2449:
who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are
Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three-manifold is either a Heegaard surface or
1949:
2074:
that any other genus 3 Heegaard splitting of the three-torus is topologically equivalent to this one. Michel
Boileau and Jean-Pierre Otal proved that in general any Heegaard splitting of the three-torus is equivalent to the result of stabilizing this
1781:
174:. This contrasts strongly with higher-dimensional manifolds which need not admit smooth or piecewise linear structures. Assuming smoothness the existence of a Heegaard splitting also follows from the work of
1073:
767:
833:
2527:
Heegaard diagrams, which are simple combinatorial descriptions of
Heegaard splittings, have been used extensively to construct invariants of three-manifolds. The most recent example of this is the
54:
2820:
643:
2406:). It follows from their work that all torus bundles have a unique splitting of minimal genus. All other splittings of the torus bundle are stabilizations of the minimal genus one.
1605:
694:
2517:
2488:
1403:
1347:
1318:
1193:
1001:
920:
161:
1435:
1115:
2342:
There are several classes of three-manifolds where the set of
Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of
2010:
1977:
501:
457:
402:
354:
310:
259:
2567:
525:
477:
422:
378:
330:
2367:
2323:
2288:
2227:
2200:
2158:
2125:
2068:
2041:
1776:
1749:
1718:
1687:
1644:
1555:
1493:
1466:
1374:
1289:
1254:
1164:
880:
1219:
2251:
940:
853:
1075:; the complexity of a disconnected surface is the sum of complexities of its components. The complexity of a generalized Heegaard splitting is the multi-set
2726:
2569:
symmetric product of a
Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the
1944:{\displaystyle \Gamma =S^{1}\times \{x_{0}\}\times \{x_{0}\}\cup \{x_{0}\}\times S^{1}\times \{x_{0}\}\cup \{x_{0}\}\times \{x_{0}\}\times S^{1}}
2802:
2627:
2536:
1021:
2519:
was given by Meeks and
Frohman. The result relied heavily on techniques developed for studying the topology of Heegaard splittings.
2093:
699:
2461:
went on to use results of
Waldhausen to prove results about the topological uniqueness of minimal surfaces of finite genus in
2682:
2602:
2089:
772:
2687:
1117:, where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by
2420:
Computational methods can be used to determine or approximate the
Heegaard genus of a 3-manifold. John Berge's software
2882:
2794:
170:
three-manifold may be so obtained; this follows from deep results on the triangulability of three-manifolds due to
1607:. It is easy to show that the stabilization procedure yields stabilized splittings. Inductively, a splitting is
2877:
2414:
584:
1560:
223:
Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with
2754:
2622:
2570:
1956:
648:
1646:
under the quotient map used to define the lens space in question. It follows from the structure of the
1118:
2493:
2464:
1379:
1323:
1294:
1169:
2872:
2594:
2097:
1257:
966:
885:
429:
282:
124:
2377:
1647:
1408:
1007:
835:. The interiors of the compression bodies must be pairwise disjoint and their union must be all of
547:
209:
96:
1078:
2852:
2832:
2389:
2092:) embedding of the two-sphere into the three-sphere. (In higher dimensions this is known as the
217:
35:
2798:
2738:
2706:
2425:
1690:
1982:
1962:
486:
442:
387:
339:
295:
244:
2842:
2778:
2696:
2632:
2542:
2451:
579:
510:
462:
407:
363:
315:
224:
2812:
2769:
2750:
2718:
2345:
2301:
2266:
2205:
2178:
2136:
2103:
2046:
2019:
1754:
1727:
1696:
1665:
1622:
1528:
1471:
1444:
1352:
1267:
1232:
1142:
858:
2808:
2782:
2746:
2714:
2442:
47:
1198:
1619:
All have a standard splitting of genus one. This is the image of the
Clifford torus in
2590:
2528:
2458:
2236:
1438:
925:
838:
171:
2866:
2701:
2678:
2598:
2597:
in the 1960s, it was not until a few decades later that the field was rejuvenated by
2582:
2532:
2446:
1522:
1010:, defined for knots, for Heegaard splittings. The complexity of a connected surface
92:
2856:
2403:
2399:
1261:
1134:
27:
Decomposition of a compact oriented 3-manifold by dividing it into two handlebodies
2589:). While Heegaard splittings were studied extensively by mathematicians such as
2666:
1657:
1496:
227:. The gluing map is between the positive boundaries of the compression bodies.
31:
2370:
1614:
1222:
167:
116:
84:
64:
60:
17:
2742:
2710:
2071:
2847:
198:
2771:
Forstudier til en topologisk Teori for de algebraiske
Fladers Sammenhang
234:
if it is not homotopic to a point, a puncture, or a boundary component.
