:
2418:
2412:
2350:
2344:
2164:{\displaystyle S_{0}\subset S}
2090:
2064:
2008:
1982:
1956:
1910:
1873:
1861:
1811:
1805:
1762:
1750:
1741:
1735:
1625:
1400:
1341:What is usually understood by
1266:{\displaystyle \partial f_{s}}
1051:
999:to be a (smooth) manifold and
963:
861:
592:
146:
63:
1:
1565:{\displaystyle \mathbb {K} .}
1343:Thom's transversality theorem
650:{\displaystyle f\pitchfork Z}
261:{\displaystyle f\pitchfork Z}
113:, and the starting point for
2755:; Guillemin, Victor (1974).
2099:{\displaystyle n\to \infty }
1965:{\displaystyle n\to \infty }
1366:Infinite-dimensional version
727:is a regular submanifold of
503:is a regular submanifold of
107:Pontryagin–Thom construction
98:{\displaystyle Z\subseteq Y}
2709:{\displaystyle s\in S_{0}.}
2626:{\displaystyle f_{s}(x)=y,}
2285:{\displaystyle s\in S_{0}.}
2247:{\displaystyle s\in S_{0}.}
1540:{\displaystyle C^{\infty }}
1470:-Banach manifolds. Assume:
1463:{\displaystyle C^{\infty }}
1331:{\displaystyle G_{\delta }}
268:, if and only if for every
29:Thom transversality theorem
2909:
2859:Zeidler, Eberhard (1997).
2430:{\displaystyle f_{s}(x)=y}
2073:{\displaystyle x_{n}\to x}
1919:{\displaystyle s_{n}\to s}
1520:are non-empty, metrizable
1163:{\displaystyle \partial F}
1127:without boundary. If both
624:{\displaystyle \partial f}
121:Finite-dimensional version
2588:for all the solutions of
2292:If there exists a number
1692:(iii) For each parameter
2777:; Pollack, Alan (1974).
2437:, then the solution set
1190:, then for almost every
2848:Bol. Soc. Mat. Mexicana
2311:{\displaystyle n\geq 0}
2128:{\displaystyle x\in X.}
1660:{\displaystyle k\geq 1}
1306:transversality theorems
2710:
2674:
2654:
2627:
2582:
2528:
2501:
2481:
2431:
2389:
2388:{\displaystyle x\in X}
2363:
2312:
2286:
2248:
2212:
2191:is a regular value of
2185:
2165:
2129:
2100:
2074:
2041:
2021:
1966:
1940:
1920:
1883:
1824:
1769:
1712:
1711:{\displaystyle s\in S}
1681:
1661:
1635:
1597:
1566:
1541:
1514:
1494:
1464:
1437:
1410:
1360:transversality theorem
1332:
1287:
1267:
1237:
1210:
1209:{\displaystyle s\in S}
1184:
1164:
1141:
1121:
1107:be any submanifold of
1101:
1087:has boundary, and let
1081:
1061:
1013:
993:
973:
934:
871:
824:
741:
721:
680:
651:
625:
602:
564:
548:manifold with boundary
540:
517:
497:
456:
436:
412:
309:
262:
236:
216:
196:
176:
156:
99:
73:
25:transversality theorem
2893:Differential geometry
2802:Differential Topology
2779:Differential Topology
2711:
2675:
2655:
2653:{\displaystyle S_{0}}
2628:
2583:
2529:
2527:{\displaystyle C^{k}}
2502:
2482:
2432:
2390:
2364:
2313:
2287:
2249:
2213:
2211:{\displaystyle f_{s}}
2186:
2166:
2130:
2101:
2075:
2042:
2022:
1967:
1941:
1921:
1893:(iv) The convergence
1884:
1825:
1770:
1713:
1682:
1662:
1636:
1598:
1596:{\displaystyle C^{k}}
1567:
1542:
1515:
1495:
1465:
1438:
1436:{\displaystyle C^{k}}
1411:
1333:
1288:
1268:
1238:
1236:{\displaystyle f_{s}}
1211:
1185:
1165:
1142:
1122:
1102:
1082:
1062:
1022:The statement of the
1014:
994:
974:
935:
872:
825:
742:
722:
681:
652:
626:
603:
565:
541:
518:
498:
457:
437:
413:
310:
263:
237:
217:
197:
177:
157:
100:
74:
21:differential topology
2684:
2664:
2637:
2592:
2541:
2511:
2491:
2441:
2399:
2373:
2322:
2296:
2260:
2256:Now, fix an element
2222:
2195:
2175:
2142:
2110:
2084:
2051:
2031:
1976:
1950:
1930:
1897:
1834:
1783:
1722:
1696:
1671:
1645:
1607:
1580:
1551:
1524:
1504:
1478:
1447:
1420:
1382:
1315:
1277:
1247:
1220:
1194:
1174:
1151:
1131:
1111:
1091:
1071:
1033:
1003:
983:
944:
881:
843:
754:
747:with boundary, and
731:
690:
661:
635:
612:
574:
570:to the boundary, as
554:
530:
507:
466:
446:
426:
322:
272:
246:
226:
206:
186:
182:be a submanifold of
166:
134:
126:Previous definitions
105:. Together with the
83:
51:
27:, also known as the
2759:. Springer-Verlag.
