2387:
1354:
2362:
2211:
1143:
1161:
521:
59:: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The
998:
2222:
1910:
have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold.
2090:
1012:
796:
1510:
which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface; in fact one can realize any finite group as the group of isometries of some compact
1811:
1707:
1349:{\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\oplus \sum _{i=0}^{n}{\binom {n}{i}}\Gamma _{i+1}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}
705:
1629:
95:. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of homeomorphisms.
439:
231:
904:
2080:; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the
2041:
acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the
1476:
1896:
382:
2357:{\displaystyle 1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} ^{*}(\Sigma )\to \operatorname {Sp} ^{\pm }(H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}^{\pm }(\mathbf {Z} )\to 1}
1568:
857:
270:
594:
306:
91:(again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called
1417:
2491:
2557:
2524:
2448:
2206:{\displaystyle 1\to \operatorname {Tor} (\Sigma )\to \operatorname {MCG} (\Sigma )\to \operatorname {Sp} (H^{1}(\Sigma ))\cong \operatorname {Sp} _{2g}(\mathbf {Z} )\to 1}
1852:
1138:{\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\oplus {\binom {n}{2}}\mathbb {Z} _{2}\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}
1384:
731:
417:
890:
2687:
1748:
1728:, and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.
1647:
647:
2869:
2371:
is well understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the
Torelli group.
3113:
3079:
3045:
3001:
2969:
2945:
2919:
2781:
2744:
1576:
516:{\displaystyle 1\rightarrow \operatorname {Aut} _{0}(X)\rightarrow \operatorname {Aut} (X)\rightarrow \operatorname {MCG} (X)\rightarrow 1.}
1503:
177:
2572:
2883:
993:{\displaystyle 0\to \mathbb {Z} _{2}^{\infty }\to \operatorname {MCG} (\mathbf {T} ^{n})\to \operatorname {GL} (n,\mathbb {Z} )\to 0}
47:
Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of
1428:
3200:
896:
711:
1440:
3134:
2580:
1861:
860:
339:
3216:
2374:
Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the
Torelli group vanishes.
1533:
3100:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 19, European Mathematical Society (EMS), ZĂĽrich,
3066:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 17, European Mathematical Society (EMS), ZĂĽrich,
3022:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 13, European Mathematical Society (EMS), ZĂĽrich,
2988:, IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, European Mathematical Society (EMS), ZĂĽrich,
2875:
1926:
the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of
807:
2006:; moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a
1004:
236:
84:
2635:
325:
1515:(which immediately implies that it injects in the mapping class group of the underlying topological surface).
567:
279:
3221:
2678:
2559:), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the
612:
admits an orientation-reversing automorphism. Similarly, the subgroup that acts as the identity on all the
2774:
Discontinuous Groups and
Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland
2729:
Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1
3156:
2810:
2026:
613:
308:. (Notice that in the compact-open topology, path components and isotopy classes coincide, i.e., two maps
1437:
have been heavily studied, and are sometimes called TeichmĂĽller modular groups (note the special case of
1528:
321:
1527:
surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the
1393:
2644:
2457:
2081:
1479:
527:
135:
3161:
2815:
2905:
2529:
2496:
2420:
2073:
1732:
139:
52:
2980:
1831:
3182:
3174:
3146:
3023:
1979:
1975:
1434:
1362:
791:{\displaystyle \operatorname {MCG} (\mathbf {T} ^{n})\simeq \operatorname {GL} (n,\mathbb {Z} ).}
88:
24:
1502:). The elements of this group have also been studied by themselves: an important result is the
1486:
of
Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to
3109:
3075:
3041:
2997:
2965:
2941:
2933:
2929:
2915:
2879:
2777:
2769:
2740:
723:
534:
395:
167:
32:
2776:. Annals of Mathematics Studies. Vol. 79. Princeton University Press. pp. 207–226.
869:
3166:
3101:
3067:
3033:
2989:
2959:
2909:
2847:
2820:
2732:
2724:
2696:
2652:
2609:
2368:
1491:
1487:
56:
3089:
3055:
3011:
2893:
2791:
2754:
2710:
2664:
3085:
3051:
3007:
2889:
2787:
2750:
2706:
2660:
2046:
1995:
1721:
1512:
1387:
1153:
385:
273:
151:
119:
1717:
2805:
Scharlemann, Martin (February 1982). "The complex of curves on nonorientable surfaces".
