3777:
The study of pseudo-Anosov diffeomorphisms of a surface is fundamental. They are the most interesting diffeomorphisms, since finite-order mapping classes are isotopic to isometries and thus well understood, and the study of reducible classes indeed essentially reduces to the study of mapping classes
4291:
This action, together with combinatorial and geometric properties of the curve complex, can be used to prove various properties of the mapping class group. In particular, it explains some of the hyperbolic properties of the mapping class group: while as mentioned in the previous section the mapping
4883:
One way to prove this theorem is to deduce it from the properties of the action of the mapping class group on the pants complex: the stabiliser of a vertex is seen to be finitely presented, and the action is cofinite. Since the complex is connected and simply connected it follows that the mapping
5772:
4650:
are contained in a subsurface homeomorphic to a torus then they intersect once, and if the surface is a four-holed sphere they intersect twice). Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible
4884:
class group must be finitely generated. There are other ways of getting finite presentations, but in practice the only one to yield explicit relations for all geni is that described in this paragraph with a slightly different complex instead of the curve complex, called the
4664:
The mapping class group is generated by the subset of Dehn twists about all simple closed curves on the surface. The Dehn–Lickorish theorem states that it is sufficient to select a finite number of those to generate the mapping class group. This generalises the fact that
1757:
A surface with punctures is a compact surface with a finite number of points removed ("punctures"). The mapping class group of such a surface is defined as above (note that the mapping classes are allowed to permute the punctures, but not the boundary components).
1751:
104:
The mapping class group appeared in the first half of the twentieth century. Its origins lie in the study of the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earliest contributors were
2815:
This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a different number of punctures. It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups. It was proven by
2806:
is strictly larger than the image of the mapping class group via the morphism defined in the previous paragraph. The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.
4799:
2974:
6040:
1051:
584:
113:: Dehn proved finite generation of the group, and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of a surface can be represented by homeomorphisms (the Dehn–Nielsen–Baer theorem).
3781:
Pseudo-Anosov mapping classes are "generic" in the mapping class group in various ways. For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.
4875:
It is possible to prove that all relations between the Dehn twists in a generating set for the mapping class group can be written as combinations of a finite number among them. This means that the mapping class group of a surface is a
5479:
5314:
708:
5659:
372:
933:
6139:
whether the mapping class group is a linear group or not. Besides the symplectic representation on homology explained above there are other interesting finite-dimensional linear representations arising from
5371:
of closed curves induces a symplectic form on the first homology, which is preserved by the action of the mapping class group. The surjectivity is proven by showing that the images of Dehn twists generate
3599:
1186:
4066:
can be endowed. In particular, the TeichmĂĽller metric can be used to establish some large-scale properties of the mapping class group, for example that the maximal quasi-isometrically embedded flats in
4141:
of TeichmĂĽller space, and the
Nielsen-Thurston classification of mapping classes can be seen in the dynamical properties of the action on TeichmĂĽller space together with its Thurston boundary. Namely:
1098:
1646:
1559:
2641:
5648:
5550:
5414:
5361:
3044:
1346:
4704:
3543:
1457:
1387:
977:
2525:
4508:
2772:
4554:
1654:
2724:
637:
3992:
2680:
1598:
457:
414:
209:
3654:
3376:
1952:
866:
2577:
1829:
3763:
The main content of the theorem is that a mapping class which is neither of finite order nor reducible must be pseudo-Anosov, which can be defined explicitly by dynamical properties.
6174:
5943:
5838:
5582:
5150:
5057:
4933:
4455:
4383:
4286:
4175:
4097:
4024:
3435:
4648:
2433:
4712:
5505:. It is a finitely generated, torsion-free subgroup and its study is of fundamental importance for its bearing on both the structure of the mapping class group itself (since the
2877:
2025:
3501:
is not null-homotopic this mapping class is nontrivial, and more generally the Dehn twists defined by two non-homotopic curves are distinct elements in the mapping class group.
2242:
1416:
1219:
1127:
765:
5911:
6210:
4200:
Pseudo-Anosov classes fix the two points on the boundary corresponding to their stable and unstable foliation and the action is minimal (has a dense orbit) on the boundary;
5948:
3307:
1851:
982:
4608:
4581:
1511:
1305:
505:
5802:
3403:
3261:
2072:
1878:
5186:
5098:
3953:
2325:
6106:
5864:
4836:
2545:
2475:
2345:
2262:
4132:
3684:
2869:
3897:
3711:
3210:
2372:
2289:
2193:
1259:
4865:
4064:
3845:
1279:
1239:
3618:
There is a classification of the mapping classes on a surface, originally due to
Nielsen and rediscovered by Thurston, which can be stated as follows. An element
831:
6127:
Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.
5209:
6067:
5503:
5249:
5229:
5021:
5001:
4981:
4961:
4423:
4351:
4327:
4254:
4234:
4195:
3917:
3865:
3813:
3754:
3734:
3499:
3479:
3455:
3327:
3281:
3230:
3183:
3163:
3143:
3123:
3103:
3064:
3000:
2843:
2800:
2166:
2146:
2122:
2045:
1485:
805:
785:
736:
497:
477:
269:
249:
229:
158:
6144:. The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of
6812:. Mathematical Notes. Vol. 48. translated from the 1979 French original by Djun M. Kim and Dan Margalit. Princeton University Press. pp. xvi+254.
4288:
on the vertices carries over to the full complex. The action is not properly discontinuous (the stabiliser of a simple closed curve is an infinite group).
5422:
5257:
649:
833:. The latter is not orientation-preserving and we see that the mapping class group of the sphere is trivial, and its extended mapping class group is
5767:{\displaystyle \Phi _{n}:\operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} /n\mathbb {Z} )}
2802:
has a non-empty boundary (except in a finite number of cases). In this case the fundamental group is a free group and the outer automorphism group
2779:
The image of the mapping class group is an index 2 subgroup of the outer automorphism group, which can be characterised by its action on homology.
5650:, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.
277:
231:. This group has a natural topology, the compact-open topology. It can be defined easily by a distance function: if we are given a metric
877:
3551:
479:
which are isotopic to the identity. It is a normal subgroup of the group of positive homeomorphisms, and the mapping class group of
1132:
6179:
In the other direction there is a lower bound for the dimension of a (putative) faithful representation, which has to be at least
120:
who gave the subject a more geometric flavour and used this work to great effect in his program for the study of three-manifolds.
6592:
5874:
The mapping class group has only finitely many classes of finite groups, as follows from the fact that the finite-index subgroup
5866:(this follows easily from a classical result of Minkowski on linear groups and the fact that the Torelli group is torsion-free).
4145:
Finite-order elements fix a point inside TeichmĂĽller space (more concretely this means that any mapping class of finite order in
1056:
4330:
1603:
1516:
6873:
6817:
2585:
5609:
5511:
5375:
5322:
3005:
1310:
642:
This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "
4668:
3507:
1421:
1351:
941:
17:
5103:
The first homology of the mapping class group is finite and it follows that the first cohomology group is finite as well.
6798:
6704:
3613:
2480:
1746:{\displaystyle \operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S,\partial S)/\operatorname {Homeo} _{0}(S,\partial S)}
5584:
boil down to a statement about its
Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.
4467:
4306:
2729:
4513:
2088:
Braid groups can be defined as the mapping class groups of a disc with punctures. More precisely, the braid group on
2690:
603:
6141:
3958:
2646:
1564:
423:
380:
175:
3621:
3332:
7147:
5913:
is torsion-free, as discussed in the previous paragraph. Moreover, this also implies that any finite subgroup of
1886:
4903:
There are other interesting systems of generators for the mapping class group besides Dehn twists. For example,
836:
2550:
1773:
1488:
6147:
6070:
5916:
5811:
5555:
5123:
5030:
4906:
4794:{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\1&1\end{pmatrix}}}
4428:
4356:
4259:
4148:
4070:
4030:
3997:
3408:
110:
6605:; Farb, Benson (2004). "Every mapping class group is generated by 3 torsion elements and by 6 involutions".
