266:
152:
2345:-handles. Whereas handle decompositions are the analogue for manifolds what cell decompositions are to topological spaces, handle presentations of cobordisms are to manifolds with boundary what relative cell decompositions are for pairs of spaces.
2690:
2184:
1710:
1345:
1914:
1523:
2543:
38:
771:
601:
556:
2857:
1008:
3008:
2453:
3314:
1771:
3081:
2097:
885:
2548:
1067:
416:
The problem with CW-decompositions is that the attaching maps for cells do not live in the world of smooth maps between manifolds. The germinal insight to correct this defect is the
3392:
1220:
411:
365:
3115:
2388:
1421:
3435:
1985:
1107:
810:
726:
640:
915:
2279:
3458:
2936:
1243:
3340:
2339:
2102:
1613:
1248:
212:
2759:
2234:
3559:
3532:
3233:
2892:
2306:
2022:
1945:
1609:
1368:
1168:
837:
687:
511:
480:
453:
315:
179:
3117:
which is Morse on the interior and constant on the boundary and satisfying the increasing index property, there is an induced handle presentation of the cobordism
3478:
3032:
1797:
1141:
660:
290:-cell. From the point of view of smooth manifolds, this is a degenerate decomposition of the sphere, as there is no natural way to see the smooth structure of
232:
242:
is to a topological space—in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of
1816:
1426:
558:
glued along their common boundary. The vital issue here is that the gluing map is a diffeomorphism. Similarly, take a smooth embedded arc in
1556:-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (i.e.: 0-handles and
3689:
1552:. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only
3585:-sphere. In the oriented case, it's conventional to reduce this framed link to a framed embedding of a disjoint union of circles.
2462:
3680:
731:
561:
516:
2764:
946:
2948:
2395:
3014:, moreover, every manifold has such Morse functions, so they have handle decompositions. Similarly, given a cobordism
3704:
2400:
3253:
1716:
147:{\displaystyle \emptyset =M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset \dots \subset M_{m-1}\subset M_{m}=M}
3161:-handlebodies along their common boundary, called the Heegaard splitting surface. Heegaard splittings arise for
1348:
3037:
2053:
842:
3194:
1013:
3345:
1173:
3627:
370:
324:
3086:
2359:
1373:
3400:
1950:
1072:
918:
776:
692:
606:
417:
890:
3240:
3189:-handlebody (from the point of view of the dual decomposition), thus a Heegaard splitting. If the
3165:-manifolds in several natural ways: given a handle decomposition of a 3-manifold, the union of the
2685:{\displaystyle t_{0}<f(p_{1})<t_{1}<f(p_{2})<\cdots <t_{k-1}<f(p_{k})<t_{k},}
2239:
3150:
3440:
2904:
1225:
3136:
is also a Morse function. The corresponding handle decomposition / presentation is called the
3685:
3319:
2311:
184:
3607:
2699:
2209:
3537:
3510:
3211:
2870:
2284:
2000:
1923:
1587:
1353:
1146:
815:
665:
489:
458:
431:
293:
265:
157:
243:
3622:
3481:
3463:
3017:
2939:
1782:
1126:
645:
217:
3698:
3602:
1808:
1804:
925:
3671:
2354:
255:
3651:
S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
3589:
2179:{\displaystyle W_{-1}\subset W_{0}\subset W_{1}\subset \cdots \subset W_{m+1}=W}
1705:{\displaystyle M\cup _{f}H^{j}=\left(M\sqcup (D^{j}\times D^{m-j})\right)/\sim }
1340:{\displaystyle M\cup _{f}H^{j}=\left(M\sqcup (D^{j}\times D^{m-j})\right)/\sim }
317:
from the eyes of this decomposition—in particular the smooth structure near the
259:
17:
3617:
3612:
1565:
279:
239:
3581:. Thus every 3-manifold can be obtained via surgery on framed links in the
3578:
662:
as the union of three manifolds, glued along parts of their boundaries: 1)
1909:{\displaystyle \{0\}^{j}\times S^{m-j-1}\subset D^{j}\times D^{m-j}=H^{j}}
1518:{\displaystyle (p,x)\in S^{j-1}\times D^{m-j}\subset D^{j}\times D^{m-j}}
773:. Notice all the gluing maps are smooth maps—in particular when we glue
29:
2901:
refers to the dimension of the maximal subspace of the tangent space
728:
and 3) the complement of the open tubular neighbourhood of the arc in
3592:
is proven by simplifying handle decompositions of smooth manifolds.
264:
3342:, i.e.: this manifold is diffeomorphic to a manifold of the form
254:-cell. Handle decompositions of manifolds arise naturally via
3316:, it is possible to switch the order of attachment, provided
258:. The modification of handle structures is closely linked to
3668:
Vol 138 Pure and
Applied Mathematics, Academic Press (1992).