2648:
2598:
2060:. Orientation-preserving maps are precisely those that act trivially on top cohomology
1999:
1920:
1499:
1495:
430:
427:
420:
155:
36:
2386:
3210:
3186:
2901:
2851:
2568:
1524:
64:
48:
2701:
2736:
2560:
2003:
1635:
1483:
123:
2630:
3170:
2955:
2865:
2838:
Kojima, S. (August 1988). "Isometry transformations of hyperbolic 3-manifolds".
2682:
2592:
2077:
1806:{\displaystyle \operatorname {MCG} (N_{3})=\operatorname {GL} (2,\mathbb {Z} ).}
1702:{\displaystyle \operatorname {MCG} (K)=\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}.}
20:
2824:
2030:
2007:
1907:
1713:
1507:
1149:
3137:(2007). "The stable moduli space of Riemann surfaces: Mumford's conjecture".
700:{\displaystyle \operatorname {MCG} (S^{2})\simeq \mathbb {Z} /2\mathbb {Z} ,}
83:
range throughout our original topological space, completed with their finite
3130:
2731:. Proceedings of Symposia in Pure Mathematics. Vol. 32. pp. 3–21.
2603:
2576:
2575:). The integral (not just rational) cohomology ring was computed in 2002 by
1725:
2052:
In the case of orientable surfaces, this is the action on first cohomology
604:(as an oriented manifold) would be index two in the mapping class group of
2025:
Notice that there is an induced action of the mapping class group on the
557:
546:
542:
108:
92:
63:
of this new function space will be made up of sets of functions that map
60:
3178:
2656:
1624:{\displaystyle \operatorname {MCG} (\mathbf {P} ^{2}(\mathbb {R} ))=1.}
3151:
3028:
3018:
Lawton, Sean; Peterson, Elisha (2009), Papadopoulos, Athanase (ed.),
2725:"Concordance spaces, higher simple-homotopy theory, and applications"
226:{\displaystyle \operatorname {Aut} (X)/\operatorname {Aut} _{0}(X)}
2493:
by attaching an additional hole on the end (i.e., gluing together
802:
2874:. Annals of Mathematical Studies. Vol. 82. Princeton, N.J.:
2685:(1967), "The diffeomorphism group of a compact Riemann surface",
1490:
and to higher rank linear groups. They have many applications in
320:
they are isotopic). For topological spaces, this is usually the
107:
has a flexible usage. Most often it is used in the context of a
3105:
3071:
3037:
2993:
2381:
384:, where one substitutes for Aut the appropriate group for the
317:
154:, the mapping class group is the group of isotopy classes of
138:, the mapping class group is the group of isotopy classes of
87:(which must be open by definition of topology) and arbitrary
16:
Group of isotopy classes of a topological automorphism group
1506:
theorem, and a generating family for the group is given by
560:
of mapping class groups that are frequently studied. If
2398:
2037:. This is because (co)homology is functorial and Homeo
1858:. As an unoriented surface, its mapping class group is
1820:
has a unique class of one-sided curves such that, when
1986:
is defined to be the mapping class group of the pair (
1471:{\displaystyle \operatorname {MCG} (\mathbf {T} ^{2})}
2532:
2499:
2460:
2423:
2225:
2093:
1864:
1834:
1751:
1650:
1579:
1536:
1443:
1396:
1365:
1164:
1015:
907:
872:
810:
734:
650:
596:
would be the orientation-preserving automorphisms of
570:
442:
398:
342:
282:
239:
180:
2911:
Automorphisms of surfaces after
Nielsen and Thurston
1742:(the connected sum of three projective planes) has:
641:
In any category (smooth, PL, topological, homotopy)
162:. Whenever the group of automorphisms of an object
1891:{\displaystyle \operatorname {GL} (2,\mathbb {Z} )}
2567:whose rational cohomology ring was conjectured by
2551:
2518:
2485:
2442:
2356:
2205:
1890:
1846:
1805:
1701:
1623:
1562:
1470:
1411:
1378:
1348:
1137:
992:
884:
851:
790:
699:
588:
515:
411:
376:
300:
264:
225:
1386:are the Kervaire–Milnor finite abelian groups of
1265:
1252:
1207:
1194:
1058:
1045:
377:{\displaystyle \pi _{0}(\operatorname {Aut} (X))}
35:. Briefly, the mapping class group is a certain
2934:"9. Mapping class groups and arithmetic groups"
892:, one has the following split-exact sequences:
1563:{\displaystyle \mathbf {P} ^{2}(\mathbb {R} )}
2688:Bulletin of the American Mathematical Society
2595:, the mapping class groups of punctured discs
8:
1716:on a two-sided curve which does not bound a
852:{\displaystyle \mathbf {T} ^{n}=(S^{1})^{n}}
2631:"Characteristic classes of surface bundles"
265:{\displaystyle \operatorname {Aut} _{0}(X)}
2807:Journal of the London Mathematical Society
2563:of these groups and inclusions yields the
336:), although it is also frequently denoted
39:corresponding to symmetries of the space.