4613:
4385:
extends to an action on this complex. This complex is quasi-isometric to TeichmĂĽller space endowed with the
2969:{\displaystyle 1\to \pi _{1}(S,x)\to \operatorname {Mod} (S\setminus \{x\})\to \operatorname {Mod} (S)\to 1}
6786:
6045:
A bound on the order of finite subgroups can also be obtained through geometric means. The solution to the
2377:
6985:
Masur, Howard A.; Minsky, Yair N. (2000). "Geometry of the complex of curves II: Hierarchical structure".
4812:
The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus
4397:
The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. The
2682:. The Dehn–Nielsen–Baer theorem states that it is in addition surjective. In particular, it implies that:
7152:
6046:
1960:
7025:
2198:
1392:
1195:
1103:
741:
165:
6753:
Farb, Benson; Lubotzky, Alexander; Minsky, Yair (2001). "Rank-1 phenomena for mapping class groups".
6049:
implies that any such group is realised as the group of isometries of an hyperbolic surface of genus
6035:{\displaystyle \operatorname {Mod} (S)/\ker(\Phi _{3})\cong \operatorname {Sp} _{2g}(\mathbb {Z} /3)}
5877:
4386:
1046:{\displaystyle \Phi :\operatorname {SL} _{2}(\mathbb {Z} )\to \operatorname {Mod} (\mathbb {T} ^{2})}
6883:
Masbaum, Gregor; Reid, Alan W. (2012). "All finite groups are involved in the mapping class group".
6182:
4877:
4806:
2580:
579:{\displaystyle \operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S)/\operatorname {Homeo} _{0}(S)}
93:
3286:
1834:
6930:
Masur, Howard A.; Minsky, Yair N. (1999). "Geometry of the complex of curves. I. Hyperbolicity".
5598:
5024:
4586:
3955:
a homeomorphism, modulo a suitable equivalence relation. There is an obvious action of the group
4559:
1493:
1284:
81:(indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in
5780:
3381:
3239:
2050:
1856:
124:
5155:
5062:
3926:
3847:
is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on
2294:
6076:
5843:
5252:
4815:
2530:
2438:
2330:
2247:
1767:
55:
4102:
3663:
2848:
7102:
7059:
6969:
6949:
6914:
6774:
6653:
6580:
4458:
3870:
3689:
3188:
2350:
2267:
2171:
1244:
4841:
4040:
3821:
3816:
1264:
1224:
8:
6714:
Eskin, Alex; Masur, Howard; Rafi, Kasra (2017). "Large-scale rank of TeichmĂĽller space".
5653:
An interesting class of finite-index subgroups is given by the kernels of the morphisms:
5368:
4610:, and this "minimally" (this is a technical condition which can be stated as follows: if
810:
74:
6953:
5191:
7063:
7037:
7012:
6994:
6973:
6939:
6918:
6892:
6741:
6723:
6632:
6614:
6052:
5488:
5234:
5214:
5006:
4986:
4966:
4946:
4408:
4336:
4312:
4239:
4219:
4180:
3902:
3850:
3798:
3739:
3719:
3484:
3464:
3440:
3312:
3266:
3215:
3168:
3148:
3128:
3108:
3088:
3049:
2985:
2828:
2785:
2151:
2131:
2107:
2030:
1648:
is the connected component of the identity. The mapping class group is then defined as
1470:
790:
770:
721:
482:
462:
417:
254:
234:
214:
143:
88:
The mapping class group of surfaces are related to various other groups, in particular
63:
39:
710:
induces an isomorphism between the quotients by their respective identity components.
7128:
7111:
6869:
6861:
6852:
6835:
6813:
6794:
6700:
6476:
5117:
4353:(isotopy classes of maximal systems of disjoint simple closed curves). The action of
4138:
4037:). It is compatible with various geometric structures (metric or complex) with which
3772:
7093:
7076:
7026:"A note on the abelianizations of finite-index subgroups of the mapping class group"
7016:
6977:
6922:
6745:
6636:
6559:
Birman, Joan (1969). "Mapping class groups and their relationship to braid groups".
4425:, which are acted upon by, and have trivial stabilisers in, the mapping class group
7123:
7088:
7067:
7047:
7004:
6957:
6902:
6847:
6831:
6762:
6733:
6677:
6628:
6624:
6568:
6117:
5506:
5364:
4892:
4891:
An example of a relation between Dehn twists occurring in this presentation is the
4292:
class group is not a hyperbolic group it has some properties reminiscent of those.
3233:
1513:
then the definition of the mapping class group needs to be more precise. The group
117:
7051:
6766:
3778:
on smaller surfaces which may themselves be either finite order or pseudo-Anosov.
7098:
7055:
6965:
6910:
6770:
6649:
6602:
6576:
5474:{\displaystyle \operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )}
5309:{\displaystyle \operatorname {Mod} (S)\to \operatorname {GL} _{2g}(\mathbb {Z} )}
3920:
2125:
703:{\displaystyle \operatorname {Diff} ^{+}(S)\subset \operatorname {Homeo} ^{+}(S)}
161:
6425:
1189:
643:
6737:
1462:
7141:
6827:
4213:
936:
591:
169:
127:, where it provides a testing ground for various conjectures and techniques.
51:
6906:
123:
More recently the mapping class group has been by itself a central topic in
6836:"A presentation for the mapping class group of a closed orientable surface"
6572:
6136:
5592:
An example of application of the
Torelli subgroup is the following result:
4236:
is a complex whose vertices are isotopy classes of simple closed curves on
2083:
82:
67:
6961:
6868:. Translations of Mathematical Monographs. American Mathematical Society.
6644:
Brock, Jeff (2002). "Pants decompositions and the Weil–Petersson metric".
6597:. Annals of Mathematics Studies. Vol. 82. Princeton University Press.
6329:
5606:
The proof proceeds first by using residual finiteness of the linear group
3545:
the Dehn twists correspond to unipotent matrices. For example, the matrix
2092:
strands is naturally isomorphic to the mapping class group of a disc with
6782:
6588:
6452:
4034:
2817:
89:
6257:
5319:
This map is in fact a surjection with image equal to the integer points
3046:
must be replaced by the finite-index subgroup of mapping classes fixing
1261:(proving in particular that it is injective) and it can be checked that
597:
If we modify the definition to include all homeomorphisms we obtain the
7008:
6682:
6665:
6121:
3080:
713:
59:
6666:"Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flächen"
6999:
6944:
6619:
3604:
corresponds to the Dehn twist about a horizontal curve in the torus.
6488:
4654:
4295:
367:{\displaystyle \delta (f,g)=\sup _{x\in S}\left(d(f(x),g(x))\right)}
211:
the group of orientation-preserving, or positive, homeomorphisms of
6661:
6464:
6401:
639:, which contains the mapping class group as a subgroup of index 2.
106:
78:
31:
7042:
6897:
6728:
6353:
6317:
6293:
6269:
2547:, but only up to conjugation. Thus we get a well-defined map from
2477:
to be the element of the fundamental group associated to the loop
928:{\displaystyle \mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}
116:
The Dehn–Nielsen theory was reinterpreted in the mid-seventies by
6524:
5587:
2803:
1600:
which restrict to the identity on the boundary, and the subgroup
6699:. translated and introduced by John Stillwell. Springer-Verlag.
6536:
6365:
2643:. This map is a morphism and its kernel is exactly the subgroup
58:) deformation. It is of fundamental importance for the study of
7112:"Mapping class group of a surface is generated by two elements"
6808:
Fathi, Albert; Laudenbach, François; Poénaru, Valentin (2012).
6228:
We describe here only "clean, complete" (in the terminology of
4029:
This action has many interesting properties; for example it is
3594:{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
1561:
of homeomorphisms relative to the boundary is the subgroup of
7077:"On the geometry and dynamics of diffeomorphisms of surfaces"
6793:. Princeton Mathematical Series. Princeton University press.
4461:
complex which is quasi-isometric to the mapping class group.