3153:
of a closed, orientable 3-manifold is a decomposition of a
2538:{\displaystyle f(p_{1})<f(p_{2})<\cdots <f(p_{k})}
839:
the equivalence relation is generated by the embedding of
3200:, there is an induced Heegaard splitting where the first
321:-cell depends on the behavior of the characteristic map
766:{\displaystyle M\setminus \operatorname {int} (N_{p})}
596:{\displaystyle M\setminus \operatorname {int} (N_{p})}
551:{\displaystyle M\setminus \operatorname {int} (N_{p})}
3540:
3513:
3466:
3443:
3403:
3348:
3322:
3256:
3214:
3089:
3040:
3020:
2951:
2907:
2873:
2852:{\displaystyle (f^{-1}(t_{j-1})\times )\cup H^{I(j)}}
2767:
2702:
2551:
2465:
2403:
2362:
2314:
2287:
2242:
2212:
2105:
2056:
2003:
1953:
1926:
1819:
1785:
1719:
1616:
1590:
1429:
1376:
1356:
1251:
1228:
1176:
1149:
1129:
1075:
1016:
1003:{\displaystyle f:S^{j-1}\times D^{m-j}\to \partial M}
949:
893:
845:
818:
779:
734:
695:
668:
648:
609:
564:
519:
492:
461:
434:
373:
327:
296:
220:
187:
160:
41:
238:. A handle decomposition is to a manifold what a
3553:
3526:
3472:
3452:
3429:
3386:
3334:
3308:
3227:
3109:
3075:
3026:
3003:{\displaystyle I(1)\leq I(2)\leq \cdots \leq I(k)}
3002:
2930:
2886:
2851:
2753:
2684:
2537:
2447:
2382:
2333:
2300:
2273:
2228:
2178:
2091:
2016:
1979:
1939:
1908:
1791:
1765:
1704:
1603:
1517:
1415:
1362:
1339:
1237:
1214:
1162:
1135:
1101:
1061:
1002:
909:
879:
831:
804:
765:
720:
681:
654:
634:
595:
550:
505:
474:
447:
405:
359:
309:
226:
206:
173:
146:
3684:), American Mathematical Society, Providence, RI
2448:{\displaystyle \{p_{1},\ldots ,p_{k}\}\subset M}
603:, its tubular neighbourhood is diffeomorphic to
3309:{\displaystyle (M\cup _{f}H^{i})\cup _{g}H^{j}}
1766:{\displaystyle f(S^{j-1}\times \{0\})\subset M}
3534:by surgery on a collection of framed links in
3239:-handlebody is a regular neighbourhood of the
3204:-handlebody is a regular neighbourhood of the
8:
2436:
2404:
1827:
1820:
1751:
1745:
943:assumes that one has a smooth embedding of
3076:{\displaystyle \partial W=M_{0}\cup M_{1}}
2092:{\displaystyle \partial W=M_{0}\cup M_{1}}
924:Handle decompositions are an invention of
880:{\displaystyle (\partial I)\times D^{m-1}}
3545:
3539:
3518:
3512:
3465:
3442:
3421:
3411:
3402:
3378:
3362:
3347:
3321:
3300:
3290:
3277:
3267:
3255:
3250:When attaching two handles in succession
3219:
3213:
3103:
3102:
3088:
3067:
3054:
3039:
3019:
2950:
2917:
2912:
2906:
2878:
2872:
2834:
2791:
2775:
2766:
2742:
2723:
2707:
2701:
2673:
2657:
2632:
2610:
2591:
2575:
2556:
2550:
2526:
2498:
2476:
2464:
2430:
2411:
2402:
2376:
2375:
2361:
2319:
2313:
2292:
2286:
2247:
2241:
2217:
2211:
2158:
2139:
2126:
2110:
2104:
2083:
2070:
2055:
2008:
2002:
1971:
1961:
1952:
1931:
1925:
1900:
1881:
1868:
1843:
1830:
1818:
1784:
1730:
1718:
1694:
1674:
1661:
1634:
1624:
1615:
1595:
1589:
1503:
1490:
1471:
1452:
1428:
1375:
1355:
1329:
1309:
1296:
1269:
1259:
1250:
1227:
1200:
1181:
1175:
1154:
1148:
1128:
1093:
1083:
1074:
1062:{\displaystyle H^{j}=D^{j}\times D^{m-j}}
1047:
1034:
1021:
1015:
979:
960:
948:
901:
892:
865:
844:
823:
817:
790:
778:
754:
733:
706:
694:
673:
667:
647:
620:
608:
584:
563:
539:
518:
497:
491:
466:
460:
439:
433:
397:
378:
372:
351:
338:
326:
301:
295:
286:-sphere, with one zero cell and a single
219:
192:
186:
165:
159:
132:
113:
94:
81:
68:
52:
40:
1803:of the attaching sphere, since it gives
3644:
3569:-manifold (similarly oriented and spin
3387:{\displaystyle (M\cup H^{j})\cup H^{i}}
738:
568:
523:
269:A 3-ball with three 1-handles attached.
3561:. For example, it's known that every
2867:) is the index of the critical point
1215:{\displaystyle S^{j-1}\times D^{m-j}}
250:-handle is the smooth analogue of an
7:
3480:. This is the primary link between
3144:Some major theorems and observations
406:{\displaystyle S^{n-1}\subset D^{n}}
360:{\displaystyle \chi :D^{n}\to S^{n}}
3573:-manifolds bound oriented and spin
3110:{\displaystyle f:W\to \mathbb {R} }
2390:on a compact boundaryless manifold
2383:{\displaystyle f:M\to \mathbb {R} }
428:, its closed tubular neighbourhood
3444:
3177:-handlebody, and the union of the
3041:
3010:this is a handle decomposition of
2057:
2044:handle presentation of a cobordism
1229:
1123:) refers to the disjoint union of
994:
894:
849:
42:
14:
3460:surgered along the framed sphere
1990:A manifold obtained by attaching
3577:-manifolds respectively) due to
3157:-manifold into the union of two
1416:{\displaystyle (p,x)\sim f(p,x)}
928:. In his original formulation,
3681:Graduate Studies in Mathematics
3430:{\displaystyle M\cup _{f}H^{j}}
1980:{\displaystyle M\cup _{f}H^{j}}
1102:{\displaystyle M\cup _{f}H^{j}}
805:{\displaystyle I\times D^{m-1}}
721:{\displaystyle I\times D^{m-1}}
635:{\displaystyle I\times D^{m-1}}
3676:4-Manifolds and Kirby Calculus
3484:, handles and Morse functions.