2490:. The final topological classification of embedded minimal surfaces in
546:
if there is no other splitting of the ambient three-manifold of lower
2837:
1721:
1651:
216:. This connection with the mapping class group was first made by
205:
175:
2298:
is a Heegaard splitting. Then there is an essential two-sphere
2100:.) This may be restated as follows: the genus zero splitting of
1121:(monotonically decreasing). A generalized Heegaard splitting is
1068:{\displaystyle \operatorname {max} \left\{0,1-\chi (S)\right\}}
2160:
is obtained by stabilizing the unique splitting of genus zero.
2043:
is a Heegaard splitting and this is the standard splitting of
204:
The gluing map ƒ need only be specified up to taking a double
2380:
are more subtle. Here, all splittings may be isotoped to be
2421:
762:{\displaystyle \partial _{+}V_{i}=\partial _{+}W_{i}=H_{i}}
1405:. Then the set of points where each coordinate has norm
2727:"The structure of a solvmanifold's Heegaard splittings"
2417:
three-manifolds which are two-bridge knot complements.
828:{\displaystyle \partial _{-}W_{i}=\partial _{-}V_{i+1}}
2821:"Heegaard splittings of exteriors of two bridge knots"
2545:
2496:
2467:
2348:
2304:
2269:
2239:
2208:
2181:
2139:
2106:
2049:
2022:
1985:
1965:
1784:
1757:
1730:
1699:
1668:
1625:
1563:
1531:
1474:
1447:
1411:
1382:
1355:
1326:
1297:
1270:
1260:, all manifolds admitting a genus zero splitting are
1235:
1201:
1172:
1145:
1081:
1024:
969:
928:
888:
861:
841:
775:
702:
651:
587:
513:
489:
465:
445:
439:
if there are disjoint essential simple closed curves
410:
390:
366:
342:
318:
298:
247:
127:
2581:The idea of a Heegaard splitting was introduced by
1654:
that only lens spaces have splittings of genus one.
1611:
if it is the stabilization of a standard splitting.
197:of the splitting. Splittings are considered up to
2561:
2511:
2482:
2361:
2317:
2282:
2245:
2221:
2194:
2152:
2119:
2062:
2035:
2004:
1971:
1943:
1770:
1743:
1712:
1681:
1638:
1599:
1549:
1487:
1460:
1429:
1397:
1368:
1341:
1312:
1283:
1248:
1213:
1187:
1158:
1109:
1067:
995:
934:
914:
874:
847:
827:
761:
688:
637:
519:
495:
471:
451:
416:
396:
372:
348:
324:
304:
253:
155:
2395:
2793:, Annals of Mathematics Studies, vol. 86,
1468:. This is the standard genus one splitting of
178:about handle decompositions from Morse theory.
2441:Heegaard splittings appeared in the theory of
2424:studies Heegaard splittings generated by the
1195:with length one. Intersecting this with the
956:is allowed to have more than one component.)
882:forms a Heegaard surface for the submanifold
241:if there is an essential simple closed curve
8:
2606:
1925:
1912:
1906:
1893:
1887:
1874:
1855:
1842:
1836:
1823:
1817:
1804:
1104:
1082:
292:if there are essential simple closed curves
2725:Cooper, Daryl; Scharlemann, Martin (1999),
959:A generalized Heegaard splitting is called
59:) is a decomposition of a compact oriented
2846:
2836:
2700:
2656:. Princeton University Press. p. 22.
2550:
2544:
2503:
2499:
2498:
2495:
2474:
2470:
2469:
2466:
2410:
2402:(which includes all three-manifolds with
2373:(as proved by Francis Bonahon and Otal).
2353:
2347:
2309:
2303:
2274:
2268:
2238:
2213:
2207:
2186:
2180:
2144:
2138:
2111:
2105:
2054:
2048:
2027:
2021:
1990:
1984:
1964:
1935:
1919:
1900:
1881:
1865:
1849:
1830:
1811:
1795:
1783:
1762:
1756:
1735:
1729:
1704:
1698:
1673:
1667:
1630:
1624:
1586:
1573:
1562:
1530:
1479:
1473:
1452:
1446:
1420:
1415:
1410:
1389:
1385:
1384:
1381:
1360:
1354:
1333:
1329:
1328:
1325:
1304:
1300:
1299:
1296:
1275:
1269:
1240:
1234:
1200:
1179:
1175:
1174:
1171:
1150:
1144:
1095:
1080:
1023:
987:
974:
968:
927:
906:
893:
887:
866:
860:
840:
813:
803:
790:
780:
774:
753:
740:
730:
717:
707:
701:
656:
650:
638:{\displaystyle V_{i},W_{i},i=1,\dotsc ,n}
605:
592:
586:
512:
488:
464:
444:
424:intersect exactly once. It follows from
409:
389:
365:
341:
317:
297:
277:if it is not reducible. It follows from
246:
141:
126:
2685:(1987), "Reducing Heegaard splittings",
2586:
1600:{\displaystyle \left(S^{3},T^{2}\right)}
91:, and let ƒ be an orientation reversing
2644:
2168:is a closed orientable three-manifold.