4177:
can be realised as an isometry for some hyperbolic metric on
1181:{\displaystyle x+\mathbb {Z} ^{2}\mapsto Ax+\mathbb {Z} ^{2}}
872:
135:
6245:
6120:: that is, any subgroup of it either contains a non-abelian
5552:
is comparatively very well understood, a lot of facts about
787:
is isotopic to either the identity or to the restriction to
6512:
6500:
6413:
3716:
reducible: there exists a set of disjoint closed curves on
1463:
Mapping class group of surfaces with boundary and punctures
1093:{\displaystyle A\in \operatorname {SL} _{2}(\mathbb {Z} )}
6341:
4805:
In particular, the mapping class group of a surface is a
6305:
6124:
subgroup or it is virtually solvable (in fact abelian).
5106:
4938:
4457:. It is (in opposition to the curve or pants complex) a
3504:
In the mapping class group of the torus identified with
2077:
1641:{\displaystyle \operatorname {Homeo} _{0}(S,\partial S)}
1554:{\displaystyle \operatorname {Homeo} ^{+}(S,\partial S)}
6389:
4203:
Reducible classes do not act minimally on the boundary.
2636:{\displaystyle \operatorname {Out} (\pi _{1}(S,x_{0}))}
1389:. In the same way, the extended mapping class group of
7074:
7023:
6807:
6458:
6440:
6359:
6335:
6263:
5643:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5545:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5409:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5356:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
4760:
4721:
3607:
3560:
3069:
3039:{\displaystyle \operatorname {Mod} (S\setminus \{x\})}
1341:{\displaystyle \operatorname {Mod} (\mathbb {T} ^{2})}
6185:
6150:
6079:
6055:
5951:
5919:
5880:
5846:
5814:
5783:
5662:
5612:
5558:
5514:
5491:
5425:
5378:
5325:
5260:
5237:
5217:
5194:
5158:
5126:
5065:
5033:
5009:
4989:
4969:
4949:
4909:
4844:
4818:
4715:
4699:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
4671:
4616:
4589:
4562:
4516:
4470:
4431:
4411:
4359:
4339:
4315:
4262:
4242:
4222:
4183:
4151:
4105:
4073:
4043:
4000:
3961:
3929:
3905:
3873:
3853:
3824:
3801:
3785:
3742:
3722:
3692:
3666:
3624:
3554:
3538:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
3510:
3487:
3467:
3443:
3411:
3384:
3335:
3315:
3289:
3269:
3242:
3218:
3191:
3171:
3151:
3131:
3111:
3091:
3052:
3008:
2988:
2880:
2851:
2831:
2788:
2732:
2693:
2649:
2588:
2553:
2533:
2483:
2441:
2380:
2353:
2333:
2297:
2270:
2250:
2201:
2174:
2154:
2134:
2110:
2053:
2033:
1963:
1889:
1859:
1837:
1776:
1657:
1606:
1567:
1519:
1496:
1473:
1452:{\displaystyle \operatorname {GL} _{2}(\mathbb {Z} )}
1424:
1395:
1382:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
1354:
1313:
1287:
1267:
1247:
1227:
1198:
1135:
1106:
1059:
985:
972:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
944:
880:
839:
813:
793:
773:
744:
724:
652:
606:
508:
485:
465:
459:. By definition it is equal to the homeomorphisms of
426:
383:
280:
257:
237:
217:
178:
146:
4935:
can be generated by two elements or by involutions.
3457:
made above, and the resulting element is called the
3236:
orientation. This is used to define a homeomorphism
1761:
714:
The mapping class groups of the sphere and the torus
377:
is a distance inducing the compact-open topology on
2520:{\displaystyle {\bar {\gamma }}*f(\alpha )*\gamma }
271:inducing its topology then the function defined by
62:via their embedded surfaces and is also studied in
6204:
6168:
6100:
6061:
6034:
5937:
5905:
5858:
5832:
5796:
5766:
5642:
5576:
5544:
5497:
5473:
5408:
5355:
5308:
5243:
5223:
5203:
5180:
5152:acts by automorphisms on the first homology group
5144:
5092:
5051:
5015:
4995:
4975:
4955:
4927:
4859:
4830:
4793:
4698:
4642:
4602:
4575:
4548:
4503:{\displaystyle \alpha _{1},\ldots ,\alpha _{\xi }}
4502:
4449:
4417:
4377:
4345:
4321:
4280:
4248:
4228:
4189:
4169:
4126:
4091:
4058:
4018:
3986:
3947:
3911:
3891:
3859:
3839:
3807:
3748:
3728:
3705:
3678:
3648:
3593:
3537:
3493:
3473:
3449:
3429:
3397:
3370:
3321:
3301:
3275:
3255:
3224:
3204:
3177:
3157:
3137:
3117:
3097:
3058:
3038:
2994:
2968:
2863:
2837:
2794:
2766:
2718:
2674:
2635:
2571:
2539:
2519:
2469:
2427:
2366:
2339:
2319:
2283:
2256:
2236:
2187:
2160:
2140:
2116:
2066:
2039:
2019:
1957:which is the identity on both boundary components
1946:
1872:
1845:
1823:
1745:
1640:
1592:
1553:
1505:
1479:
1451:
1410:
1381:
1340:
1299:
1273:
1253:
1233:
1213:
1180:
1121:
1092:
1045:
971:
927:
860:
825:
799:
779:
759:
730:
702:
646:" we obtain the same group, that is the inclusion
631:
578:
491:
471:
451:
408:
366:
263:
243:
223:
203:
152:
6377:
6281:
4655:Generators and relations for mapping class groups
4464:A marking is determined by a pants decomposition
4296:Other complexes with a mapping class group action
2782:The conclusion of the theorem does not hold when
2767:{\displaystyle \operatorname {Out} (\pi _{1}(S))}
7139:
7030:Proceedings of the American Mathematical Society
6073:then implies that the maximal order is equal to
4549:{\displaystyle \beta _{1},\ldots ,\beta _{\xi }}
3766:
2099:
303:
6826:
6419:
3125:and one chooses a closed tubular neighbourhood
54:of the surface viewed up to continuous (in the
6713:
6561:Communications on Pure and Applied Mathematics
6347:
5588:Residual finiteness and finite-index subgroups
4898:
4207:
3994:on such pairs, which descends to an action of
3790:
2726:is isomorphic to the outer automorphism group
2719:{\displaystyle \operatorname {Mod} ^{\pm }(S)}
632:{\displaystyle \operatorname {Mod} ^{\pm }(S)}
7081:Bulletin of the American Mathematical Society
6929:
6752:
6542:
6371:
6111:
4659:
3987:{\displaystyle \operatorname {Homeo} ^{+}(S)}
3002:itself has punctures the mapping class group
2675:{\displaystyle \operatorname {Homeo} _{0}(S)}
2527:. This automorphism depends on the choice of
1593:{\displaystyle \operatorname {Homeo} ^{+}(S)}
1188:. The action of diffeomorphisms on the first
452:{\displaystyle \operatorname {Homeo} _{0}(S)}
409:{\displaystyle \operatorname {Homeo} ^{+}(S)}
204:{\displaystyle \operatorname {Homeo} ^{+}(S)}
6882:
6781:
6530:
6494:
6482:
6470:
6407:
6323:
6299:
6275:
3649:{\displaystyle g\in \operatorname {Mod} (S)}
3371:{\displaystyle f^{-1}\circ \tau _{0}\circ f}
3030:
3024:
2936:
2930:
2820:in 1969. The exact statement is as follows.