3368:
3349:
3283:
3257:
3099:
2997:
2991:
2976:
2970:
2961:
2955:
2844:
2838:
2824:
2821:
2809:
2803:
2784:
2768:
2748:
2716:
2663:
2650:
2616:
2603:
2581:
2568:
2532:
2519:
2504:
2491:
2482:
2469:
2372:
2268:
2256:
1754:
1723:
1686:
1654:
1442:
1430:
1410:
1398:
1389:
1377:
1321:
1289:
991:
910:{\displaystyle \partial D^{m}}
855:
846:
760:
747:
590:
577:
545:
532:
344:
1:
3579:René Thom's work on cobordism
2945:Provided the indices satisfy
1548:-handles is diffeomorphic to
919:tubular neighbourhood theorem
418:tubular neighbourhood theorem
3394:for suitable attaching maps.
2274:{\displaystyle M_{0}\times }
1170:with the identification of
930:the process of attaching a
642:. This allows us to write
486:into the disjoint union of
3721:
3453:{\displaystyle \partial M}
2931:{\displaystyle T_{p_{j}}M}
1238:{\displaystyle \partial M}
482:, thus we have decomposed
2349:Morse theoretic viewpoint
1540:-handles if the union of
917:, which is smooth by the
2046:consists of a cobordism
1799:is sometimes called the
1560:-handles for some fixed
3678:, (1999) (Volume 20 in
3335:{\displaystyle j\leq i}
3128:is a Morse function on
2334:{\displaystyle W_{i-1}}
2099:and an ascending union
2038:Cobordism presentations
207:{\displaystyle M_{i-1}}
3666:Differential Manifolds
3628:Manifold decomposition
3555:
3528:
3495:is the boundary of an
3474:
3454:
3431:
3388:
3336:
3310:
3229:
3111:
3077:
3028:
3004:
2942:is negative definite.
2932:
2888:
2853:
2755:
2754:{\displaystyle f^{-1}}
2686:
2539:
2449:
2384:
2335:
2302:
2275:
2230:
2229:{\displaystyle W_{-1}}
2180:
2093:
2018:
1981:
1941:
1910:
1793:
1767:
1706:
1605:
1519:
1417:
1364:
1341:
1239:
1216:
1164:
1137:
1103:
1063:
1004:
911:
881:
833:
806:
767:
722:
683:
656:
636:
597:
552:
507:
476:
449:
407:
367:in a neighbourhood of
361:
311:
278:Consider the standard
270:
228:
208:
175:
148:
3674:and Andras Stipsicz,
3556:
3554:{\displaystyle S^{m}}
3529:
3527:{\displaystyle S^{m}}
3507:can be obtained from
3487:As a consequence, an
3475:
3455:
3432:
3389:
3337:
3311:
3230:
3228:{\displaystyle T^{1}}
3112:
3078:
3029:
3005:
2933:
2889:
2887:{\displaystyle p_{j}}
2854:
2756:
2687:
2540:
2450:
2385:
2341:by the attachment of
2336:
2303:
2301:{\displaystyle W_{i}}
2276:
2231:
2181:
2094:
2029:-handlebody of genus
2019:
2017:{\displaystyle D^{m}}
1997:-handles to the disc
1982:
1942:
1940:{\displaystyle H^{j}}
1911:
1794:
1768:
1707:
1606:
1604:{\displaystyle H^{j}}
1520:
1418:
1365:
1363:{\displaystyle \sim }
1342:
1240:
1217:
1165:
1163:{\displaystyle H^{j}}
1138:
1104:
1064:
1005:
912:
882:
834:
832:{\displaystyle D^{m}}
807:
768:
723:
684:
682:{\displaystyle D^{m}}
657:
637:
598:
553:
508:
506:{\displaystyle N_{p}}
477:
475:{\displaystyle D^{m}}
450:
448:{\displaystyle N_{p}}
408:
362:
312:
310:{\displaystyle S^{n}}
268:
229:
209:
176:
174:{\displaystyle M_{i}}
149:
3538:
3511:
3464:
3441:
3437:is diffeomorphic to
3401:
3346:
3320:
3254:
3212:
3087:
3038:
3018:
2949:
2905:
2871:
2765:
2761:is diffeomorphic to
2700:
2549:
2463:
2401:
2360:
2312:
2285:
2240:
2236:is diffeomorphic to
2210:
2103:
2054:
2001:
1951:
1924:
1817:
1783:
1717:
1614:
1588:
1528:One says a manifold
1427:
1374:
1354:
1349:equivalence relation
1249:
1226:
1174:
1147:
1127:
1073:
1014:
947:
891:
843:
816:
777:
732:
693:
666:
646:
607:
562:
517:
490:
459:
455:is diffeomorphic to
432:
371:
325:
294:
218:
214:by the attaching of
185:
158:
39:
22:handle decomposition
3590:H-cobordism theorem
3565:-manifold bounds a
3185:-handles is also a
1544:with finitely many
3705:Geometric topology
3659:General references
3551:
3524:
3470:
3450:
3427:
3384:
3332:
3306:
3225:
3151:Heegaard splitting
3138:dual decomposition
3107:
3073:
3024:
3000:
2928:
2884:
2849:
2751:
2682:
2535:
2445:
2380:
2331:
2298:
2271:
2226:
2176:
2089:
2014:
1977:
1937:
1906:
1789:
1763:
1702:
1601:
1515:
1413:
1360:
1337:
1235:
1222:with its image in
1212:
1160:
1133:
1099:
1059:
1000:
907:
877:
829:
802:
763:
718:
679:
652:
632:
593:
548:
503:
472:
445:
403:
357:
307:
271:
224:
204:
171:
144:
3473:{\displaystyle f}
3027:{\displaystyle W}
2308:is obtained from
1792:{\displaystyle f}
1532:is obtained from
1136:{\displaystyle M}
655:{\displaystyle M}
420:. Given a point
227:{\displaystyle i}
181:is obtained from
3712:
3652:
3649:
3608:Cobordism theory
3560:
3558:
3557:
3552:
3550:
3549:
3533:
3531:
3530:
3525:
3523:
3522:
3479:
3477:
3476:
3471:
3459:
3457:
3456:
3451:
3436:
3434:
3433:
3428:
3426:
3425:
3416:
3415:
3397:The boundary of
3393:
3391:
3390:
3385:
3383:
3382:
3367:
3366:
3341:
3339:
3338:
3333:
3315:
3313:
3312:
3307:
3305:
3304:
3295:
3294:
3282:
3281:
3272:
3271:
3235:, and the other
3234:
3232:
3231:
3226:
3224:
3223:
3193:-manifold has a
3116:
3114:
3113:
3108:
3106:
3082:
3080:
3079:
3074:
3072:
3071:
3059:
3058:
3033:
3031:
3030:
3025:
3009:
3007:
3006:
3001:
2937:
2935:
2934:
2929:
2924:
2923:
2922:
2921:
2893:
2891:
2890:
2885:
2883:
2882:
2858:
2856:
2855:
2850:
2848:
2847:
2802:
2801:
2783:
2782:
2760:
2758:
2757:
2752:
2747:
2746:
2734:
2733:
2715:
2714:
2691:
2689:
2688:
2683:
2678:
2677:
2662:
2661:
2643:
2642:
2615:
2614:
2596:
2595:
2580:
2579:
2561:
2560:
2544:
2542:
2541:
2536:
2531:
2530:
2503:
2502:
2481:
2480:
2454:
2452:
2451:
2446:
2435:
2434:
2416:
2415:
2394:, such that the
2389:
2387:
2386:
2381:
2379:
2340:
2338:
2337:
2332:
2330:
2329:
2307:
2305:
2304:
2299:
2297:
2296:
2280:
2278:
2277:
2272:
2252:
2251:
2235:
2233:
2232:
2227:
2225:
2224:
2201:
2195:
2191:
2185:
2183:
2182:
2177:
2169:
2168:
2144:
2143:
2131:
2130:
2118:
2117:
2098:
2096:
2095:
2090:
2088:
2087:
2075:
2074:
2023:
2021:
2020:
2015:
2013:
2012:
1986:
1984:
1983:
1978:
1976:
1975:
1966:
1965:
1946:
1944:
1943:
1938:
1936:
1935:
1915:
1913:
1912:
1907:
1905:
1904:
1892:
1891:
1873:
1872:
1860:
1859:
1835:
1834:
1798:
1796:
1795:
1790:
1775:attaching sphere
1773:is known as the
1772:
1770:
1769:
1764:
1741:
1740:
1711:
1709:
1708:
1703:
1698:
1693:
1689:
1685:
1684:
1666:
1665:
1639:
1638:
1629:
1628:
1610:
1608:
1607:
1602:
1600:
1599:
1524:
1522:
1521:
1516:
1514:
1513:
1495:
1494:
1482:
1481:
1463:
1462:
1422:
1420:
1419:
1414:
1370:is generated by
1369:
1367:
1366:
1361:
1346:
1344:
1343:
1338:
1333:
1328:
1324:
1320:
1319:
1301:
1300:
1274:
1273:
1264:
1263:
1244:
1242:
1241:
1236:
1221:
1219:
1218:
1213:
1211:
1210:
1192:
1191:
1169:
1167:
1166:
1161:
1159:
1158:
1142:
1140:
1139:
1134:
1108:
1106:
1105:
1100:
1098:
1097:
1088:
1087:
1068:
1066:
1065:
1060:
1058:
1057:
1039:
1038:
1026:
1025:
1009:
1007:
1006:
1001:
990:
989:
971:
970:
916:
914:
913:
908:
906:
905:
886:
884:
883:
878:
876:
875:
838:
836:
835:
830:
828:
827:
811:
809:
808:
803:
801:
800:
772:
770:
769:
764:
759:
758:
727:
725:
724:
719:
717:
716:
688:
686:
685:
680:
678:
677:
661:
659:
658:
653:
641:
639:
638:
633:
631:
630:
602:
600:
599:
594:
589:
588:
557:
555:
554:
549:
544:
543:
512:
510:
509:
504:
502:
501:
481:
479:
478:
473:
471:
470:
454:
452:
451:
446:
444:
443:
412:
410:
409:
404:
402:
401:
389:
388:
366:
364:
363:
358:
356:
355:
343:
342:
316:
314:
313:
308:
306:
305:
280:CW-decomposition
244:smooth manifolds
240:CW-decomposition
233:
231:
230:
225:
213:
211:
210:
205:
203:
202:
180:
178:
177:
172:
170:
169:
153:
151:
150:
145:
137:
136:
124:
123:
99:
98:
86:
85:
73:
72:
60:
59:
3720:
3719:
3715:
3714:
3713:
3711:
3710:
3709:
3695:
3694:
3661:
3656:
3655:
3650:
3646:
3641:
3636:
3599:
3541:
3536:
3535:
3514:
3509:
3508:
3503:if and only if
3462:
3461:
3439:
3438:
3417:
3407:
3399:
3398:
3374:
3358:
3344:
3343:
3318:
3317:
3296:
3286:
3273:
3263:
3252:
3251:
3215:
3210:
3209:
3146:
3085:
3084:
3083:and a function
3063:
3050:
3036:
3035:
3016:
3015:
2947:
2946:
2913:
2908:
2903:
2902:
2874:
2869:
2868:
2830:
2787:
2771:
2763:
2762:
2738:
2719:
2703:
2698:
2697:
2669:
2653:
2628:
2606:
2587:
2571:
2552:
2547:
2546:
2545:, and provided
2522:
2494:
2472:
2461:
2460:
2426:
2407:
2399:
2398:
2396:critical points
2358:
2357:
2351:
2315:
2310:
2309:
2288:
2283:
2282:
2243:
2238:
2237:
2213:
2208:
2207:
2197:
2193:
2187:
2154:
2135:
2122:
2106:
2101:
2100:
2079:
2066:
2052:
2051:
2040:
2004:
1999:
1998:
1967:
1957:
1949:
1948:
1927:
1922:
1921:
1896:
1877:
1864:
1839:
1826:
1815:
1814:
1781:
1780:
1726:
1715:
1714:
1670:
1657:
1647:
1643:
1630:
1620:
1612:
1611:
1591:
1586:
1585:
1574:
1499:
1486:
1467:
1448:
1425:
1424:
1372:
1371:
1352:
1351:
1305:
1292:
1282:
1278:
1265:
1255:
1247:
1246:
1224:
1223:
1196:
1177:
1172:
1171:
1150:
1145:
1144:
1125:
1124:
1089:
1079:
1071:
1070:
1069:. The manifold
1043:
1030:
1017:
1012:
1011:
975:
956:
945:
944:
897:
889:
888:
861:
841:
840:
819:
814:
813:
786:
775:
774:
750:
730:
729:
702:
691:
690:
669:
664:
663:
644:
643:
616:
605:
604:
580:
560:
559:
535:
515:
514:
493:
488:
487:
462:
457:
456:
435:
430:
429:
393:
374:
369:
368:
347:
334:
323:
322:
297:
292:
291:
276:
216:
215:
188:
183:
182:
161:
156:
155:
128:
109:
90:
77:
64:
48:
37:
36:
12:
11:
5:
3718:
3716:
3708:
3707:
3697:
3696:
3693:
3692:
3669:
3660:
3657:
3654:
3653:
3643:
3642:
3640:
3637:
3635:
3632:
3631:
3630:
3625:
3623:Kirby calculus
3620:
3615:
3610:
3605:
3598:
3595:
3594:
3593:
3586:
3548:
3544:
3521:
3517:
3485:
3469:
3449:
3446:
3424:
3420:
3414:
3410:
3406:
3395:
3381:
3377:
3373:
3370:
3365:
3361:
3357:
3354:
3351:
3331:
3328:
3325:
3303:
3299:
3293:
3289:
3285:
3280:
3276:
3270:
3266:
3262:
3259:
3248:
3222:
3218:
3173:-handles is a
3145:
3142:
3105:
3101:
3098:
3095:
3092:
3070:
3066:
3062:
3057:
3053:
3049:
3046:
3043:
3023:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2927:
2920:
2916:
2911:
2881:
2877:
2846:
2843:
2840:
2837:
2833:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2800:
2797:
2794:
2790:
2786:
2781:
2778:
2774:
2770:
2750:
2745:
2741:
2737:
2732:
2729:
2726:
2722:
2718:
2713:
2710:
2706:
2681:
2676:
2672:
2668:
2665:
2660:
2656:
2652:
2649:
2646:
2641:
2638:
2635:
2631:
2627:
2624:
2621:
2618:
2613:
2609:
2605:
2602:
2599:
2594:
2590:
2586:
2583:
2578:
2574:
2570:
2567:
2564:
2559:
2555:
2534:
2529:
2525:
2521:
2518:
2515:
2512:
2509:
2506:
2501:
2497:
2493:
2490:
2487:
2484:
2479:
2475:
2471:
2468:
2444:
2441:
2438:
2433:
2429:
2425:
2422:
2419:
2414:
2410:
2406:
2378:
2374:
2371:
2368:
2365:
2355:Morse function
2350:
2347:
2328:
2325:
2322:
2318:
2295:
2291:
2270:
2267:
2264:
2261:
2258:
2255:
2250:
2246:
2223:
2220:
2216:
2206:-dimensional,
2196:-dimensional,
2175:
2172:
2167:
2164:
2161:
2157:
2153:
2150:
2147:
2142:
2138:
2134:
2129:
2125:
2121:
2116:
2113:
2109:
2086:
2082:
2078:
2073:
2069:
2065:
2062:
2059:
2039:
2036:
2011:
2007:
1974:
1970:
1964:
1960:
1956:
1934:
1930:
1920:of the handle
1903:
1899:
1895:
1890:
1887:
1884:
1880:
1876:
1871:
1867:
1863:
1858:
1855:
1852:
1849:
1846:
1842:
1838:
1833:
1829:
1825:
1822:
1805:trivialization
1788:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1739:
1736:
1733:
1729:
1725:
1722:
1701:
1697:
1692:
1688:
1683:
1680:
1677:
1673:
1669:
1664:
1660:
1656:
1653:
1650:
1646:
1642:
1637:
1633:
1627:
1623:
1619:
1598:
1594:
1573:
1570:
1564:) is called a
1512:
1509:
1506:
1502:
1498:
1493:
1489:
1485:
1480:
1477:
1474:
1470:
1466:
1461:
1458:
1455:
1451:
1447:
1444:
1441:
1438:
1435:
1432:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1359:
1336:
1332:
1327:
1323:
1318:
1315:
1312:
1308:
1304:
1299:
1295:
1291:
1288:
1285:
1281:
1277:
1272:
1268:
1262:
1258:
1254:
1234:
1231:
1209:
1206:
1203:
1199:
1195:
1190:
1187:
1184:
1180:
1157:
1153:
1132:
1118:-handle along
1096:
1092:
1086:
1082:
1078:
1056:
1053:
1050:
1046:
1042:
1037:
1033:
1029:
1024:
1020:
999:
996:
993:
988:
985:
982:
978:
974:
969:
966:
963:
959:
955:
952:
934:-handle to an
904:
900:
896:
874:
871:
868:
864:
860:
857:
854:
851:
848:
826:
822:
799:
796:
793:
789:
785:
782:
762:
757:
753:
749:
746:
743:
740:
737:
715:
712:
709:
705:
701:
698:
676:
672:
651:
629:
626:
623:
619:
615:
612:
592:
587:
583:
579:
576:
573:
570:
567:
547:
542:
538:
534:
531:
528:
525:
522:
500:
496:
469:
465:
442:
438:
424:in a manifold
400:
396:
392:
387:
384:
381:
377:
354:
350:
346:
341:
337:
333:
330:
304:
300:
275:
272:
223:
201:
198:
195:
191:
168:
164:
143:
140:
135:
131:
127:
122:
119:
116:
112:
108:
105:
102:
97:
93:
89:
84:
80:
76:
71:
67:
63:
58:
55:
51:
47:
44:
13:
10:
9:
6:
4:
3:
2:
3717:
3706:
3703:
3702:
3700:
3691:
3690:0-8218-0994-6
3687:
3683:
3682:
3677:
3673:
3670:
3667:
3664:A. Kosinski,
3663:
3662:
3658:
3648:
3645:
3638:
3633:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3603:Casson handle
3601:
3600:
3596:
3591:
3587:
3584:
3580:
3576:
3572:
3568:
3564:
3546:
3542:
3519:
3515:
3506:
3502:
3498:
3494:
3490:
3486:
3483:
3467:
3447:
3422:
3418:
3412:
3408:
3404:
3396:
3379:
3375:
3371:
3363:
3359:
3355:
3352:
3329:
3326:
3323:
3301:
3297:
3291:
3287:
3278:
3274:
3268:
3264:
3260:
3249:
3246:
3244:
3238:
3220:
3216:
3207:
3203:
3199:
3196:
3195:triangulation
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3147:
3143:
3141:
3139:
3135:
3131:
3127:
3122:
3120:
3096:
3093:
3090:
3068:
3064:
3060:
3055:
3051:
3047:
3044:
3021:
3013:
2994:
2988:
2985:
2982:
2979:
2973:
2967:
2964:
2958:
2952:
2943:
2941:
2925:
2918:
2914:
2909:
2900:
2897:
2879:
2875:
2866:
2862:
2841:
2835:
2831:
2827:
2818:
2815:
2812:
2806:
2798:
2795:
2792:
2788:
2779:
2776:
2772:
2743:
2739:
2735:
2730:
2727:
2724:
2720:
2711:
2708:
2704:
2695:
2692:then for all
2679:
2674:
2670:
2666:
2658:
2654:
2647:
2644:
2639:
2636:
2633:
2629:
2625:
2622:
2619:
2611:
2607:
2600:
2597:
2592:
2588:
2584:
2576:
2572:
2565:
2562:
2557:
2553:
2527:
2523:
2516:
2513:
2510:
2507:
2499:
2495:
2488:
2485:
2477:
2473:
2466:
2458:
2442:
2439:
2431:
2427:
2423:
2420:
2417:
2412:
2408:
2397:
2393:
2369:
2366:
2363:
2356:
2348:
2346:
2344:
2326:
2323:
2320:
2316:
2293:
2289:
2265:
2262:
2259:
2253:
2248:
2244:
2221:
2218:
2214:
2205:
2200:
2190:
2173:
2170:
2165:
2162:
2159:
2155:
2151:
2148:
2145:
2140:
2136:
2132:
2127:
2123:
2119:
2114:
2111:
2107:
2084:
2080:
2076:
2071:
2067:
2063:
2060:
2049:
2045:
2037:
2035:
2033:
2032:
2028:
2009:
2005:
1996:
1993:
1988:
1972:
1968:
1962:
1958:
1954:
1932:
1928:
1919:
1901:
1897:
1893:
1888:
1885:
1882:
1878:
1874:
1869:
1865:
1861:
1856:
1853:
1850:
1847:
1844:
1840:
1836:
1831:
1823:
1812:
1810:
1809:normal bundle
1806:
1802:
1786:
1778:
1776:
1760:
1757:
1748:
1742:
1737:
1734:
1731:
1727:
1720:
1712:
1699:
1695:
1690:
1681:
1678:
1675:
1671:
1667:
1662:
1658:
1651:
1648:
1644:
1640:
1635:
1631:
1625:
1621:
1617:
1596:
1592:
1583:
1579:
1576:When forming
1571:
1569:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1536:by attaching
1535:
1531:
1526:
1510:
1507:
1504:
1500:
1496:
1491:
1487:
1483:
1478:
1475:
1472:
1468:
1464:
1459:
1456:
1453:
1449:
1445:
1439:
1436:
1433:
1407:
1404:
1401:
1395:
1392:
1386:
1383:
1380:
1357:
1350:
1334:
1330:
1325:
1316:
1313:
1310:
1306:
1302:
1297:
1293:
1286:
1283:
1279:
1275:
1270:
1266:
1260:
1256:
1252:
1232:
1207:
1204:
1201:
1197:
1193:
1188:
1185:
1182:
1178:
1155:
1151:
1130:
1122:
1121:
1117:
1113:
1094:
1090:
1084:
1080:
1076:
1054:
1051:
1048:
1044:
1040:
1035:
1031:
1027:
1022:
1018:
997:
986:
983:
980:
976:
972:
967:
964:
961:
957:
953:
950:
942:
941:
937:
933:
927:
926:Stephen Smale
922:
920:
902:
898:
872:
869:
866:
862:
858:
852:
824:
820:
797:
794:
791:
787:
783:
780:
755:
751:
744:
741:
735:
713:
710:
707:
703:
699:
696:
674:
670:
649:
627:
624:
621:
617:
613:
610:
585:
581:
574:
571:
565:
540:
536:
529:
526:
520:
498:
494:
485:
467:
463:
440:
436:
427:
423:
419:
414:
398:
394:
390:
385:
382:
379:
375:
352:
348:
339:
335:
331:
328:
320:
302:
298:
289:
285:
281:
273:
267:
263:
261:
257:
253:
249:
245:
241:
237:
221:
199:
196:
193:
189:
166:
162:
141:
138:
133:
129:
125:
120:
117:
114:
110:
106:
103:
100:
95:
91:
87:
82:
78:
74:
69:
65:
61:
56:
53:
49:
45:
34:
31:
27:
23:
19:
3679:
3675:
3672:Robert Gompf
3665:
3647:
3582:
3574:
3570:
3566:
3562:
3504:
3500:
3496:
3492:
3488:
3242:
3236:
3205:
3201:
3197:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3137:
3133:
3129:
3125:
3123:
3118:
3011:
2944:
2898:
2895:
2864:
2860:
2693:
2456:
2391:
2352:
2342:
2203:
2198:
2188:
2047:
2043:
2041:
2030:
2026:
2025:
1994:
1991:
1989:
1917:
1813:
1800:
1779:
1774:
1713:
1581:
1577:
1575:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1527:
1119:
1115:
1111:
1110:
939:
935:
931:
929:
923:
483:
425:
421:
415:
318:
287:
283:
277:
256:Morse theory
251:
247:
235:
32:
25:
21:
15:
1918:belt sphere
1572:Terminology
1109:(in words,
260:Cerf theory
246:. Thus an
154:where each
35:is a union
18:mathematics
3634:References
3618:Handlebody
3613:CW complex
3499:-manifold
3491:-manifold
3208:-skeleton
2938:where the
1566:handlebody
1347:where the
938:-manifold
274:Motivation
3445:∂
3409:∪
3372:∪
3356:∪
3327:≤
3288:∪
3265:∪
3245:-skeleton
3100:→
3061:∪
3042:∂
2986:≤
2983:⋯
2980:≤
2965:≤
2828:∪
2807:×
2796:−
2777:−
2728:−
2709:−
2637:−
2623:⋯
2511:⋯
2440:⊂
2421:…
2373:→
2324:−
2254:×
2219:−
2152:⊂
2149:⋯
2146:⊂
2133:⊂
2120:⊂
2112:−
2077:∪
2058:∂
1959:∪
1886:−
1875:×
1862:⊂
1854:−
1848:−
1837:×
1758:⊂
1743:×
1735:−
1700:∼
1679:−
1668:×
1652:⊔
1622:∪
1508:−
1497:×
1484:⊂
1476:−
1465:×
1457:−
1446:∈
1393:∼
1358:∼
1335:∼
1314:−
1303:×
1287:⊔
1257:∪
1230:∂
1205:−
1194:×
1186:−
1081:∪
1052:−
1041:×
995:∂
992:→
984:−
973:×
965:−
895:∂
870:−
859:×
850:∂
795:−
784:×
745:
739:∖
711:−
700:×
625:−
614:×
575:
569:∖
530:
524:∖
391:⊂
383:−
345:→
329:χ
197:−
126:⊂
118:−
107:⊂
104:⋯
101:⊂
88:⊂
75:⊂
62:⊂
54:−
43:∅
3699:Category
3597:See also
2459:satisfy
2353:Given a
1584:-handle
1580:union a
1423:for all
1245:, i.e.,
1114:union a
30:manifold
3482:surgery
2940:Hessian
1916:is the
1807:of its
1801:framing
282:of the
236:handles
3688:
2894:. The
2859:where
2186:where
2050:where
2024:is an
1010:. Let
24:of an
3639:Notes
3241:dual
3237:(3,1)
3202:(3,1)
3187:(3,1)
3175:(3,1)
3159:(3,1)
3124:When
3034:with
2896:index
2027:(m,k)
3686:ISBN
3588:The
3181:and
3169:and
2899:I(j)
2667:<
2645:<
2626:<
2620:<
2598:<
2585:<
2563:<
2514:<
2508:<
2486:<
2281:and
1143:and
513:and
20:, a
3497:m+1
3132:, -
2455:of
2204:m+1
2202:is
2192:is
1947:in
887:in
812:to
742:int
689:2)
572:int
527:int
16:In
3701::
3149:A
3140:.