2070:. It was proved by Charles Frohman and
1951:. It is an easy exercise to show that
115:along ƒ we obtain the compact oriented
63:that results from dividing it into two
2609:), primarily through their concept of
2413:classifies the Heegaard splittings of
428:that every reducible splitting of an
46:
7:
2628:Handle decompositions of 3-manifolds
1291:. Under the usual identification of
689:{\displaystyle H_{i},i=1,\dotsc ,n}
2369:are standard. The same holds for
2088:Up to isotopy, there is a unique (
1966:
1785:
800:
777:
727:
704:
185:into two handlebodies is called a
25:
2257:which is a stabilization of both.
2671:A Primer on Mapping Class Groups
2654:A Primer on Mapping Class Groups
2512:{\displaystyle \mathbb {R} ^{3}}
2483:{\displaystyle \mathbb {R} ^{3}}
2096:. In dimension two this is the
1398:{\displaystyle \mathbb {C} ^{2}}
1342:{\displaystyle \mathbb {C} ^{2}}
1313:{\displaystyle \mathbb {R} ^{4}}
1188:{\displaystyle \mathbb {R} ^{4}}
1006:There is an analogous notion of
554:of the splitting surface is the
2396:Cooper & Scharlemann (1999)
1495:. (See also the discussion at
996:{\displaystyle V_{i}\cup W_{i}}
915:{\displaystyle V_{i}\cup W_{i}}
566:Generalized Heegaard splittings
535:if it is not weakly reducible.
2731:Turkish Journal of Mathematics
2388:(as proved by Yoav Moriah and
2290:is an essential two-sphere in
1544:
1532:
1125:if its complexity is minimal.
1101:
1088:
1057:
1051:
572:generalized Heegaard splitting
285:every splitting is reducible.
156:{\displaystyle M=V\cup _{f}W.}
1:
2688:Topology and Its Applications
1430:{\displaystyle 1/{\sqrt {2}}}
425:
278:
2819:Kobayashi, Tsuyoshi (2001),
2702:10.1016/0166-8641(87)90092-7
2673:, Princeton University Press
2432:Applications and connections
1662:Recall that the three-torus
1110:{\displaystyle \{c(S_{i})\}}
265:which bounds a disk in both
189:, and their common boundary
2233:there is a third splitting
2175:For any pair of splittings
2172:Reidemeister–Singer theorem
1505:Given a Heegaard splitting
2899:
2795:Princeton University Press
2398:classified splittings of
1166:is the set of vectors in
1003:is strongly irreducible.
578:is a decomposition into
230:A closed curve is called
2012:. Thus the boundary of
1979:, is a handlebody as is
1521:is formed by taking the
1229:genus zero splitting of
1119:lexicographical ordering
942:. (Note that here each
538:A Heegaard splitting is
435:A Heegaard splitting is
288:A Heegaard splitting is
237:A Heegaard splitting is
2768:Heegaard, Poul (1898),
2652:Farb, B.; Margalit, D.
2571:Lagrangian submanifolds
2529:Heegaard Floer homology
2523:Heegaard Floer homology
2005:{\displaystyle T^{3}-V}
1972:{\displaystyle \Gamma }
1778:and consider the graph
496:{\displaystyle \alpha }
452:{\displaystyle \alpha }
397:{\displaystyle \alpha }
349:{\displaystyle \alpha }
305:{\displaystyle \alpha }
254:{\displaystyle \alpha }
2777:, Thesis (in Danish),
2623:Manifold decomposition
2563:
2562:{\displaystyle g^{th}}
2539:. The theory uses the
2513:
2484:
2363:
2319:
2284:
2247:
2223:
2196:
2154:
2121:
2064:
2037:
2006:
1973:
1945:
1772:
1745:
1714:
1683:
1640:
1601:
1551:
1489:
1462:
1431:
1399:
1370:
1343:
1314:
1285:
1250:
1215:
1189:
1160:
1111:
1069:
997:
936:
916:
876:
849:
829:
763:
690:
639:
521:
520:{\displaystyle \beta }
497:
473:
472:{\displaystyle \beta }
453:
418:
417:{\displaystyle \beta }
398:
374:
373:{\displaystyle \beta }
350:
326:
325:{\displaystyle \beta }
306:
255:
157:
2848:10.2140/gt.2001.5.609
2825:Geometry and Topology
2789:Hempel, John (1976),
2611:strong irreducibility
2564:
2514:
2485:
2445:first in the work of
2364:
2362:{\displaystyle S^{3}}
2320:
2318:{\displaystyle S_{2}}
2285:
2283:{\displaystyle S_{1}}
2248:
2224:
2222:{\displaystyle H_{2}}
2197:
2195:{\displaystyle H_{1}}
2155:
2153:{\displaystyle S^{3}}
2122:
2120:{\displaystyle S^{3}}
2065:
2063:{\displaystyle T^{3}}
2038:
2036:{\displaystyle T^{3}}
2007:
1974:
1946:
1773:
1771:{\displaystyle S^{1}}
1746:
1744:{\displaystyle x_{0}}
1715:
1713:{\displaystyle S^{1}}
1684:
1682:{\displaystyle T^{3}}
1641:
1639:{\displaystyle S^{3}}
1602:
1552:
1550:{\displaystyle (M,H)}
1490:
1488:{\displaystyle S^{3}}
1463:
1461:{\displaystyle T^{2}}
1432:
1400:
1371:
1369:{\displaystyle S^{3}}
1344:
1315:
1286:
1284:{\displaystyle S^{3}}
1251:
1249:{\displaystyle S^{3}}
1216:
1190:
1161:
1159:{\displaystyle S^{3}}
1112:
1070:
998:
937:
917:
877:
875:{\displaystyle H_{i}}
850:
830:
764:
691:
640:
522:
498:
474:
454:
419:
399:
375:
351:
327:
307:
256:
181:The decomposition of
158:
2683:Gordon, Cameron McA.
2595:Friedhelm Waldhausen
2543:
2494:
2465:
2378:Seifert fiber spaces
2346:
2302:
2267:
2237:
2206:
2179:
2137:
2130:Waldhausen's theorem
2104:
2098:Jordan curve theorem
2047:
2020:
1983:
1963:
1957:regular neighborhood
1782:
1755:
1728:
1697:
1666:
1623:
1561:
1529:
1472:
1445:
1409:
1380:
1353:
1324:
1295:
1268:
1233:
1199:
1170:
1143:
1079:
1022:
967:
961:strongly irreducible
926:
886:
859:
839:
773:
700:
649:
585:
550:. The minimal value
533:strongly irreducible
511:
487:
463:
443:
430:irreducible manifold
426:Waldhausen's Theorem
408:
388:
364:
340:
316:
296:
245:
125:
2133:Every splitting of
2094:Schoenflies theorem
1693:of three copies of
1648:mapping class group
1221:hyperplane gives a
1214:{\displaystyle xyz}
1018:, is defined to be
210:mapping class group
103:to the boundary of
48:[ˈhe̝ˀˌkɒˀ]
2883:Geometric topology
2559:
2509:
2480:
2390:Jennifer Schultens
2359:
2333:in a single curve.
2315:
2280:
2243:
2219:
2192:
2150:
2117:
2060:
2033:
2002:
1969:
1941:
1768:
1741:
1710:
1679:
1636:
1597:
1547:
1485:
1458:
1427:
1395:
1366:
1339:
1310:
1281:
1256:. Conversely, by
1246:
1211:
1185:
1156:
1107:
1065:
993:
932:
912:
872:
845:
825:
759:
686:
635:
580:compression bodies
531:. A splitting is
517:
493:
469:
449:
414:
394:
370:
346:
322:
302:
283:reducible manifold
273:. A splitting is
251:
225:compression bodies
218:W. B. R. Lickorish
187:Heegaard splitting
153:
40:Heegaard splitting
36:geometric topology
2804:978-0-8218-3695-8
2679:Casson, Andrew J.