3160:
3150:
3027:
2814:
2700:
2537:
2531:
2504:
2498:
2465:
2459:
2428:
2422:
2340:
2328:
2320:
2295:
2279:
2254:
2224:
2189:
2174:
2149:
2092:
1881:
1880:
1863:
1833:
1793:
1792:
1765:
1750:
1690:
1686:
1685:
1675:
1671:
1670:
1649:
1605:
1604:
1595:
1590:
1578:
1553:
1552:
1543:
1538:
1535:
1459:
1454:
1442:
1403:
1399:
1398:
1395:
1370:
1364:
1333:
1332:
1305:
1300:
1275:
1264:
1251:
1249:
1243:
1232:
1219:
1215:
1214:
1206:
1193:
1191:
1182:
1177:
1173:
1172:
1163:
1122:
1121:
1094:
1089:
1070:
1066:
1065:
1057:
1044:
1042:
1033:
1028:
1024:
1023:
1014:
977:
976:
949:
944:
925:
920:
916:
915:
906:
871:
843:
833:
817:
812:
809:
778:
777:
750:
745:
733:
690:
689:
681:
677:
676:
664:
649:
569:
453:
441:
403:
397:
347:
341:
281:
244:
238:
205:
196:
179:
31:is an important algebraic invariant of a
3064:Handbook of TeichmĂĽller theory. Vol. III
3098:Handbook of TeichmĂĽller theory. Vol. IV
3020:Handbook of TeichmĂĽller theory. Vol. II
2621:
1984:symmetry group of the knot (resp. link)
1838:
589:{\displaystyle \operatorname {Aut} (M)}
328:literature, the mapping class group of
301:{\displaystyle \operatorname {Aut} (X)}
2982:Handbook of TeichmĂĽller theory. Vol. I
2871:Braids, links and mapping class groups
1712:The four elements are the identity, a
608:(as an unoriented manifold) provided
51:from the space into itself, that is,
7:
3096:Papadopoulos, Athanase, ed. (2014),
3062:Papadopoulos, Athanase, ed. (2012),
2979:Papadopoulos, Athanase, ed. (2007),
2534:
2501:
2462:
2425:
2304:
2266:
2241:
2158:
2127:
2109:
1367:
1272:
1256:
1198:
1183:
1049:
1034:
926:
600:and so the mapping class group of
14:
1930:is defined as an automorphism of
2961:A Primer on Mapping Class Groups
2583:, proving Mumford's conjecture.
2385:
2341:
2190:
1962:Symmetry group of knot and links
1591:
1539:
1455:
1429:Mapping class group of a surface
1412:{\displaystyle \mathbb {Z} _{2}}
1301:
1090:
945:
813:
746:
526:Frequently this sequence is not
2770:"Maximal groups and signatures"
2702:10.1090/S0002-9904-1967-11746-4
2571:(one of conjectures called the
2486:{\displaystyle \Sigma _{g+1,1}}
1824:is cut open along such a curve
1731:We also remark that the closed
1634:The mapping class group of the
1504:Nielsen–Thurston classification
316:are in the same path-component
118:is interpreted as the group of
2964:. Princeton University Press.
2940:. Elsevier. pp. 618–624.
2938:Handbook of Geometric Topology
2914:. Cambridge University Press.