2727:Arnold, Vladimir I.
2461:
2218:for each parameter
1860:
1687:as a regular value.
1493:{\displaystyle X,S}
2839:10.1007/BF02566923
2753:Golubitsky, Martin
2706:
2670:
2650:
2623:
2578:
2524:
2497:
2477:
2444:
2427:
2385:
2369:for all solutions
2359:
2308:
2282:
2244:
2208:
2181:
2161:
2125:
2096:
2070:
2037:
2017:
1962:
1936:
1916:
1879:
1843:
1820:
1765:
1708:
1677:
1657:
1631:
1593:
1562:
1537:
1510:
1490:
1460:
1433:
1406:
1328:
1283:
1273:are transverse to
1263:
1233:
1206:
1180:
1170:are transverse to
1160:
1137:
1117:
1097:
1077:
1057:
1009:
989:
969:
930:
867:
820:
737:
717:
676:
647:
621:
598:
560:
536:
513:
493:
452:
432:
408:
305:
258:
232:
212:
192:
172:
152:
95:
69:
2797:Hirsch, Morris W.
2781:. Prentice-Hall.
2775:Guillemin, Victor
2673:{\displaystyle S}
2500:{\displaystyle n}
2184:{\displaystyle y}
2040:{\displaystyle n}
1939:{\displaystyle S}
1680:{\displaystyle y}
1513:{\displaystyle Y}
1286:{\displaystyle Z}
1183:{\displaystyle Z}
1140:{\displaystyle F}
1120:{\displaystyle Y}
1100:{\displaystyle Z}
1080:{\displaystyle X}
1012:{\displaystyle F}
992:{\displaystyle S}
839:Consider the map
740:{\displaystyle X}
563:{\displaystyle f}
539:{\displaystyle X}
516:{\displaystyle X}
455:{\displaystyle Z}
442:is transverse to
435:{\displaystyle f}
235:{\displaystyle Z}
222:is transverse to
215:{\displaystyle f}
195:{\displaystyle Y}
175:{\displaystyle Z}
47:: any smooth map
2900:
2874:
2855:
2842:
2816:
2792:
2770:
2748:
2736:
2715:
2713:
2712:
2707:
2702:
2701:
2679:
2677:
2676:
2671:
2659:
2657:
2656:
2651:
2649:
2648:
2632:
2630:
2629:
2624:
2604:
2603:
2587:
2585:
2584:
2579:
2562:
2561:
2533:
2531:
2530:
2525:
2523:
2522:
2506:
2504:
2503:
2498:
2486:
2484:
2483:
2478:
2460:
2452:
2436:
2434:
2433:
2428:
2411:
2410:
2394:
2392:
2391:
2386:
2368:
2366:
2365:
2360:
2343:
2342:
2317:
2315:
2314:
2309:
2291:
2289:
2288:
2283:
2278:
2277:
2253:
2251:
2250:
2245:
2240:
2239:
2217:
2215:
2214:
2209:
2207:
2206:
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2154:
2153:
2134:
2132:
2131:
2126:
2105:
2103:
2102:
2097:
2079:
2077:
2076:
2071:
2063:
2062:
2046:
2044:
2043:
2038:
2026:
2024:
2023:
2018:
2007:
2006:
1994:
1993:
1971:
1969:
1968:
1963:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1909:
1908:
1888:
1886:
1885:
1880:
1859:
1851:
1829:
1827:
1826:
1821:
1804:
1803:
1774:
1772:
1771:
1766:
1734:
1733:
1717:
1715:
1714:
1709:
1686:
1684:
1683:
1678:
1666:
1664:
1663:
1658:
1640:
1638:
1637:
1632:
1602:
1600:
1599:
1594:
1592:
1591:
1571:
1569:
1568:
1563:
1558:
1546:
1544:
1543:
1538:
1536:
1535:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1469:
1467:
1466:
1461:
1459:
1458:
1442:
1440:
1439:
1434:
1432:
1431:
1415:
1413:
1412:
1407:
1374:Formal statement
1337:
1335:
1334:
1329:
1327:
1326:
1292:
1290:
1289:
1284:
1272:
1270:
1269:
1264:
1262:
1261:
1242:
1240:
1239:
1234:
1232:
1231:
1215:
1213:
1212:
1207:
1189:
1187:
1186:
1181:
1169:
1167:
1166:
1161:
1146:
1144:
1143:
1138:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1066:
1064:
1063:
1058:
1018:
1016:
1015:
1010:
998:
996:
995:
990:
978:
976:
975:
970:
956:
955:
939:
937:
936:
931:
929:
925:
904:
893:
892:
876:
874:
873:
868:
829:
827:
826:
821:
810:
799:
798:
783:
772:
771:
746:
744:
743:
738:
726:
724:
723:
718:
716:
705:
704:
685:
683:
682:
677:
656:
654:
653:
648:
630:
628:
627:
622:
607:
605:
604:
599:
569:
567:
566:
561:
545:
543:
542:
537:
522:
520:
519:
514:
502:
500:
499:
494:
492:
481:
480:
461:
459:
458:
453:
441:
439:
438:
433:
417:
415:
414:
409:
404:
403:
402:
377:
376:
375:
353:
349:
348:
347:
314:
312:
311:
306:
304:
293:
292:
267:
265:
264:
259:
241:
239:
238:
233:
221:
219:
218:
213:
201:
199:
198:
193:
181:
179:
178:
173:
161:
159:
158:
153:
111:cobordism theory
104:
102:
101:
96:
78:
76:
75:
70:
45:generic property
2908:
2907:
2903:
2902:
2901:
2899:
2898:
2897:
2878:
2877:
2871:
2858:
2845:
2819:
2813:
2795:
2789:
2773:
2767:
2751:
2745:
2725:
2722:
2693:
2682:
2681:
2662:
2661:
2640:
2635:
2634:
2595:
2590:
2589:
2553:
2539:
2538:
2514:
2509:
2508:
2489:
2488:
2487:consists of an
2439:
2438:
2402:
2397:
2396:
2371:
2370:
2334:
2320:
2319:
2294:
2293:
2269:
2258:
2257:
2231:
2220:
2219:
2198:
2193:
2192:
2173:
2172:
2145:
2140:
2139:
2108:
2107:
2082:
2081:
2054:
2049:
2048:
2029:
2028:
1998:
1985:
1974:
1973:
1948:
1947:
1928:
1927:
1900:
1895:
1894:
1832:
1831:
1795:
1781:
1780:
1725:
1720:
1719:
1694:
1693:
1669:
1668:
1643:
1642:
1605:
1604:
1583:
1578:
1577:
1549:
1548:
1527:
1522:
1521:
1502:
1501:
1476:
1475:
1450:
1445:
1444:
1423:
1418:
1417:
1380:
1379:
1376:
1368:
1318:
1313:
1312:
1299:
1275:
1274:
1253:
1245:
1244:
1223:
1218:
1217:
1192:
1191:
1172:
1171:
1149:
1148:
1129:
1128:
1109:
1108:
1089:
1088:
1069:
1068:
1031:
1030:
1001:
1000:
981:
980:
947:
942:
941:
915:
911:
894:
884:
879:
878:
841:
840:
837:
800:
787:
773:
760:
752:
751:
729:
728:
706:
693:
688:
687:
659:
658:
633:
632:
610:
609:
572:
571:
552:
551:
528:
527:
505:
504:
482:
469:
464:
463:
444:
443:
424:
423:
392:
384:
365:
357:
339:
335:
331:
320:
319:
294:
281:
270:
269:
244:
243:
224:
223:
204:
203:
184:
183:
164:
163:
132:
131:
128:
123:
81:
80:
49:
48:
17:
12:
11:
5:
2906:
2904:
2896:
2895:
2890:
2880:
2879:
2876:
2875:
2869:
2856:
2843:
2817:
2811:
2793:
2787:
2771:
2765:
2749:
2743:
2721:
2718:
2705:
2700:
2696:
2692:
2689:
2669:
2647:
2643:
2622:
2619:
2616:
2613:
2610:
2607:
2602:
2598:
2577:
2574:
2571:
2568:
2565:
2560:
2556:
2552:
2549:
2546:
2521:
2517:
2496:
2476:
2473:
2470:
2467:
2464:
2459:
2456:
2451:
2447:
2426:
2423:
2420:
2417:
2414:
2409:
2405:
2384:
2381:
2378:
2358:
2355:
2352:
2349:
2346:
2341:
2337:
2333:
2330:
2327:
2307:
2304:
2301:
2281:
2276:
2272:
2268:
2265:
2243:
2238:
2234:
2230:
2227:
2205:
2201:
2180:
2160:
2157:
2152:
2148:
2136:
2135:
2124:
2121:
2118:
2115:
2095:
2092:
2089:
2069:
2066:
2061:
2057:
2036:
2016:
2013:
2010:
2005:
2001:
1997:
1992:
1988:
1984:
1981:
1961:
1958:
1955:
1935:
1915:
1912:
1907:
1903:
1890:
1889:
1878:
1875:
1872:
1869:
1866:
1863:
1858:
1855:
1850:
1846:
1842:
1839:
1819:
1816:
1813:
1810:
1807:
1802:
1798:
1794:
1791:
1788:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1732:
1728:
1707:
1704:
1701:
1689:
1688:
1676:
1656:
1653:
1650:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1590:
1586:
1573:
1572:
1561:
1557:
1534:
1530:
1509:
1489:
1486:
1483:
1457:
1453:
1430:
1426:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1375:
1372:
1367:
1364:
1325:
1321:
1298:
1295:
1282:
1260:
1256:
1252:
1230:
1226:
1205:
1202:
1199:
1179:
1159:
1156:
1136:
1116:
1096:
1076:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1019:to be smooth.
1008:
988:
968:
965:
962:
959:
954:
950:
928:
924:
921:
918:
914:
910:
907:
903:
900:
897:
891:
887:
866:
863:
860:
857:
854:
851:
848:
836:
833:
832:
831:
819:
816:
813:
809:
806:
803:
797:
794:
790:
786:
782:
779:
776:
770:
767:
763:
759:
736:
715:
712:
709:
703:
700:
696:
675:
672:
669:
666:
646:
643:
640:
620:
617:
597:
594:
591:
588:
585:
582:
579:
559:
535:
512:
491:
488:
485:
479:
476:
472:
451:
431:
420:
419:
407:
401:
398:
395:
391:
387:
383:
380:
374:
371:
368:
364:
360:
356:
352:
346:
342:
338:
334:
330:
327:
315:we have that
303:
300:
297:
291:
288:
284:
280:
277:
257:
254:
251:
231:
211:
202:. We say that
191:
171:
151:
148:
145:
142:
139:
127:
124:
122:
119:
115:surgery theory
94:
91:
88:
68:
65:
62:
59:
56:
41:transversality
35:mathematician
15:
13:
10:
9:
6:
4:
3:
2:
2905:
2894:
2891:
2889:
2886:
2885:
2883:
2872:
2870:0-387-96499-1
2866:
2862:
2857:
2853:
2849:
2844:
2840:
2836:
2832:
2828:
2827:
2822:
2818:
2814:
2812:0-387-90148-5
2808:
2804:
2803:
2798:
2794:
2790:
2788:0-13-212605-2
2784:
2780:
2776:
2772:
2768:
2766:0-387-90073-X
2762:
2758:
2754:
2750:
2746:
2744:0-387-96649-8
2740:
2735:
2734:
2728:
2724:
2723:
2719:
2717:
2703:
2698:
2694:
2690:
2687:
2667:
2645:
2641:
2620:
2617:
2614:
2608:
2600:
2596:
2575:
2572:
2566:
2558:
2554:
2550:
2547:
2544:
2537:Note that if
2535:
2519:
2515:
2507:-dimensional
2494:
2468:
2457:
2454:
2449:
2445:
2424:
2421:
2415:
2407:
2403:
2382:
2379:
2376:
2356:
2353:
2347:
2339:
2335:
2331:
2328:
2325:
2305:
2302:
2299:
2279:
2274:
2270:
2266:
2263:
2254:
2241:
2236:
2232:
2228:
2225:
2203:
2199:
2178:
2158:
2155:
2150:
2146:
2122:
2119:
2116:
2113:
2087:
2067:
2059:
2055:
2034:
2014:
2011:
2003:
1999:
1995:
1990:
1986:
1979:
1953:
1933:
1913:
1905:
1901:
1892:
1891:
1876:
1867:
1856:
1853:
1848:
1844:
1840:
1837:
1817:
1814:
1808:
1800:
1796:
1792:
1789:
1786:
1778:
1759:
1756:
1753:
1747:
1744:
1738:
1730:
1726:
1705:
1702:
1699:
1691:
1690:
1674:
1654:
1651:
1648:
1628:
1622:
1619:
1616:
1613:
1610:
1588:
1584:
1575:
1574:
1559:
1528:
1507:
1487:
1484:
1481:
1473:
1472:
1471:
1451:
1428:
1424:
1403:
1397:
1394:
1391:
1388:
1385:
1373:
1371:
1365:
1363:
1361:
1359:
1354:
1350:
1348:
1344:
1339:
1323:
1319:
1309:
1307:
1302:
1296:
1294:
1280:
1258:
1254:
1228:
1224:
1203:
1200:
1197:
1177:
1157:
1134:
1114:
1094:
1074:
1054:
1048:
1045:
1042:
1039:
1036:
1029:Suppose that
1027:
1025:
1020:
1006:
986:
966:
960:
957:
952:
948:
926:
922:
919:
916:
912:
908:
905:
901:
898:
895:
889:
885:
864:
858:
855:
852:
849:
846:
834:
817:
811:
807:
804:
801:
795:
792:
788:
784:
780:
777:
774:
768:
765:
761:
750:
749:
748:
734:
713:
710:
707:
701:
698:
694:
673:
670:
667:
644:
641:
638:
618:
595:
589:
583:
580:
557:
549:
533:
524:
510:
489:
486:
483:
477:
474:
470:
449:
429:
405:
399:
396:
393:
389:
385:
381:
378:
372:
369:
366:
362:
358:
354:
350:
344:
340:
336:
332:
328:
325:
318:
317:
316:
301:
298:
295:
289:
286:
282:
278:
275:
255:
252:
249:
242:, denoted as
229:
209:
189:
169:
149:
143:
140:
137:
125:
120:
118:
116:
112:
108:
92:
89:
86:
66:
60:
57:
54:
46:
42:
38:
34:
30:
26:
22:
2863:. Springer.
2860:
2851:
2847:
2833:(1): 17–86.
2830:
2824:
2805:. Springer.
2801:
2778:
2756:
2737:. Springer.
2732:
2536:
2255:
2137:
1777:Fredholm map
1377:
1369:
1356:
1351:
1342:
1340:
1310:
1305:
1303:
1300:
1028:
1023:
1021:
838:
525:
421:
129:
28:
24:
18:
2854:(1): 59–71.
1353:John Mather
877:and define
2882:Categories
2821:Thom, René
2720:References
2171:such that
1830:for every
1718:, the map
608:. The map
2691:∈
2548:
2455:−
2380:∈
2329:
2303:≥
2267:∈
2229:∈
2156:⊂
2117:∈
2094:∞
2091:→
2065:→
1960:∞
1957:→
1911:→
1854:−
1841:∈
1790:
1703:∈
1652:≥
1626:→
1620:×
1576:(ii) The
1533:∞
1456:∞
1401:→
1395:×
1324:δ
1251:∂
1201:∈
1155:∂
1052:→
1046:×
1040::
964:→
958::
862:→
856:×
850::
815:∂
812:∩
793:−
766:−
758:∂
699:−
671:⋔
665:∂
642:⋔
616:∂
593:→
587:∂
584::
578:∂
475:−
329:
287:−
279:∈
253:⋔
147:→
141::
90:⊆
64:→
58::
37:René Thom
2799:(1976).