2669:; Margalit, Dan,
2599:Andrew Casson
2583:Poul Heegaard
2426:fundamental group
2246:{\displaystyle H}
2164:Suppose now that
2085:Alexander's lemma
1691:Cartesian product
1425:
1258:Alexander's Trick
1139:The three-sphere
935:{\displaystyle M}
848:{\displaystyle M}
527:bounds a disk in
503:bounds a disk in
380:bounds a disk in
356:bounds a disk in
16:(Redirected from
2890:
2878:Minimal surfaces
2859:
2850:
2840:
2815:
2785:
2776:
2764:
2763:
2762:
2753:, archived from
2721:
2704:
2674:
2658:
2657:
2649:
2633:Compression body
2568:
2566:
2565:
2560:
2558:
2557:
2518:
2516:
2515:
2510:
2508:
2507:
2502:
2489:
2487:
2486:
2481:
2479:
2478:
2473:
2452:totally geodesic
2443:minimal surfaces
2437:Minimal surfaces
2411:Kobayashi (2001)
2368:
2366:
2365:
2360:
2358:
2357:
2324:
2322:
2321:
2316:
2314:
2313:
2289:
2287:
2286:
2281:
2279:
2278:
2252:
2250:
2249:
2244:
2228:
2226:
2225:
2220:
2218:
2217:
2201:
2199:
2198:
2193:
2191:
2190:
2159:
2157:
2156:
2151:
2149:
2148:
2126:
2124:
2123:
2118:
2116:
2115:
2090:piecewise linear
2069:
2067:
2066:
2061:
2059:
2058:
2042:
2040:
2039:
2034:
2032:
2031:
2011:
2009:
2008:
2003:
1995:
1994:
1978:
1976:
1975:
1970:
1950:
1948:
1947:
1942:
1940:
1939:
1924:
1923:
1905:
1904:
1886:
1885:
1870:
1869:
1854:
1853:
1835:
1834:
1816:
1815:
1800:
1799:
1777:
1775:
1774:
1769:
1767:
1766:
1750:
1748:
1747:
1742:
1740:
1739:
1719:
1717:
1716:
1711:
1709:
1708:
1688:
1686:
1685:
1680:
1678:
1677:
1645:
1643:
1642:
1637:
1635:
1634:
1606:
1604:
1603:
1598:
1596:
1592:
1591:
1590:
1578:
1577:
1556:
1554:
1553:
1548:
1494:
1492:
1491:
1486:
1484:
1483:
1467:
1465:
1464:
1459:
1457:
1456:
1436:
1434:
1433:
1428:
1426:
1421:
1419:
1404:
1402:
1401:
1396:
1394:
1393:
1388:
1375:
1373:
1372:
1367:
1365:
1364:
1348:
1346:
1345:
1340:
1338:
1337:
1332:
1319:
1317:
1316:
1311:
1309:
1308:
1303:
1290:
1288:
1287:
1282:
1280:
1279:
1255:
1253:
1252:
1247:
1245:
1244:
1220:
1218:
1217:
1212:
1194:
1192:
1191:
1186:
1184:
1183:
1178:
1165:
1163:
1162:
1157:
1155:
1154:
1116:
1114:
1113:
1108:
1100:
1099:
1074:
1072:
1071:
1066:
1064:
1060:
1002:
1000:
999:
994:
992:
991:
979:
978:
941:
939:
938:
933:
921:
919:
918:
913:
911:
910:
898:
897:
881:
879:
878:
873:
871:
870:
854:
852:
851:
846:
834:
832:
831:
826:
824:
823:
808:
807:
795:
794:
785:
784:
768:
766:
765:
760:
758:
757:
745:
744:
735:
734:
722:
721:
712:
711:
695:
693:
692:
687:
661:
660:
644:
642:
641:
636:
610:
609:
597:
596:
526:
524:
523:
518:
502:
500:
499:
494:
478:
476:
475:
470:
458:
456:
455:
450:
437:weakly reducible
423:
421:
420:
415:
403:
401:
400:
395:
379:
377:
376:
371:
355:
353:
352:
347:
331:
329:
328:
323:
311:
309:
308:
303:
260:
258:
257:
252:
195:Heegaard surface
162:
160:
159:
154:
146:
145:
58:
57:
56:
50:
45:
21:
2898:
2897:
2893:
2892:
2891:
2889:
2888:
2887:
2863:
2862:
2818:
2805:
2788:
2774:
2767:
2760:
2758:
2724:
2677:
2665:
2662:
2661:
2651:
2650:
2646:
2641:
2619:
2579:
2546:
2541:
2540:
2525:
2497:
2492:
2491:
2468:
2463:
2462:
2439:
2434:
2428:of a manifold.
2349:
2344:
2343:
2340:
2338:Classifications
2305:
2300:
2299:
2270:
2265:
2264:
2235:
2234:
2209:
2204:
2203:
2182:
2177:
2176:
2140:
2135:
2134:
2107:
2102:
2101:
2082:
2050:
2045:
2044:
2023:
2018:
2017:
1986:
1981:
1980:
1961:
1960:
1931:
1915:
1896:
1877:
1861:
1845:
1826:
1807:
1791:
1780:
1779:
1758:
1753:
1752:
1731:
1726:
1725:
1700:
1695:
1694:
1669:
1664:
1663:
1626:
1621:
1620:
1582:
1569:
1568:
1564:
1559:
1558:
1527:
1526:
1475:
1470:
1469:
1448:
1443:
1442:
1407:
1406:
1383:
1378:
1377:
1356:
1351:
1350:
1327:
1322:
1321:
1298:
1293:
1292:
1271:
1266:
1265:
1236:
1231:
1230:
1225:. This is the
1197:
1196:
1173:
1168:
1167:
1146:
1141:
1140:
1131:
1091:
1077:
1076:
1035:
1031:
1020:
1019:
983:
970:
965:
964:
954:
947:
924:
923:
902:
889:
884:
883:
862:
857:
856:
855:. The surface
837:
836:
809:
799:
786:
776:
771:
770:
749:
736:
726:
713:
703:
698:
697:
652:
647:
646:
601:
588:
583:
582:
568:
509:
508:
485:
484:
461:
460:
441:
440:
432:is stabilized.