2810:
2014:
1992:
1986:
1964:
1818:
1790:
6984:
6395:
6229:
1947:{\displaystyle \tau _{0}(z)=e^{2i\pi |z|}z}
6660:
6601:
6446:
6251:
861:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
136:Mapping class group of orientable surfaces
130:
7127:
7092:
7041:
6998:
6943:
6896:
6851:
6727:
6681:
6646:Complex Manifolds and Hyperbolic Geometry
6618:
6130:
6017:
5757:
5744:
5714:
5633:
5535:
5464:
5399:
5346:
5299:
4689:
4256:. The action of the mapping class groups
3528:
2572:{\displaystyle \operatorname {Homeo} (S)}
1839:
1824:{\displaystyle A_{0}=\{1\leq |z|\leq 2\}}
1442:
1398:
1372:
1325:
1201:
1168:
1144:
1109:
1083:
1030:
1009:
962:
915:
898:
883:
854:
841:
747:
7109:
6431:
4870:
6866:Subgroups of TeichmĂĽller Modular Groups
6594:Braids, links, and mapping class groups
6169:{\displaystyle \operatorname {Mod} (S)}
5938:{\displaystyle \operatorname {Mod} (S)}
5833:{\displaystyle \operatorname {Mod} (S)}
5577:{\displaystyle \operatorname {Mod} (S)}
5188:. This is a free abelian group of rank
5145:{\displaystyle \operatorname {Mod} (S)}
5120:is functorial, the mapping class group
5111:
5052:{\displaystyle \operatorname {Mod} (S)}
4928:{\displaystyle \operatorname {Mod} (S)}
4450:{\displaystyle \operatorname {Mod} (S)}
4378:{\displaystyle \operatorname {Mod} (S)}
4281:{\displaystyle \operatorname {Mod} (S)}
4170:{\displaystyle \operatorname {Mod} (S)}
4092:{\displaystyle \operatorname {Mod} (S)}
4019:{\displaystyle \operatorname {Mod} (S)}
3430:{\displaystyle \operatorname {Mod} (S)}
14:
7140:
6860:
6587:
6558:
6518:
6506:
6311:
6287:
4867:; this was proven later by Humphries.
4643:{\displaystyle \alpha _{i},\beta _{i}}
4510:and a collection of transverse curves
3105:is an oriented simple closed curve on
1487:is a compact surface with a non-empty
30:In mathematics, and more precisely in
6643:
6383:
6116:The mapping class groups satisfy the
5840:. It is a torsion-free group for all
5107:Subgroups of the mapping class groups
4939:Cohomology of the mapping class group
2428:{\displaystyle \in \pi _{1}(S,x_{0})}
2078:Braid groups and mapping class groups
979:. It is easy to construct a morphism
6694:
6688:
6360:Fathi, Laudenbach & Poénaru 2012
6336:Fathi, Laudenbach & Poénaru 2012
5367:. This comes from the fact that the
4329:is a complex whose vertices are the
3736:which is preserved by the action of
6987:Geometric & Functional Analysis
6697:Papers on group theory and topology
5869:
4392:
3660:of finite order (i.e. there exists
3608:The Nielsen–Thurston classification
3070:Elements of the mapping class group
2168:then we can define an automorphism
2020:{\displaystyle \{|z|=1\},\{|z|=2\}}
418:connected component of the identity
24:
5982:
5945:is a subgroup of the finite group
5891:
5785:
5664:
3786:Actions of the mapping class group
2047:is then generated by the class of
1853:. One can define a diffeomorphism
1734:
1698:
1629:
1542:
1497:
1294:
1288:
1268:
1248:
1228:
986:
25:
7164:
6648:. American Mathematical Society.
3867:. These are represented by pairs
3437:does not depend on the choice of
3293:
3021:
2927:
2687:The extended mapping class group
2237:{\displaystyle \pi _{1}(S,x_{0})}
1762:Mapping class group of an annulus
1307:are inverse isomorphisms between
935:is naturally identified with the
6791:A primer on mapping class groups
6485:, Theorem 6.15 and Theorem 6.12.
6142:topological quantum field theory
4401:is a complex whose vertices are
4300:
1411:{\displaystyle \mathbb {T} ^{2}}
1214:{\displaystyle \mathbb {T} ^{2}}
1122:{\displaystyle \mathbb {T} ^{2}}
760:{\displaystyle \mathbb {R} ^{3}}
7094:10.1090/s0273-0979-1988-15685-6
6222:
5906:{\displaystyle \ker(\Phi _{3})}
3815:(usually without boundary) the
3614:Nielsen–Thurston classification
871:The mapping class group of the
868:, the cyclic group of order 2.
6629:10.1016/j.jalgebra.2004.02.019
6205:{\displaystyle 2{\sqrt {g-1}}}
6163:
6157:
6095:
6083:
6029:
6013:
5991:
5978:
5964:
5958:
5932:
5926:
5900:
5887:
5827:
5821:
5761:
5740:
5721:
5718:
5710:
5691:
5688:
5682:
5637:
5629:
5571:
5565:
5539:
5531:
5468:
5460:
5441:
5438:
5432:
5403:
5395:
5350:
5342:
5303:
5295:
5276:
5273:
5267:
5175:
5169:
5139:
5133:
5046:
5040:
4922:
4916:
4693:
4685:
4651:higher-dimensional simplices.
4583:intersects at most one of the
4444:
4438:
4372:
4366:
4275:
4269:
4164:
4158:
4086:
4080:
4053:
4047:
4013:
4007:
3981:
3975:
3939:
3886:
3874:
3834:
3828:
3643:
3637:
3532:
3524:
3424:
3418:
3145:then there is a homeomorphism
3074:
3033:
3015:
2960:
2957:
2951:
2942:
2939:
2921:
2912:
2909:
2897:
2884:
2761:
2758:
2752:
2739:
2713:
2707:
2669:
2663:
2630:
2627:
2608:
2595:
2566:
2560:
2508:
2502:
2490:
2464:
2461:
2455:
2452:
2422:
2403:
2387:
2381:
2314:
2301:
2231:
2212:
2004:
1996:
1976:
1968:
1935:
1927:
1906:
1900:
1808:
1800:
1770:is homeomorphic to the subset
1740:
1725:
1704:
1689:
1670:
1664:
1635:
1620:
1587:
1581:
1548:
1533:
1446:
1438:
1376:
1368:
1335:
1320:
1154:
1087:
1079:
1040:
1025:
1016:
1013:
1005:
966:
958:
697:
691:
672:
666:
626:
620:
573:
567:
546:
540:
521:
515:
446:
440:
403:
397:
356:
353:
347:
338:
332:
326:
296:
284:
198:
192:
13:
1:
7075:Thurston, William P. (1988).