2454:and 1 boundary component into
2348:
2345:
2337:
2310:
2307:
2301:
2288:
2272:
2269:
2263:
2247:
2244:
2238:
2229:
2197:
2194:
2186:
2164:
2161:
2155:
2142:
2133:
2130:
2124:
2115:
2112:
2106:
2097:
1885:
1871:
1797:
1783:
1771:
1758:
1663:
1657:
1612:
1609:
1601:
1586:
1557:
1549:
1465:
1450:
1340:
1337:
1323:
1314:
1311:
1296:
1287:
1168:
1129:
1126:
1112:
1103:
1100:
1085:
1076:
1019:
984:
981:
967:
958:
955:
940:
931:
911:
897:category of topological spaces
840:
826:
782:
768:
756:
741:
670:
657:
583:
577:
507:
504:
498:
489:
486:
480:
471:
468:
462:
446:
371:
368:
362:
353:
295:
289:
259:
253:
220:
214:
193:
187:
1:
2840:Topology and Its Applications
2552:{\displaystyle \Sigma _{1,2}}
2519:{\displaystyle \Sigma _{g,1}}
2443:{\displaystyle \Sigma _{g,1}}
1915:Mapping class groups of pairs
1735:three non-orientable surface
1570:is isotopic to the identity:
537:, the mapping class group of
170:, the mapping class group of
114:. The mapping class group of
2852:10.1016/0166-8641(88)90027-2
2010:is known to be of order two
1994:). The symmetry group of a
1847:{\displaystyle N\setminus C}
1816:This is because the surface
1433:The mapping class groups of
3171:10.4007/annals.2007.165.843
2565:stable mapping class group,
2076:structure, coming from the
1856:a torus with a disk removed
1379:{\displaystyle \Gamma _{i}}
3238:
3124:Stable mapping class group
2876:Princeton University Press
2737:10.1090/pspum/032.1/520490
2629:Morita, Shigeyuki (1987).
2417:One can embed the surface
2378:Stable mapping class group
1478:above), since they act on
1426:
426:So in general, there is a
1419:is the group of order 2.
710:corresponding to maps of
564:is an oriented manifold,
3201:Madsen-Weiss MCG Seminar
2958:; Margalit, Dan (2012).
2825:10.1112/jlms/s2-25.1.171
2768:Greenberg, Leon (1974).
2636:Inventiones Mathematicae
1906:Mapping class groups of
1828:, the resulting surface
1482:and the quotient is the
866:For other categories if
412:{\displaystyle \pi _{0}}
326:low-dimensional topology
2216:One can extend this to
1519:Non-orientable surfaces
1494:'s theory of geometric
885:{\displaystyle n\geq 5}
861:Eilenberg–MacLane space
332:is usually denoted MCG(
2809:. s2-25 (1): 171–184.
2723:Hatcher, A.E. (1978).
2553:
2520:
2487:
2444:
2358:
2207:
1892:
1848:
1807:
1703:
1625:
1564:
1472:
1413:
1380:
1350:
1248:
1139:
994:
886:
853:
792:
701:
590:
517:
413:
378:
302:
266:
227:
3139:Annals of Mathematics
2554:
2521:
2488:
2445:
2359:
2208:
1893:
1849:
1808:
1704:
1626:
1565:
1529:real projective plane
1473:
1414:
1381:
1351:
1228:
1140:
995:
887:
854:
793:
702:
591:
547:homotopy equivalences
518:
414:
379:
322:compact-open topology
303:
267:
228:
55:maps with continuous
23:, in the subfield of
2530:
2497:
2458:
2421:
2223:
2091:
2082:short exact sequence
1862:
1832:
1749:
1648:
1577:
1534:
1441:
1394:
1363:
1162:
1013:
905:
870:
808:
801:This is because the
732:
648:
568:
440:
396:
340:
280:
237:
178:
136:topological manifold
2649:1987InMat..90..551M
2573:Mumford conjectures
2333:
1187:
1038:
930:
803:n-dimensional torus
276:of the identity in
105:mapping class group
29:mapping class group
3217:Geometric topology
2930:Ivanov, Nikolai V.
2679:Earle, Clifford J.