406:
405:
386:
385:
362:
361:
338:
337:
314:
313:
294:
293:
243:
242:
137:
123:
122:
73:
53:
52:
51:
43:
28:
23:
22:
15:
12:
11:
5:
2896:
2894:
2886:
2885:
2880:
2875:
2865:
2864:
2861:
2860:
2831:(2): 609–650,
2816:
2803:
2786:
2765:
2722:
2695:(3): 275–283,
2675:
2660:
2659:
2643:
2642:
2640:
2637:
2636:
2635:
2630:
2625:
2618:
2615:
2603:Cameron Gordon
2591:Wolfgang Haken
2578:
2575:
2556:
2553:
2549:
2524:
2521:
2506:
2501:
2477:
2472:
2459:Shing-Tung Yau
2438:
2435:
2433:
2430:
2376:Splittings of
2356:
2352:
2339:
2336:
2335:
2334:
2312:
2308:
2277:
2273:
2261:
2258:
2242:
2216:
2212:
2189:
2185:
2173:
2162:
2161:
2147:
2143:
2131:
2128:
2114:
2110:
2086:
2081:
2078:
2077:
2076:
2057:
2053:
2030:
2026:
2001:
1998:
1993:
1989:
1968:
1938:
1934:
1930:
1927:
1922:
1918:
1914:
1911:
1908:
1903:
1899:
1895:
1892:
1889:
1884:
1880:
1876:
1873:
1868:
1864:
1860:
1857:
1852:
1848:
1844:
1841:
1838:
1833:
1829:
1825:
1822:
1819:
1814:
1810:
1806:
1803:
1798:
1794:
1790:
1787:
1765:
1761:
1751:be a point of
1738:
1734:
1707:
1703:
1676:
1672:
1660:
1655:
1633:
1629:
1617:
1612:
1595:
1589:
1585:
1581:
1576:
1572:
1567:
1557:with the pair
1546:
1543:
1540:
1537:
1534:
1503:
1500:
1482:
1478:
1455:
1451:
1439:Clifford torus
1424:
1418:
1414:
1392:
1387:
1363:
1359:
1336:
1331:
1307:
1302:
1278:
1274:
1243:
1239:
1210:
1207:
1204:
1182:
1177:
1153:
1149:
1137:
1130:
1127:
1106:
1103:
1098:
1094:
1090:
1087:
1084:
1063:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1034:
1030:
1027:
990:
986:
982:
977:
973:
952:
945:
931:
909:
905:
901:
896:
892:
869:
865:
844:
822:
819:
816:
812:
806:
802:
798:
793:
789:
783:
779:
756:
752:
748:
743:
739:
733:
729:
725:
720:
716:
710:
706:
685:
682:
679:
676:
673:
670:
667:
664:
659:
655:
634:
631:
628:
625:
622:
619:
616:
613:
608:
604:
600:
595:
591:
567:
564:
556:Heegaard genus
516:
492:
468:
448:
413:
393:
369:
345:
321:
301:
250:
193:is called the
166:Every closed,
164:
163:
152:
149:
144:
140:
136:
133:
130:
72:
69:
26:
24:
18:Heegaard genus
14:
13:
10:
9:
6:
4:
3:
2:
2895:
2884:
2881:
2879:
2876:
2874:
2871:
2870:
2868:
2858:
2854:
2849:
2844:
2839:
2834:
2830:
2826:
2822:
2817:
2814:
2810:
2806:
2800:
2796:
2792:
2787:
2784:
2780:
2773:
2772:
2766:
2757:on 2011-08-22
2756:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2723:
2720:
2716:
2712:
2708:
2703:
2698:
2694:
2690:
2689:
2684:
2680:
2676:
2672:
2668:
2664:
2663:
2655:
2648:
2645:
2638:
2634:
2631:
2629:
2626:
2624:
2621:
2620:
2616:
2614:
2612:
2608:
2604:
2601: and
2600:
2596:
2592:
2588:
2584:
2576:
2574:
2572:
2554:
2551:
2547:
2538:
2534:
2533:Peter Ozsvath
2530:
2522:
2520:
2504:
2475:
2460:
2455:
2453:
2448:
2447:Blaine Lawson
2444:
2436:
2431:
2429:
2427:
2423:
2418:
2416:
2412:
2407:
2405:
2401:
2400:torus bundles
2397:
2393:
2391:
2387:
2383:
2379:
2374:
2372:
2354:
2350:
2337:
2332:
2328:
2310:
2306:
2297:
2293:
2275:
2271:
2263:Suppose that
2262:
2260:Haken's lemma
2259:
2256:
2240:
2232:
2214:
2210:
2187:
2183:
2174:
2171:
2170:
2169:
2167:
2145:
2141:
2132:
2129:
2112:
2108:
2099:
2095:
2091:
2087:
2084:
2083:
2079:
2073:
2055:
2051:
2028:
2024:
2015:
1999:
1996:
1991:
1987:
1958:
1954:
1936:
1932:
1928:
1920:
1916:
1909:
1901:
1897:
1890:
1882:
1878:
1871:
1866:
1862:
1858:
1850:
1846:
1839:
1831:
1827:
1820:
1812:
1808:
1801:
1796:
1792:
1788:
1763:
1759:
1736:
1732:
1723:
1705:
1701:
1692:
1674:
1670:
1661:
1659:
1656:
1653:
1649:
1631:
1627:
1618:
1616:
1613:
1610:
1593:
1587:
1583:
1579:
1574:
1570:
1565:
1541:
1538:
1535:
1524:
1523:connected sum
1520:
1516:
1515:stabilization
1512:
1508:
1504:
1502:Stabilization
1501:
1498:
1480:
1476:
1453:
1449:
1440:
1422:
1416:
1412:
1390:
1376:as living in
1361:
1357:
1334:
1305:
1276:
1272:
1263:
1259:
1241:
1237:
1228:
1224:
1208:
1205:
1202:
1180:
1151:
1147:
1138:
1136:
1133:
1132:
1128:
1126:
1124:
1120:
1096:
1092:
1085:
1061:
1054:
1048:
1045:
1042:
1039:
1036:
1032:
1028:
1025:
1017:
1013:
1009:
1008:thin position
1004:
988:
984:
980:
975:
971:
962:
957:
955:
948:
929:
907:
903:
899:
894:
890:
867:
863:
842:
820:
817:
814:
810:
804:
796:
791:
787:
781:
754:
750:
746:
741:
737:
731:
723:
718:
714:
708:
683:
680:
677:
674:
671:
668:
665:
662:
657:
653:
645:and surfaces
632:
629:
626:
623:
620:
617:
614:
611:
606:
602:
598:
593:
589:
581:
577:
573:
565:
563:
561:
557:
553:
549:
545:
544:minimal genus
541:
536:
534:
530:
514:
506:
490:
482:
466:
446:
438:
433:
431:
427:
411:
391:
383:
367:
359:
343:
335:
319:
299:
291:
286:
284:
280:
279:Haken's Lemma
276:
272:
268:
264:
248:
240:
235:
233:
228:
226:
221:
219:
215:
211:
207:
202:
200:
196:
192:
188:
184:
179:
177:
173:
169:
150:
147:
142:
138:
134:
131:
128:
121:
120:
119:
118:
114:
110:
107:. By gluing
106:
102:
98:
94:
93:homeomorphism
90:
86:
82:
78:
70:
68:
66:
62:
55:
49:
41:
37:
33:
19:
2838:math/0101148
2828:
2824:
2790:
2770:
2759:, retrieved
2755:the original
2734:
2730:
2692:
2686:
2670:
2667:Farb, Benson
2653:
2647:
2610:
2580:
2537:Zoltán Szabó
2526:
2456:
2440:
2419:
2408:
2404:Sol geometry
2394:
2385:
2381:
2375:
2341:
2330:
2326:
2295:
2291:
2254:
2230:
2165:
2163:
2013:
1952:
1608:
1525:of the pair
1518:
1514:
1510:
1506:
1349:we may view
1262:homeomorphic
1226:
1135:Three-sphere
1122:
1015:
1011:
1005:
960:
958:
950:
943:
575:
571:
569:
559:
555:
551:
543:
539:
537:
532:
528:
504:
480:
436:
434:
381:
357:
333:
289:
287:
274:
270:
266:
262:
238:
236:
231:
229:
222:
213:
203:
194:
190:
186:
182:
180:
165:
112:
108:
104:
100:
88:
85:handlebodies
80:
76:
74:
65:handlebodies
39:
32:mathematical
29:
2873:3-manifolds
2791:3-manifolds
2737:(1): 1–18,
2409:A paper of
2371:lens spaces
1658:Three-torus
1615:Lens spaces
1497:Hopf bundle
275:irreducible
71:Definitions
2867:Categories
2783:29.0417.02
2761:2020-01-11
2639:References
2457:Meeks and
2415:hyperbolic
2386:horizontal
2127:is unique.