3121:.
2696:,
2042:A
2034:.
1987:.
1811:.
1777:.
1568:.
1525:.
921:.
413:.
262:.
3583:3
3575:4
3571:3
3567:4
3563:3
3547:m
3543:S
3520:m
3516:S
3505:M
3501:W
3493:M
3489:m
3468:f
3448:M
3423:j
3419:H
3413:f
3405:M
3380:i
3376:H
3369:)
3364:j
3360:H
3353:M
3350:(
3330:i
3324:j
3302:j
3298:H
3292:g
3284:)
3279:i
3275:H
3269:f
3261:M
3258:(
3247:.
3243:1
3221:1
3217:T
3206:1
3198:T
3191:3
3183:2
3179:3
3171:1
3167:0
3163:3
3155:3
3134:f
3130:M
3126:f
3119:W
3104:R
3097:W
3094::
3091:f
3069:1
3065:M
3056:0
3052:M
3048:=
3045:W
3022:W
3012:M
2998:)
2995:k
2992:(
2989:I
2977:)
2974:2
2971:(
2968:I
2962:)
2959:1
2956:(
2953:I
2926:M
2919:j
2915:p
2910:T
2880:j
2876:p
2865:j
2863:(
2861:I
2845:)
2842:j
2839:(
2836:I
2832:H
2825:)
2822:]
2819:1
2816:,
2813:0
2810:[
2804:)
2799:1
2793:j
2789:t
2785:(
2780:1
2773:f
2769:(
2749:]
2744:j
2740:t
2736:,
2731:1
2725:j
2721:t
2717:[
2712:1
2705:f
2694:j
2680:,
2675:k
2671:t
2664:)
2659:k
2655:p
2651:(
2648:f
2640:1
2634:k
2630:t
2617:)
2612:2
2608:p
2604:(
2601:f
2593:1
2589:t
2582:)
2577:1
2573:p
2569:(
2566:f
2558:0
2554:t
2533:)
2528:k
2524:p
2520:(
2517:f
2505:)
2500:2
2496:p
2492:(
2489:f
2483:)
2478:1
2474:p
2470:(
2467:f
2457:f
2443:M
2437:}
2432:k
2428:p
2424:,
2418:,
2413:1
2409:p
2405:{
2392:M
2377:R
2370:M
2367::
2364:f
2343:i
2327:1
2321:i
2317:W
2294:i
2290:W
2269:]
2266:1
2263:,
2260:0
2257:[
2249:0
2245:M
2222:1
2215:W
2199:W
2194:m
2189:M
2174:W
2171:=
2166:1
2163:+
2160:m
2156:W
2141:1
2137:W
2128:0
2124:W
2115:1
2108:W
2085:1
2081:M
2072:0
2068:M
2064:=
2061:W
2048:W
2031:g
2010:m
2006:D
1995:k
1992:g
1973:j
1969:H
1963:f
1955:M
1933:j
1929:H
1902:j
1898:H
1894:=
1889:j
1883:m
1879:D
1870:j
1866:D
1857:1
1851:j
1845:m
1841:S
1832:j
1828:}
1824:0
1821:{
1787:f
1761:M
1755:)
1752:}
1749:0
1746:{
1738:1
1732:j
1728:S
1724:(
1721:f
1696:/
1691:)
1687:)
1682:j
1676:m
1672:D
1663:j
1659:D
1655:(
1649:M
1645:(
1641:=
1636:j
1632:H
1626:f
1618:M
1597:j
1593:H
1582:j
1578:M
1562:j
1558:j
1554:0
1550:N
1546:j
1542:M
1538:j
1534:M
1530:N
1511:j
1505:m
1501:D
1492:j
1488:D
1479:j
1473:m
1469:D
1460:1
1454:j
1450:S
1443:)
1440:x
1437:,
1434:p
1431:(
1411:)
1408:x
1405:,
1402:p
1399:(
1396:f
1390:)
1387:x
1384:,
1381:p
1378:(
1331:/
1326:)
1322:)
1317:j
1311:m
1307:D
1298:j
1294:D
1290:(
1284:M
1280:(
1276:=
1271:j
1267:H
1261:f
1253:M
1233:M
1208:j
1202:m
1198:D
1189:1
1183:j
1179:S
1156:j
1152:H
1131:M
1120:f
1116:j
1112:M
1095:j
1091:H
1085:f
1077:M
1055:j
1049:m
1045:D
1036:j
1032:D
1028:=
1023:j
1019:H
998:M
987:j
981:m
977:D
968:1
962:j
958:S
954::
951:f
940:M
936:m
932:j
903:m
899:D
873:1
867:m
863:D
856:)
853:I
847:(
825:m
821:D
798:1
792:m
788:D
781:I
761:)
756:p
752:N
748:(
736:M
714:1
708:m
704:D
697:I
675:m
671:D
650:M
628:1
622:m
618:D
611:I
591:)
586:p
582:N
578:(
566:M
546:)
541:p
537:N
533:(
521:M
499:p
495:N
484:M
468:m
464:D
441:p
437:N
426:M
422:p
399:n
395:D
386:1
380:n
376:S
353:n
349:S
340:n
336:D
332::
319:0
303:n
299:S
288:n
284:n
252:i
248:i
234:-
222:i
200:1
194:i
190:M
167:i
163:M
142:M
139:=
134:m
130:M
121:1
115:m
111:M
96:2
92:M
83:1
79:M
70:0
66:M
57:1
50:M
46:=
33:M
28:-
26:m
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