2729:(1988).
2027:for all
1779:, where
1378:Suppose
1358:multijet
1443:map of
1216:, both
686:, then
462:, then
2867:
2809:
2785:
2763:
2741:
33:French
31:after
23:, the
2318:with
2106:with
1775:is a
1641:with
1603:-map
1416:is a
546:is a
43:is a
2865:ISBN
2807:ISBN
2783:ISBN
2761:ISBN
2739:ISBN
1972:and
1815:<
1667:has
1500:and
1474:(i)
1243:and
1147:and
1026:is:
657:and
130:Let
2835:doi
2660:of
2545:ind
2395:of
2326:ind
2080:as
1946:as
1926:on
1787:ind
1347:jet
526:If
19:In
2884::
2850:.
2831:28
2829:.
1293:.
523:.
326:im
2873:.
2852:2
2841:.
2837::
2815:.
2791:.
2769:.
2747:.
2704:.
2699:0
2695:S
2688:s
2668:S
2646:0
2642:S
2621:,
2618:y
2615:=
2612:)
2609:x
2606:(
2601:s
2597:f
2576:0
2573:=
2570:)
2567:x
2564:(
2559:s
2555:f
2551:D
2520:k
2516:C
2495:n
2475:)
2472:}
2469:y
2466:{
2463:(
2458:1
2450:s
2446:f
2425:y
2422:=
2419:)
2416:x
2413:(
2408:s
2404:f
2383:X
2377:x
2357:n
2354:=
2351:)
2348:x
2345:(
2340:s
2336:f
2332:D
2306:0
2300:n
2280:.
2275:0
2271:S
2264:s
2242:.
2237:0
2233:S
2226:s
2204:s
2200:f
2179:y
2159:S
2151:0
2147:S
2123:.
2120:X
2114:x
2088:n
2068:x
2060:n
2056:x
2035:n
2015:y
2012:=
2009:)
2004:n
2000:s
1996:,
1991:n
1987:x
1983:(
1980:F
1954:n
1934:S
1914:s
1906:n
1902:s
1877:.
1874:)
1871:}
1868:y
1865:{
1862:(
1857:1
1849:s
1845:f
1838:x
1818:k
1812:)
1809:x
1806:(
1801:s
1797:f
1793:D
1763:)
1760:s
1757:,
1754:x
1751:(
1748:F
1745:=
1742:)
1739:x
1736:(
1731:s
1727:f
1706:S
1700:s
1675:y
1655:1
1649:k
1629:Y
1623:S
1617:X
1614::
1611:F
1589:k
1585:C
1560:.
1556:K
1529:C
1508:Y
1488:S
1485:,
1482:X
1452:C
1429:k
1425:C
1404:Y
1398:S
1392:X
1389::
1386:F
1320:G
1281:Z
1259:s
1255:f
1229:s
1225:f
1204:S
1198:s
1178:Z
1158:F
1135:F
1115:Y
1095:Z
1075:X
1055:Y
1049:S
1043:X
1037:F
1007:F
987:S
967:Y
961:X
953:s
949:f
927:)
923:s
920:,
917:x
913:(
909:F
906:=
902:)
899:x
896:(
890:s
886:f
865:Y
859:S
853:X
847:F
830:.
818:X
808:)
805:Z
802:(
796:1
789:f
785:=
781:)
778:Z
775:(
769:1
762:f
735:X
714:)
711:Z
708:(
702:1
695:f
674:Z
668:f
645:Z
639:f
619:f
596:Y
590:X
581:f
558:f
534:X
511:X
490:)
487:Z
484:(
478:1
471:f
450:Z
430:f
418:.
406:Y
400:)
397:x
394:(
390:f
386:T
382:=
379:Z
373:)
370:x
367:(
363:f
359:T
355:+
351:)
345:x
341:f
337:d
333:(
302:)
299:Z
296:(
290:1
283:f
276:x
256:Z
250:f
230:Z
210:f
190:Y
170:Z
150:Y
144:X
138:f
93:Y
87:Z
67:Y
61:X
55:f
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