1223:two-sphere
696:such that
290:stabilized
281:that in a
168:orientable
117:3-manifold
61:3-manifold
2743:1300-0098
2711:0166-8641
2072:Joel Hass
1997:−
1967:Γ
1929:×
1910:×
1891:∪
1872:×
1859:×
1840:∪
1821:×
1802:×
1786:Γ
1652:two-torus
1049:χ
1046:−
1029:
981:∪
900:∪
805:−
801:∂
782:−
778:∂
728:∂
705:∂
678:…
627:…
515:β
491:α
467:β
447:α
412:β
392:α
368:β
344:α
320:β
300:α
249:α
239:reducible
232:essential
139:∪
95:from the
87:of genus
34:field of
2857:13991798
2617:See also
2422:Heegaard
2382:vertical
2329:meeting
2080:Theorems
2075:example.
1724:). Let
1609:standard
1437:forms a
1227:standard
1129:Examples
963:if each
97:boundary
2813:0415619
2751:1701636
2719:0918537
2605: (
2585: (
2577:History
1722:circles
1689:is the
1650:of the
540:minimal
269:and in
208:in the
199:isotopy
44:Danish:
30:In the
2855:
2811:
2801:
2781:
2749:
2741:
2717:
2709:
483:where
384:, and
336:where
2853:S2CID
2833:arXiv
2775:(PDF)
1320:with
548:genus
206:coset
176:Smale
172:Moise
2799:ISBN
2739:ISSN
2707:ISSN
2607:1987
2593:and
2587:1898
2535:and
2294:and
2202:and
1955:, a
1513:the
1123:thin
1016:c(S)
949:and
769:and
507:and
459:and
404:and
312:and
79:and
75:Let
38:, a
2843:doi
2779:JFM
2697:doi
2531:of
2392:).
2384:or
2325:in
2253:in
2229:in
2016:in
1959:of
1517:of
1509:in
1264:to
1026:max
922:of
574:of
558:of
542:or
479:on
332:on
261:on
212:of
111:to
99:of
83:be
2869::
2851:,
2841:,
2827:,
2823:,
2809:MR
2807:,
2797:,
2747:MR
2745:,
2735:23
2733:,
2729:,
2715:MR
2713:,
2705:,
2693:27
2691:,
2681:;
2613:.
2573:.
2454:.
1499:.)
1441:,
1014:,
570:A
562:.
360:,
220:.
201:.
67:.
2845::
2835::
2829:5
2699::
2555:h
2552:t
2548:g
2505:3
2500:R
2476:3
2471:R
2355:3
2351:S
2331:H
2327:M
2311:2
2307:S
2296:H
2292:M
2276:1
2272:S
2255:M
2241:H
2231:M
2215:2
2211:H
2188:1
2184:H
2166:M
2146:3
2142:S
2113:3
2109:S
2056:3
2052:T
2029:3
2025:T
2014:V
2000:V
1992:3
1988:T
1953:V
1937:1
1933:S
1926:}
1921:0
1917:x
1913:{
1907:}
1902:0
1898:x
1894:{
1888:}
1883:0
1879:x
1875:{
1867:1
1863:S
1856:}
1851:0
1847:x
1843:{
1837:}
1832:0
1828:x
1824:{
1818:}
1813:0
1809:x
1805:{
1797:1
1793:S
1789:=
1764:1
1760:S
1737:0
1733:x
1720:(
1706:1
1702:S
1675:3
1671:T
1632:3
1628:S
1594:)
1588:2
1584:T
1580:,
1575:3
1571:S
1566:(
1545:)
1542:H
1539:,
1536:M
1533:(
1519:H
1511:M
1507:H
1481:3
1477:S
1454:2
1450:T
1423:2
1417:/
1413:1
1391:2
1386:C
1362:3
1358:S
1335:2
1330:C
1306:4
1301:R
1277:3
1273:S
1242:3
1238:S
1209:z
1206:y
1203:x
1181:4
1176:R
1152:3
1148:S
1105:}
1102:)
1097:i
1093:S
1089:(
1086:c
1083:{
1062:}
1058:)
1055:S
1052:(
1043:1
1040:,
1037:0
1033:{
1012:S
989:i
985:W
976:i
972:V
953:i
951:W
946:i
944:V
930:M
908:i
904:W
895:i
891:V
868:i
864:H
843:M
821:1
818:+
815:i
811:V
797:=
792:i
788:W
755:i
751:H
747:=
742:i
738:W
732:+
724:=
719:i
715:V
709:+
684:n
681:,
675:,
672:1
669:=
666:i
663:,
658:i
654:H
633:n
630:,
624:,
621:1
618:=
615:i
612:,
607:i
603:W
599:,
594:i
590:V
576:M
560:M
552:g
529:W
505:V
481:H
382:W
358:V
334:H
271:W
267:V
263:H
214:H
191:H
183:M
151:.
148:W
143:f
135:V
132:=
129:M
113:W
109:V
105:W
101:V
89:g
81:W
77:V
42:(
20:)
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