7052:10.1090/s0002-9939-09-10124-7
6767:10.1215/s0012-7094-01-10636-4
4706:is generated by the matrices
3767:Pseudo-Anosov diffeomorphisms
2871:. There is an exact sequence
2100:The Dehn–Nielsen–Baer theorem
2027:. The mapping class group of
807:of the symmetry in the plane
420:for this topology is denoted
77:can be defined for arbitrary
7129:10.1016/0040-9383(95)00037-2
6853:10.1016/0040-9383(80)90009-9
6348:Eskin, Masur & Rafi 2017
6239:
3302:{\displaystyle S\setminus A}
1846:{\displaystyle \mathbb {C} }
1100:induces a diffeomorphism of
767:. Then any homeomorphism of
599:extended mapping class group
7:
6810:Thurston's work on surfaces
6459:Proc. Amer. Math. Soc. 2010
6420:Hatcher & Thurston 1980
6264:Bull. Amer. Math. Soc. 1988
6047:Nielsen realisation problem
5597:The mapping class group is
5419:The kernel of the morphism
5251:. This action thus gives a
5023:punctures then the virtual
4899:Other systems of generators
4603:{\displaystyle \alpha _{i}}
4556:such that every one of the
4208:Action on the curve complex
3791:Action on TeichmĂĽller space
3405:in the mapping class group
3309:it is the identity, and on
10:
7169:
6551:
6112:General facts on subgroups
4660:The Dehn–Lickorish theorem
4576:{\displaystyle \beta _{i}}
4137:The action extends to the
3795:Given a punctured surface
3770:
3611:
3078:
2081:
1880:by the following formula:
1506:{\displaystyle \partial S}
1300:{\displaystyle \Pi ,\Phi }
99:
6755:Duke Mathematical Journal
6738:10.1215/00127094-0000006X
6716:Duke Mathematical Journal
6230:Masur & Minsky (2000)
5797:{\displaystyle \Phi _{n}}
3398:{\displaystyle \tau _{c}}
3256:{\displaystyle \tau _{c}}
3185:to the canonical annulus
2845:be a compact surface and
2811:The Birman exact sequence
2195:of the fundamental group
2067:{\displaystyle \tau _{0}}
1873:{\displaystyle \tau _{0}}
94:outer automorphism groups
48:TeichmĂĽller modular group
6932:Inventiones Mathematicae
6495:Farb & Margalit 2012
6483:Farb & Margalit 2012
6471:Farb & Margalit 2012
6408:Farb & Margalit 2012
6324:Farb & Margalit 2012
6300:Farb & Margalit 2012
6276:Farb & Margalit 2012
6215:
5181:{\displaystyle H_{1}(S)}
5093:{\displaystyle 4g-4+b+k}
5003:boundary components and
4878:finitely presented group
4807:finitely generated group
3948:{\displaystyle f:S\to X}
2581:outer automorphism group
2374:representing an element
2320:{\displaystyle f(x_{0})}
6907:10.2140/gt.2012.16.1393
6885:Geometry & Topology
6396:Masur & Minsky 2000
6101:{\displaystyle 84(g-1)}
5859:{\displaystyle n\geq 3}
5025:cohomological dimension
4831:{\displaystyle g\geq 2}
3212:defined above, sending
2540:{\displaystyle \gamma }
2470:{\displaystyle f_{*}()}
2340:{\displaystyle \alpha }
2257:{\displaystyle \gamma }
2244:as follows: fix a path
131:Definition and examples
42:, sometimes called the
7148:Geometric group theory
6573:10.1002/cpa.3160220206
6206:
6170:
6131:Linear representations
6102:
6063:
6036:
5939:
5907:
5860:
5834:
5798:
5768:
5644:
5578:
5546:
5499:
5475:
5410:
5357:
5310:
5245:
5225:
5205:
5182:
5146:
5094:
5053:
5017:
4997:
4977:
4963:is a surface of genus
4957:
4929:
4861:
4832:
4795:
4700:
4644:
4604:
4577:
4550:
4504:
4451:
4419:
4379:
4347:
4323:
4282:
4250:
4230:
4191:
4171:
4128:
4127:{\displaystyle 3g-3+k}
4093:
4060:
4031:properly discontinuous
4026:on TeichmĂĽller space.
4020:
3988:
3949:
3913:
3893:
3861:
3841:
3809:
3750:
3730:
3707:
3680:
3679:{\displaystyle n>0}
3650:
3595:
3539:
3495:
3475:
3451:
3431:
3399:
3372:
3323:
3303:
3277:
3257:
3226:
3206:
3179:
3159:
3139:
3119:
3099:
3060:
3040:
2996:
2970:
2865:
2864:{\displaystyle x\in S}
2839:
2796:
2768:
2720:
2676:
2637:
2573:
2541:
2521:
2471:
2429:
2368:
2341:
2321:
2285:
2258:
2238:
2189:
2162:
2148:is a homeomorphism of
2142:
2118:
2068:
2041:
2021:
1948:
1874:
1847:
1825:
1747:
1642:
1594:
1555:
1507:
1481:
1453:
1412:
1383:
1342:
1301:
1281:is injective, so that
1275:
1255:
1235:
1215:
1182:
1123:
1094:
1047:
973:
929:
862:
827:
801:
781:
761:
738:is the unit sphere in
732:
704:
633:
580:
493:
473:
453:
410:
368:
265:
245:
225:
205:
154:
125:geometric group theory
27:Concept in mathematics
7024:Putman, Andy (2010).
6962:10.1007/s002220050343
6533:, pp. 1393–1411.
6207:
6171:
6103:
6064:
6037:
5940:
5908:
5861:
5835:
5799:
5769:
5645:
5579:
5547:
5500:
5476:
5411:
5358:
5311:
5253:linear representation
5246:
5226:
5206:
5183:
5147:
5095:
5054:
5018:
4998:
4978:
4958:
4930:
4871:Finite presentability
4862:
4833:
4796:
4701:
4645:
4605:
4578:
4551:
4505:
4452:
4420:
4387:Weil–Petersson metric
4380:
4348:
4324:
4309:of a compact surface
4283:
4251:
4231:
4192:
4172:
4129:
4094:
4061:
4021:
3989:
3950:
3914:
3894:
3892:{\displaystyle (X,f)}
3862:
3842:
3810:
3751:
3731:
3708:
3706:{\displaystyle g^{n}}
3681:
3651:
3596:
3540:
3496:
3476:
3452:
3432:
3400:
3373:
3324:
3304:
3278:
3258:
3232:to a circle with the
3227:
3207:
3205:{\displaystyle A_{0}}
3180:
3160:
3140:
3120:
3100:
3061:
3041:
2997:
2971:
2866:
2840:
2797:
2769:
2721:
2677:
2638:
2574:
2542:
2522:
2472:
2430:
2369:
2367:{\displaystyle x_{0}}
2342:
2322:
2286:
2284:{\displaystyle x_{0}}
2259:
2239:
2190:
2188:{\displaystyle f_{*}}
2163:
2143:
2119:
2069:
2042:
2022:
1949:
1875:
1848:
1826:
1748:
1643:
1595:
1556:
1508:
1482:
1454:
1413:
1384:
1343:
1302:
1276:
1256:
1254:{\displaystyle \Phi }
1236:
1221:gives a left-inverse
1216:
1183:
1124:
1095:
1048:
974:
930:
863:
828:
802:
782:
762:
733:
705:
634:
581:
494:
474:
454:
411:
369:
266:
246:
226:
206:
155:
70:problems for curves.
56:compact-open topology
7110:Wajnryb, B. (1996).
6183:
6148:
6077:
6053:
5949:
5917:
5878:
5844:
5812:
5804:is usually called a
5781:
5660:
5610:
5556:
5512:
5489:
5423:
5376:
5323:
5258:
5235:
5215:
5192:
5156:
5124:
5112:The Torelli subgroup
5063:
5031:
5007:
4987:
4967:
4947:
4907:
4860:{\displaystyle 2g+1}
4842:
4816:
4713:
4669:
4614:
4587:
4560:
4514:
4468:
4429:
4409:
4357:
4337:
4331:pants decompositions
4313:
4260:
4240:
4220:
4181:
4149:
4103:
4071:
4059:{\displaystyle T(S)}
4041:
3998:
3959:
3927:
3903:
3871:
3851:
3840:{\displaystyle T(S)}
3822:
3799:
3740:
3720:
3690:
3664:
3622:
3552:
3508:
3485:
3465:
3441:
3409:
3382:
3333:
3313:
3287:
3267:
3240:
3216:
3189:
3169:
3149:
3129:
3109:
3089:
3050:
3006:
2986:
2878:
2849:
2829:
2786:
2730:
2691:
2647:
2586:
2551:
2531:
2481:
2439:
2378:
2351:
2331:
2295:
2268:
2248:
2199:
2172:
2152:
2132:
2108:
2051:
2031:
1961:
1887:
1857:
1835:
1774:
1655:
1604:
1565:
1517:
1494:
1471:
1422:
1393:
1352:
1311:
1285:
1274:{\displaystyle \Pi }
1265:
1245:
1234:{\displaystyle \Pi }
1225:
1196:
1133:
1104:
1057:
983:
942:
878:
837:
811:
791:
771:
742:
722:
650:
604:
506:
483:
463:
424:
381:
278:
255:
235:
215:
176:
144:
18:Dehn-Nielsen theorem
6954:1999InMat.138..103M
6545:, pp. 581–597.
6461:, pp. 753–758.
6437:, pp. 377–383.
6374:, pp. 103–149.
6314:, pp. 213–238.
6266:, pp. 417–431.
6254:, pp. 135–206.
5806:congruence subgroup
5369:intersection number
5231:is closed of genus
826:{\displaystyle z=0}
75:mapping class group
36:mapping class group
7153:Geometric topology
7009:10.1007/pl00001643
6695:Dehn, Max (1987).