2657:10.1007/bf01389178
2549:
2516:
2483:
2440:
2397:. You can help by
2354:
2316:
2203:
2045:, named after the
1950:is invertible and
1888:
1844:
1803:
1699:
1621:
1560:
1468:
1409:
1376:
1346:
1171:
1135:
1022:
990:
914:
882:
849:
788:
697:
586:
533:If working in the
513:
409:
374:
298:
262:
223:
71:into open subsets
25:geometric topology
3203:; many references
3115:978-3-03719-117-0
3081:978-3-03719-103-3
3047:978-3-03719-055-5
3003:978-3-03719-029-6
2971:978-0-691-14794-9
2947:978-0-08-053285-1
2921:978-1-299-70610-1
2783:978-1-4008-8164-2
2746:978-0-8218-9320-3
2415:
2414:
1498:(for example, to
1488:hyperbolic groups
1480:TeichmĂĽller space
1263:
1205:
1056:
724:homotopy category
535:homotopy category
419:denotes the 0-th
33:topological space
3229:
3190:
3164:
3154:
3118:
3092:
3058:
3031:
3014:
2987:
2975:
2951:
2925:
2897:
2856:
2855:
2835:
2829:
2828:
2818:
2802:
2796:
2795:
2765:
2759:
2758:
2720:
2714:
2713:
2704:
2675:
2669:
2668:
2626:
2610:Lantern relation
2558:
2556:
2555:
2550:
2548:
2547:
2525:
2523:
2522:
2517:
2515:
2514:
2492:
2490:
2489:
2484:
2482:
2481:
2449:
2447:
2446:
2441:
2439:
2438:
2410:
2407:
2389:
2382:
2369:symplectic group
2363:
2361:
2360:
2355:
2344:
2332:
2327:
2300:
2299:
2284:
2283:
2259:
2258:
2212:
2210:
2209:
2204:
2193:
2182:
2181:
2154:
2153:
1897:
1895:
1894:
1889:
1884:
1853:
1851:
1850:
1845:
1812:
1810:
1809:
1804:
1796:
1770:
1769:
1708:
1706:
1705:
1700:
1695:
1694:
1689:
1680:
1679:
1674:
1630:
1628:
1627:
1622:
1608:
1600:
1599:
1594:
1569:
1567:
1566:
1561:
1556:
1548:
1547:
1542:
1477:
1475:
1474:
1469:
1464:
1463:
1458:
1418:
1416:
1415:
1410:
1408:
1407:
1402:
1388:homotopy spheres
1385:
1383:
1382:
1377:
1375:
1374:
1355:
1353:
1352:
1347:
1336:
1310:
1309:
1304:
1286:
1285:
1270:
1269:
1268:
1255:
1247:
1242:
1224:
1223:
1218:
1212:
1211:
1210:
1197:
1186:
1181:
1176:
1148:(⊕ representing
1144:
1142:
1141:
1136:
1125:
1099:
1098:
1093:
1075:
1074:
1069:
1063:
1062:
1061:
1048:
1037:
1032:
1027:
999:
997:
996:
991:
980:
954:
953:
948:
929:
924:
919:
891:
889:
888:
883:
858:
856:
855:
850:
848:
847:
838:
837:
822:
821:
816:
797:
795:
794:
789:
781:
755:
754:
749:
706:
704:
703:
698:
693:
685:
680:
669:
668:
595:
593:
592:
587:
543:homotopy classes
541:is the group of
522:
520:
519:
514:
458:
457:
418:
416:
415:
410:
408:
407:
383:
381:
380:
375:
352:
351:
307:
305:
304:
299:
271:
269:
268:
263:
249:
248:
232:
230:
229:
224:
210:
209:
200:
3237:
3236:
3232:
3231:
3230:
3228:
3227:
3226:
3207:
3206:
3197:
3162:10.1.1.236.2025
3129:
3126:
3121:
3116:
3095:
3082:
3061:
3048:
3017:
3004:
2985:
2978:
2972:
2954:
2948:
2928:
2922:
2900:
2886:
2864:
2860:
2859:
2837:
2836:
2832:
2816:10.1.1.591.2588
2804:
2803:
2799:
2784:
2767:
2766:
2762:
2747:
2722:
2721:
2717:
2677:
2676:
2672:
2628:
2627:
2623:
2618:
2599:Homotopy groups
2589:
2533:
2528:
2527:
2500:
2495:
2494:
2461:
2456:
2455:
2424:
2419:
2418:
2411:
2405:
2402:
2395:needs expansion
2380:
2291:
2275:
2250:
2221:
2220:
2170:
2145:
2089:
2088:
2047:Torelli theorem
2040:
2033:) of the space
2023:
2016:
1998:is known to be
1996:hyperbolic knot
1964:
1934:that preserves
1917:
1904:
1898:. (Lemma 2.1).
1860:
1859:
1830:
1829:
1761:
1747:
1746:
1741:
1722:y-homeomorphism
1684:
1669:
1646:
1645:
1589:
1575:
1574:
1537:
1532:
1531:
1521:
1513:Riemann surface
1500:surface bundles
1496:three-manifolds
1453:
1439:
1438:
1431:
1425:
1397:
1392:
1391:
1366:
1361:
1360:
1299:
1271:
1250:
1213:
1192:
1160:
1159:
1154:smooth category
1088:
1064:
1043:
1011:
1010:
943:
903:
902:
868:
867:
839:
829:
811:
806:
805:
744:
730:
729:
720:
660:
646:
645:
639:
634:
614:homology groups
566:
565:
556:There are many
449:
438:
437:
399:
394:
393:
343:
338:
337:
278:
277:
240:
235:
234:
201:
176:
175:
156:diffeomorphisms
152:smooth manifold
120:isotopy classes
101:
45:
17:
12:
11:
5:
3235:
3233:
3225:
3224:
3222:Homeomorphisms
3219:
3209:
3208:
3205:
3204:
3196:
3195:External links
3193:
3192:
3191:
3145:(3): 843–941.
3135:Weiss, Michael
3125:
3122:
3120:
3119:
3114:
3093:
3080:
3059:
3046:
3015:
3002:
2976:
2970:
2952:
2946:
2926:
2920:
2906:Bleiler, Steve
2902:Casson, Andrew
2898:
2885:978-0691081496
2884:
2861:
2858:
2857:
2846:(3): 297–307.
2830:
2797:
2782:
2760:
2745:
2715:
2695:(4): 557–559,
2670:
2643:(3): 551–577.
2620:
2619:
2617:
2614:
2613:
2612:
2607:
2601:
2596:
2588:
2585:
2546:
2543:
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2513:
2510:
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2503:
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2477:
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2468:
2464:
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2413:
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2392:
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2323:
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2298:
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2290:
2287:
2282:
2278:
2274:
2271:
2268:
2265:
2262:
2257:
2253:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2214:
2213:
2202:
2199:
2196:
2192:
2188:
2185:
2180:
2177:
2173:
2169:
2166:
2163:
2160:
2157:
2152:
2148:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2038:
2022:
2019:
2014:
1963:
1960:
1921:pair of spaces
1916:
1913:
1903:
1900:
1887:
1883:
1879:
1876:
1873:
1870:
1867:
1843:
1840:
1837:
1814:
1813:
1802:
1799:
1795:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1768:
1764:
1760:
1757:
1754:
1739:
1710:
1709:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1665:
1662:
1659:
1656:
1653:
1632:
1631:
1620:
1617:
1614:
1611:
1607:
1603:
1598:
1593:
1588:
1585:
1582:
1559:
1555:
1551:
1546:
1541:
1525:non-orientable
1520:
1517:
1467:
1462:
1457:
1452:
1449:
1446:
1427:Main article:
1424:
1421:
1406:
1401:
1373:
1369:
1357:
1356:
1345:
1342:
1339:
1335:
1331:
1328:
1325:
1322:
1319:
1316:
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1308:
1303:
1298:
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1292:
1289:
1284:
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1278:
1274:
1267:
1262:
1259:
1254:
1246:
1241:
1238:
1235:
1231:
1227:
1222:
1217:
1209:
1204:
1201:
1196:
1190:
1185:
1180:
1175:
1170:
1167:
1146:
1145:
1134:
1131:
1128:
1124:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1097:
1092:
1087:
1084:
1081:
1078:
1073:
1068:
1060:
1055:
1052:
1047:
1041:
1036:
1031:
1026:
1021:
1018:
1001:
1000:
989:
986:
983:
979:
975:
972:
969:
966:
963:
960:
957:
952:
947:
942:
939:
936:
933:
928:
923:
918:
913:
910:
881:
878:
875:
846:
842:
836:
832:
828:
825:
820:
815:
799:
798:
787:
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770:
767:
764:
761:
758:
753:
748:
743:
740:
737:
719:
716:
708:
707:
696:
692:
688:
684:
679:
675:
672:
667:
663:
659:
656:
653:
638:
635:
633:
630:
620:is called the
585:
582:
579:
576:
573:
524:
523:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
456:
452:
448:
445:
431:exact sequence
421:homotopy group
406:
402:
392:belongs. Here
373:
370:
367:
364:
361:
358:
355:
350:
346:
297:
294:
291:
288:
285:
274:path-component
261:
258:
255:
252:
247:
243:
222:
219:
216:
213:
208:
204:
199:
195:
192:
189:
186:
183:
174:is defined as
166:has a natural
140:homeomorphisms
100:
97:
49:homeomorphisms
44:
41:
37:discrete group
15:
13:
10:
9:
6:
4:
3:
2:
3234:
3223:
3220:
3218:
3215:
3214:
3212:
3202:
3199:
3198:
3194:
3188:
3184:
3180:
3176:
3172:
3168:
3163:
3158:
3153:
3148:
3144:
3140:
3136:
3132:
3128:
3127:
3123:
3117:
3111:
3107:
3103:
3099:
3094:
3091:
3087:
3083:
3077:
3073:
3069:
3065:
3060:
3057:
3053:
3049:
3043:
3039:
3035:
3030:
3025:
3021:
3016:
3013:
3009:
3005:
2999:
2995:
2991:
2984:
2983:
2977:
2973:
2967:
2963:
2962:
2957:
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2949:
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2927:
2923:
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2826:
2822:
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2812:
2808:
2801:
2798:
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2779:
2775:
2771:
2764:
2761:
2756:
2752:
2748:
2742:
2738:
2734:
2730:
2726:
2719:
2716:
2712:
2708:
2703:
2698:
2694:
2690:
2689:
2684:
2680:
2674:
2671:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2638:
2637:
2632:
2625:
2622:
2615:
2611:
2608:
2605:
2602:
2600:
2597:
2594:
2591:
2590:
2586:
2584:
2582:
2581:Michael Weiss
2578:
2574:
2570:
2569:David Mumford
2566:
2562:
2544:
2541:
2538:
2511:
2508:
2505:
2478:
2475:
2472:
2469:
2466:
2453:
2435:
2432:
2429:
2409:
2406:December 2009
2400:
2396:
2393:This section
2391:
2388:
2384:
2383:
2377:
2375:
2372:
2370:
2351:
2334:
2329:
2324:
2321:
2317:
2313:
2296:
2292:
2285:
2280:
2276:
2260:
2255:
2251:
2235:
2232:
2226:
2219:
2218:
2217:
2200:
2183:
2178:
2175:
2171:
2167:
2150:
2146:
2139:
2136:
2121:
2118:
2103:
2100:
2094:
2087:
2086:
2085:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2050:
2048:
2044:
2043:Torelli group
2036:
2032:
2028:
2021:Torelli group
2020:
2018:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1961:
1959:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1922:
1914:
1912:
1909:
1901:
1899:
1877:
1874:
1868:
1865:
1857:
1841:
1835:
1827:
1823:
1819:
1800:
1789:
1786:
1780:
1777:
1774:
1766:
1762:
1755:
1752:
1745:
1744:
1743:
1738:
1734:
1729:
1727:
1723:
1719:
1715:
1696:
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1681:
1676:
1666:
1660:
1654:
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1644:
1643:
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1640:
1637:
1618:
1615:
1596:
1583:
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1573:
1572:
1571:
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1530:
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1518:
1516:
1514:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
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1460:
1447:
1444:
1436:
1430:
1422:
1420:
1404:
1389:
1371:
1343:
1329:
1326:
1320:
1317:
1306:
1293:
1290:
1282:
1279:
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1257:
1244:
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1236:
1233:
1229:
1225:
1220:
1202:
1199:
1188:
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1157:
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1155:
1151:
1132:
1118:
1115:
1109:
1106:
1095:
1082:
1079:
1071:
1053:
1050:
1039:
1029:
1016:
1009:
1008:
1007:
1006:
987:
973:
970:
964:
961:
950:
937:
934:
921:
908:
901:
900:
899:
898:
893:
879:
876:
873:
864:
862:
844:
834:
830:
823:
818:
804:
785:
774:
771:
765:
762:
759:
751:
738:
735:
728:
727:
726:
725:
717:
715:
713:
694:
686:
682:
673:
665:
661:
654:
651:
644:
643:
642:
636:
631:
629:
627:
623:
622:Torelli group
619:
615:
611:
607:
603:
599:
580:
574:
571:
563:
559:
554:
552:
548:
544:
540:
536:
531:
529:
510:
501:
495:
492:
483:
477:
474:
465:
459:
454:
450:
443:
436:
435:
434:
432:
429:
424:
422:
404:
400:
391:
387:
365:
359:
356:
348:
344:
335:
331:
327:
323:
319:
315:
311:
292:
286:
283:
275:
256:
250:
245:
241:
217:
211:
206:
202:
197:
190:
184:
181:
173:
169:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
124:automorphisms
121:
117:
113:
110:
106:
98:
96:
94:
90:
86:
85:intersections
82:
78:
74:
70:
66:
62:
58:
54:
50:
42:
40:
38:
34:
30:
26:
22:
3152:math/0212321
3142:
3138:
3097:
3063:
3029:math/0511271
3019:
2981:
2960:
2956:Farb, Benson
2937:
2910:
2870:
2866:Birman, Joan
2843:
2839:
2833:
2806:
2800:
2773:
2763:
2728:
2718:
2692:
2686:
2683:Eells, James
2673:
2640:
2634:
2624:
2593:Braid groups
2564:
2561:direct limit
2451:
2416:
2403:
2399:adding to it
2394:
2373:
2366:
2215:
2069:
2065:
2061:
2057:
2053:
2051:
2042:
2034:
2024:
2011:
1991:
1987:
1983:
1971:
1967:
1965:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1918:
1905:
1855:
1825:
1821:
1817:
1815:
1736:
1730:
1718:Möbius strip
1711:
1638:
1636:Klein bottle
1633:
1522:
1484:moduli space
1432:
1358:
1147:
1002:
894:
865:
800:
721:
709:
640:
625:
621:
617:
609:
605:
601:
597:
561:
555:
550:
538:
532:
525:
425:
423:of a space.
389:
333:
329:
313:
309:
171:
163:
159:
147:
143:
131:
127:
115:
111:
104:
102:
80:
76:
72:
68:
46:
28:
18:
3106:10.4171/117
3072:10.4171/103
3038:10.4171/055
2994:10.4171/029
2078:cup product
1908:3-manifolds
1902:3-Manifolds
1508:Dehn twists
1005:PL-category
433:of groups:
21:mathematics
3211:Categories
3131:Madsen, Ib
2616:References
2074:symplectic
2072:(ÎŁ) has a
2031:cohomology
2008:torus knot
1714:Dehn twist
1152:). In the
1150:direct sum
714: ±1.
324:. In the
99:Definition
93:homotopies
53:continuous
43:Motivation
3187:119721243
3157:CiteSeerX
2908:(2014) .
2811:CiteSeerX
2604:Homeotopy
2577:Ib Madsen
2535:Σ
2502:Σ
2463:Σ
2450:of genus
2426:Σ
2349:→
2335:
2330:±
2314:≅
2305:Σ
2286:
2281:±
2273:→
2267:Σ
2261:
2256:∗
2248:→
2242:Σ
2236:
2230:→
2198:→
2184:
2168:≅
2159:Σ
2140:
2134:→
2128:Σ
2122:
2116:→
2110:Σ
2104:
2098:→
1869:
1839:∖
1781:
1756:
1726:Lickorish
1682:⊕
1655:
1584:
1448:
1368:Γ
1341:→
1321:
1315:→
1294:
1288:→
1273:Γ
1230:∑
1226:⊕
1189:⊕
1184:∞
1169:→
1130:→
1110:
1104:→
1083:
1077:→
1040:⊕
1035:∞
1020:→
985:→
965:
959:→
938:
932:→
927:∞
912:→
877:≥
766:
760:≃
739:
674:≃
655:
575:
558:subgroups
508:→
496:
490:→
478:
472:→
460:
447:→
401:π
388:to which
360:
345:π
287:
251:
212:
185:
130:. So if
103:The term
61:open sets
3179:20160047
2932:(2001).
2868:(1974).
2587:See also
2027:homology
2000:dihedral
1919:Given a
1492:Thurston
1435:surfaces
1423:Surfaces
632:Examples
386:category
233:, where
168:topology
109:manifold
67:subsets
57:inverses
3090:2961353
3056:2524085
3012:2284826
2894:0375281
2792:0379835
2755:0520490
2711:0212840
2665:0914849
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