6683:10.1007/bf02547712
6543:Duke Math. J. 2001
6372:Invent. Math. 1999
6202:
6166:
6098:
6059:
6032:
5935:
5903:
5856:
5830:
5794:
5764:
5640:
5574:
5542:
5495:
5471:
5406:
5353:
5306:
5241:
5221:
5204:{\displaystyle 2g}
5201:
5178:
5142:
5090:
5049:
5013:
4993:
4973:
4953:
4925:
4886:cut system complex
4857:
4828:
4791:
4785:
4746:
4696:
4640:
4600:
4573:
4546:
4500:
4447:
4415:
4375:
4343:
4319:
4278:
4246:
4226:
4187:
4167:
4124:
4089:
4056:
4016:
3984:
3945:
3909:
3889:
3857:
3837:
3805:
3746:
3726:
3703:
3676:
3646:
3591:
3585:
3535:
3491:
3471:
3447:
3427:
3395:
3368:
3319:
3299:
3273:
3253:
3222:
3202:
3175:
3155:
3135:
3115:
3095:
3056:
3036:
2992:
2982:In the case where
2966:
2861:
2835:
2792:
2764:
2716:
2672:
2633:
2569:
2537:
2517:
2467:
2425:
2364:
2337:
2317:
2281:
2254:
2234:
2185:
2158:
2138:
2114:
2064:
2037:
2017:
1944:
1870:
1843:
1821:
1743:
1638:
1590:
1551:
1503:
1477:
1467:In the case where
1449:
1408:
1379:
1338:
1297:
1271:
1251:
1231:
1211:
1178:
1119:
1090:
1043:
969:
925:
858:
823:
797:
777:
757:
728:
700:
629:
576:
489:
469:
449:
406:
364:
317:
261:
241:
221:
201:
150:
64:algebraic geometry
50:, is the group of
6875:978-1-4704-4526-3
6832:Thurston, William
6819:978-0-691-14735-2
6531:Geom. Topol. 2012
6200:
6062:{\displaystyle g}
5599:residually finite
5498:{\displaystyle S}
5244:{\displaystyle g}
5224:{\displaystyle S}
5118:singular homology
5016:{\displaystyle k}
4996:{\displaystyle b}
4976:{\displaystyle g}
4956:{\displaystyle S}
4418:{\displaystyle S}
4346:{\displaystyle S}
4322:{\displaystyle S}
4249:{\displaystyle S}
4229:{\displaystyle S}
4190:{\displaystyle S}
4139:Thurston boundary
4099:are of dimension
3912:{\displaystyle X}
3860:{\displaystyle S}
3817:TeichmĂĽller space
3808:{\displaystyle S}
3773:Pseudo-Anosov map
3759:or pseudo-Anosov.
3749:{\displaystyle g}
3729:{\displaystyle S}
3713:is the identity),
3494:{\displaystyle c}
3474:{\displaystyle c}
3450:{\displaystyle f}
3322:{\displaystyle A}
3276:{\displaystyle S}
3225:{\displaystyle c}
3178:{\displaystyle A}
3158:{\displaystyle f}
3138:{\displaystyle A}
3118:{\displaystyle S}
3098:{\displaystyle c}
3059:{\displaystyle x}
2995:{\displaystyle S}
2838:{\displaystyle S}
2795:{\displaystyle S}
2493:
2161:{\displaystyle S}
2141:{\displaystyle f}
2117:{\displaystyle S}
2040:{\displaystyle A}
1480:{\displaystyle S}
800:{\displaystyle S}
780:{\displaystyle S}
731:{\displaystyle S}
492:{\displaystyle S}
472:{\displaystyle S}
302:
264:{\displaystyle S}
244:{\displaystyle d}
224:{\displaystyle S}
153:{\displaystyle S}
16:(Redirected from
7160:
7133:
7131:
7106:
7096:
7071:
7045:
7020:
7002:
6981:
6947:
6926:
6900:
6891:(3): 1393–1411.
6879:
6857:
6855:
6823:
6804:
6778:
6749:
6731:
6710:
6687:, translated in
6686:
6685:
6670:Acta Mathematica
6657:
6640:
6622:
6603:Brendle, Tara E.
6598:
6584:
6546:
6540:
6534:
6528:
6522:
6516:
6510:
6504:
6498:
6492:
6486:
6480:
6474:
6468:
6462:
6456:
6450:
6444:
6438:
6429:
6423:
6417:
6411:
6405:
6399:
6393:
6387:
6381:
6375:
6369:
6363:
6357:
6351:
6345:
6339:
6333:
6327:
6321:
6315:
6309:
6303:
6297:
6291:
6285:
6279:
6273:
6267:
6261:
6255:
6249:
6233:
6226:
6211:
6209:
6208:
6203:
6201:
6190:
6175:
6173:
6172:
6167:
6118:Tits alternative
6107:
6105:
6104:
6099:
6068:
6066:
6065:
6060:
6041:
6039:
6038:
6033:
6025:
6020:
6009:
6008:
5990:
5989:
5971:
5944:
5942:
5941:
5936:
5912:
5910:
5909:
5904:
5899:
5898:
5870:Finite subgroups
5865:
5863:
5862:
5857:
5839:
5837:
5836:
5831:
5803:
5801:
5800:
5795:
5793:
5792:
5773:
5771:
5770:
5765:
5760:
5752:
5747:
5736:
5735:
5717:
5706:
5705:
5672:
5671:
5649:
5647:
5646:
5641:
5636:
5625:
5624:
5583:
5581:
5580:
5575:
5551:
5549:
5548:
5543:
5538:
5527:
5526:
5507:arithmetic group
5504:
5502:
5501:
5496:
5480:
5478:
5477:
5472:
5467:
5456:
5455:
5415:
5413:
5412:
5407:
5402:
5391:
5390:
5365:symplectic group
5362:
5360:
5359:
5354:
5349:
5338:
5337:
5315:
5313:
5312:
5307:
5302:
5291:
5290:
5250:
5248:
5247:
5242:
5230:
5228:
5227:
5222:
5210:
5208:
5207:
5202:
5187:
5185:
5184:
5179:
5168:
5167:
5151:
5149:
5148:
5143:
5099:
5097:
5096:
5091:
5058:
5056:
5055:
5050:
5022:
5020:
5019:
5014:
5002:
5000:
4999:
4994:
4982:
4980:
4979:
4974:
4962:
4960:
4959:
4954:
4934:
4932:
4931:
4926:
4893:lantern relation
4866:
4864:
4863:
4858:
4837:
4835:
4834:
4829:
4800:
4798:
4797:
4792:
4790:
4789:
4751:
4750:
4705:
4703:
4702:
4697:
4692:
4681:
4680:
4649:
4647:
4646:
4641:
4639:
4638:
4626:
4625:
4609:
4607:
4606:
4601:
4599:
4598:
4582:
4580:
4579:
4574:
4572:
4571:
4555:
4553:
4552:
4547:
4545:
4544:
4526:
4525:
4509:
4507:
4506:
4501:
4499:
4498:
4480:
4479:
4456:
4454:
4453:
4448:
4424:
4422:
4421:
4416:
4399:markings complex
4393:Markings complex
4384:
4382:
4381:
4376:
4352:
4350:
4349:
4344:
4328:
4326:
4325:
4320:
4287:
4285:
4284:
4279:
4255:
4253:
4252:
4247:
4235:
4233:
4232:
4227:
4196:
4194:
4193:
4188:
4176:
4174:
4173:
4168:
4133:
4131:
4130:
4125:
4098:
4096:
4095:
4090:
4065:
4063:
4062:
4057:
4025:
4023:
4022:
4017:
3993:
3991:
3990:
3985:
3971:
3970:
3954:
3952:
3951:
3946:
3918:
3916:
3915:
3910:
3898:
3896:
3895:
3890:
3866:
3864:
3863:
3858:
3846:
3844:
3843:
3838:
3814:
3812:
3811:
3806:
3755:
3753:
3752:
3747:
3735:
3733:
3732:
3727:
3712:
3710:
3709:
3704:
3702:
3701:
3685:
3683:
3682:
3677:
3655:
3653:
3652:
3647:
3600:
3598:
3597:
3592:
3590:
3589:
3544:
3542:
3541:
3536:
3531:
3520:
3519:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3456:
3454:
3453:
3448:
3436:
3434:
3433:
3428:
3404:
3402:
3401:
3396:
3394:
3393:
3377:
3375:
3374:
3369:
3361:
3360:
3348:
3347:
3328:
3326:
3325:
3320:
3308:
3306:
3305:
3300:
3282:
3280:
3279:
3274:
3262:
3260:
3259:
3254:
3252:
3251:
3234:counterclockwise
3231:
3229:
3228:
3223:
3211:
3209:
3208:
3203:
3201:
3200:
3184:
3182:
3181:
3176:
3164:
3162:
3161:
3156:
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3104:
3102:
3101:
3096:
3065:
3063:
3062:
3057:
3045:
3043:
3042:
3037:
3001:
2999:
2998:
2993:
2975:
2973:
2972:
2967:
2896:
2895:
2870:
2868:
2867:
2862:
2844:
2842:
2841:
2836:
2801:
2799:
2798:
2793:
2773:
2771:
2770:
2765:
2751:
2750:
2725:
2723:
2722:
2717:
2703:
2702:
2681:
2679:
2678:
2673:
2659:
2658:
2642:
2640:
2639:
2634:
2626:
2625:
2607:
2606:
2578:
2576:
2575:
2570:
2546:
2544:
2543:
2538:
2526:
2524:
2523:
2518:
2495:
2494:
2486:
2476:
2474:
2473:
2468:
2451:
2450:
2434:
2432:
2431:
2426:
2421:
2420:
2402:
2401:
2373:
2371:
2370:
2365:
2363:
2362:
2346:
2344:
2343:
2338:
2326:
2324:
2323:
2318:
2313:
2312:
2290:
2288:
2287:
2282:
2280:
2279:
2263:
2261:
2260:
2255:
2243:
2241:
2240:
2235:
2230:
2229:
2211:
2210:
2194:
2192:
2191:
2186:
2184:
2183:
2167:
2165:
2164:
2159:
2147:
2145:
2144:
2139:
2123:
2121:
2120:
2115:
2073:
2071:
2070:
2065:
2063:
2062:
2046:
2044:
2043:
2038:
2026:
2024:
2023:
2018:
2007:
1999:
1979:
1971:
1953:
1951:
1950:
1945:
1940:
1939:
1938:
1930:
1899:
1898:
1879:
1877:
1876:
1871:
1869:
1868:
1852:
1850:
1849:
1844:
1842:
1830:
1828:
1827:
1822:
1811:
1803:
1786:
1785:
1752:
1750:
1749:
1744:
1721:
1720:
1711:
1685:
1684:
1647:
1645:
1644:
1639:
1616:
1615:
1599:
1597:
1596:
1591:
1577:
1576:
1560:
1558:
1557:
1552:
1529:
1528:
1512:
1510:
1509:
1504:
1486:
1484:
1483:
1478:
1458:
1456:
1455:
1450:
1445:
1434:
1433:
1417:
1415:
1414:
1409:
1407:
1406:
1401:
1388:
1386:
1385:
1380:
1375:
1364:
1363:
1347:
1345:
1344:
1339:
1334:
1333:
1328:
1306:
1304:
1303:
1298:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1241:to the morphism
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1210:
1209:
1204:
1187:
1185:
1184:
1179:
1177:
1176:
1171:
1153:
1152:
1147:
1128:
1126:
1125:
1120:
1118:
1117:
1112:
1099:
1097:
1096:
1091:
1086:
1075:
1074:
1052:
1050:
1049:
1044:
1039:
1038:
1033:
1012:
1001:
1000:
978:
976:
975:
970:
965:
954:
953:
934:
932:
931:
926:
924:
923:
918:
912:
907:
906:
901:
892:
891:
886:
867:
865:
864:
859:
857:
849:
844:
832:
830:
829:
824:
806:
804:
803:
798:
786:
784:
783:
778:
766:
764:
763:
758:
756:
755:
750:
737:
735:
734:
729:
709:
707:
706:
701:
687:
686:
662:
661:
638:
636:
635:
630:
616:
615:
585:
583:
582:
577:
563:
562:
553:
536:
535:
498:
496:
495:
490:
478:
476:
475:
470:
458:
456:
455:
450:
436:
435:
415:
413:
412:
407:
393:
392:
373:
371:
370:
365:
363:
359:
316:
270:
268:
267:
262:
250:
248:
247:
242:
230:
228:
227:
222:
210:
208:
207:
202:
188:
187:
159:
157:
156:
151:
21:
7168:
7167:
7163:
7162:
7161:
7159:
7158:
7157:
7138:
7137:
7136:
6876:
6862:Ivanov, Nikolai
6820:
6801:
6800:978-069114794-9
6707:
6706:978-038796416-4
6589:Birman, Joan S.
6554:
6549:
6541:
6537:
6529:
6525:
6517:
6513:
6505:
6501:
6497:, Theorem 6.11.
6493:
6489:
6481:
6477:
6469:
6465:
6457:
6453:
6447:J. Algebra 2004
6445:
6441:
6430:
6426:
6418:
6414:
6406:
6402:
6394:
6390:
6382:
6378:
6370:
6366:
6358:
6354:
6346:
6342:
6334:
6330:
6322:
6318:
6310:
6306:
6298:
6294:
6286:
6282:
6274:
6270:
6262:
6258:
6252:Acta Math. 1938
6250:
6246:
6242:
6237:
6236:
6227:
6223:
6218:
6189:
6184:
6181:
6180:
6149:
6146:
6145:
6133:
6114:
6078:
6075:
6074:
6071:Hurwitz's bound
6054:
6051:
6050:
6021:
6016:
6001:
5997:
5985:
5981:
5967:
5950:
5947:
5946:
5918:
5915:
5914:
5894:
5890:
5879:
5876:
5875:
5872:
5845:
5842:
5841:
5813:
5810:
5809:
5788:
5784:
5782:
5779:
5778:
5756:
5748:
5743:
5728:
5724:
5713:
5698:
5694:
5667:
5663:
5661:
5658:
5657:
5632:
5617:
5613:
5611:
5608:
5607:
5590:
5557:
5554:
5553:
5534:
5519:
5515:
5513:
5510:
5509:
5490:
5487:
5486:
5463:
5448:
5444:
5424:
5421:
5420:
5398:
5383:
5379:
5377:
5374:
5373:
5345:
5330:
5326:
5324:
5321:
5320:
5298:
5283:
5279:
5259:
5256:
5255:
5236:
5233:
5232:
5216:
5213:
5212:
5193:
5190:
5189:
5163:
5159:
5157:
5154:
5153:
5125:
5122:
5121:
5114:
5109:
5064:
5061:
5060:
5032:
5029:
5028:
5008:
5005:
5004:
4988:
4985:
4984:
4968:
4965:
4964:
4948:
4945:
4944:
4941:
4908:
4905:
4904:
4901:
4873:
4843:
4840:
4839:
4817:
4814:
4813:
4784:
4783:
4778:
4772:
4771:
4766:
4756:
4755:
4745:
4744:
4739:
4733:
4732:
4727:
4717:
4716:
4714:
4711:
4710:
4688:
4676:
4672:
4670:
4667:
4666:
4662:
4657:
4634:
4630:
4621:
4617:
4615:
4612:
4611:
4594:
4590:
4588:
4585:
4584:
4567:
4563:
4561:
4558:
4557:
4540:
4536:
4521:
4517:
4515:
4512:
4511:
4494:
4490:
4475:
4471:
4469:
4466:
4465:
4430:
4427:
4426:
4410:
4407:
4406:
4395:
4358:
4355:
4354:
4338:
4335:
4334:
4314:
4311:
4310:
4303:
4298:
4261:
4258:
4257:
4241:
4238:
4237:
4221:
4218:
4217:
4210:
4182:
4179:
4178:
4150:
4147:
4146:
4104:
4101:
4100:
4072:
4069:
4068:
4042:
4039:
4038:
3999:
3996:
3995:
3966:
3962:
3960:
3957:
3956:
3928:
3925:
3924:
3921:Riemann surface
3904:
3901:
3900:
3872:
3869:
3868:
3852:
3849:
3848:
3823:
3820:
3819:
3800:
3797:
3796:
3793:
3788:
3775:
3769:
3741:
3738:
3737:
3721:
3718:
3717:
3697:
3693:
3691:
3688:
3687:
3665:
3662:
3661:
3623:
3620:
3619:
3616:
3610:
3584:
3583:
3578:
3572:
3571:
3566:
3556:
3555:
3553:
3550:
3549:
3527:
3515:
3511:
3509:
3506:
3505:
3486:
3483:
3482:
3466:
3463:
3462:
3442:
3439:
3438:
3410:
3407:
3406:
3389:
3385:
3383:
3380:
3379:
3378:. The class of
3356:
3352:
3340:
3336:
3334:
3331:
3330:
3329:it is equal to
3314:
3311:
3310:
3288:
3285:
3284:
3283:as follows: on
3268:
3265:
3264:
3247:
3243:
3241:
3238:
3237:
3217:
3214:
3213:
3196:
3192:
3190:
3187:
3186:
3170:
3167:
3166:
3150:
3147:
3146:
3130:
3127:
3126:
3110:
3107:
3106:
3090:
3087:
3086:
3083:
3077:
3072:
3051:
3048:
3047:
3007:
3004:
3003:
2987:
2984:
2983:
2891:
2887:
2879:
2876:
2875:
2850:
2847:
2846:
2830:
2827:
2826:
2813:
2787:
2784:
2783:
2746:
2742:
2731:
2728:
2727:
2698:
2694:
2692:
2689:
2688:
2654:
2650:
2648:
2645:
2644:
2621:
2617:
2602:
2598:
2587:
2584:
2583:
2552:
2549:
2548:
2532:
2529:
2528:
2485:
2484:
2482:
2479:
2478:
2446:
2442:
2440:
2437:
2436:
2416:
2412:
2397:
2393:
2379:
2376:
2375:
2358:
2354:
2352:
2349:
2348:
2332:
2329:
2328:
2327:and for a loop
2308:
2304:
2296:
2293:
2292:
2275:
2271:
2269:
2266:
2265:
2249:
2246:
2245:
2225:
2221:
2206:
2202:
2200:
2197:
2196:
2179:
2175:
2173:
2170:
2169:
2153:
2150:
2149:
2133:
2130:
2129:
2109:
2106:
2105:
2102:
2086:
2080:
2058:
2054:
2052:
2049:
2048:
2032:
2029:
2028:
2003:
1995:
1975:
1967:
1962:
1959:
1958:
1934:
1926:
1916:
1912:
1894:
1890:
1888:
1885:
1884:
1864:
1860:
1858:
1855:
1854:
1838:
1836:
1833:
1832:
1807:
1799:
1781:
1777:
1775:
1772:
1771:
1764:
1716:
1712:
1707:
1680:
1676:
1656:
1653:
1652:
1611:
1607:
1605:
1602:
1601:
1572:
1568:
1566:
1563:
1562:
1524:
1520:
1518:
1515:
1514:
1495:
1492:
1491:
1472:
1469:
1468:
1465:
1441:
1429:
1425:
1423:
1420:
1419:
1402:
1397:
1396:
1394:
1391:
1390:
1371:
1359:
1355:
1353:
1350:
1349:
1329:
1324:
1323:
1312:
1309:
1308:
1286:
1283:
1282:
1266:
1263:
1262:
1246:
1243:
1242:
1226:
1223:
1222:
1205:
1200:
1199:
1197:
1194:
1193:
1172:
1167:
1166:
1148:
1143:
1142:
1134:
1131:
1130:
1113:
1108:
1107:
1105:
1102:
1101:
1082:
1070:
1066:
1058:
1055:
1054:
1034:
1029:
1028:
1008:
996:
992:
984:
981:
980:
961:
949:
945:
943:
940:
939:
919:
914:
913:
908:
902:
897:
896:
887:
882:
881:
879:
876:
875:
853:
845:
840:
838:
835:
834:
812:
809:
808:
792:
789:
788:
772:
769:
768:
751:
746:
745:
743:
740:
739:
723:
720:
719:
716:
682:
678:
657:
653:
651:
648:
647:
611:
607:
605:
602:
601:
558:
554:
549:
531:
527:
507:
504:
503:
484:
481:
480:
464:
461:
460:
431:
427:
425:
422:
421:
388:
384:
382:
379:
378:
322:
318:
306:
279:
276:
275:
256:
253:
252:
236:
233:
232:
216:
213:
212:
183:
179:
177:
174:
173:
145:
142:
141:
138:
133:
102:
66:in relation to
28:
23:
22:
15:
12:
11:
5:
7166:
7156:
7155:
7150:
7135:
7134:
7122:(2): 377–383.
7107:
7087:(2): 417–431.
7072:
7036:(2): 753–758.
7021:
6993:(4): 902–974.
6982:
6938:(1): 103–149.
6927:
6880:
6874:
6858:
6846:(3): 221–237.
6828:Hatcher, Allen
6824:
6818:
6805:
6799:
6779:
6761:(3): 581–597.
6750:
6711:
6705:
6692:
6658:
6641:
6599:
6585:
6567:(2): 213–238.
6555:
6553:
6550:
6548:
6547:
6535:
6523:
6511:
6499:
6487:
6475:
6473:, Theorem 6.4.
6463:
6451:
6439:
6424:
6412:
6410:, Theorem 4.1.
6400:
6388:
6376:
6364:
6352:
6340:
6328:
6326:, Theorem 4.6.
6316:
6304:
6302:, Theorem 8.1.
6292:
6280:
6278:, Theorem 2.5.
6268:
6256:
6243:
6241:
6238:
6235:
6234:
6220:
6219:
6217:
6214:
6199:
6196:
6193:
6188:
6165:
6162:
6159:
6156:
6153:
6132:
6129:
6113:
6110:
6097:
6094:
6091:
6088:
6085:
6082:
6058:
6031:
6028:
6024:
6019:
6015:
6012:
6007:
6004:
6000:
5996:
5993:
5988:
5984:
5980:
5977:
5974:
5970:
5966:
5963:
5960:
5957:
5954:
5934:
5931:
5928:
5925:
5922:
5902:
5897:
5893:
5889:
5886:
5883:
5871:
5868:
5855:
5852:
5849:
5829:
5826:
5823:
5820:
5817:
5791:
5787:
5777:The kernel of
5775:
5774:
5763:
5759:
5755:
5751:
5746:
5742:
5739:
5734:
5731:
5727:
5723:
5720:
5716:
5712:
5709:
5704:
5701:
5697:
5693:
5690:
5687:
5684:
5681:
5678:
5675:
5670:
5666:
5639:
5635:
5631:
5628:
5623:
5620:
5616:
5604:
5603:
5589:
5586:
5573:
5570:
5567:
5564:
5561:
5541:
5537:
5533:
5530:
5525:
5522:
5518:
5494:
5481:is called the
5470:
5466:
5462:
5459:
5454:
5451:
5447:
5443:
5440:
5437:
5434:
5431:
5428:
5405:
5401:
5397:
5394:
5389:
5386:
5382:
5352:
5348:
5344:
5341:
5336:
5333:
5329:
5305:
5301:
5297:
5294:
5289:
5286:
5282:
5278:
5275:
5272:
5269:
5266:
5263:
5240:
5220:
5200:
5197:
5177:
5174:
5171:
5166:
5162:
5141:
5138:
5135:
5132:
5129:
5113:
5110:
5108:
5105:
5089:
5086:
5083:
5080:
5077:
5074:
5071:
5068:
5048:
5045:
5042:
5039:
5036:
5012:
4992:
4972:
4952:
4940:
4937:
4924:
4921:
4918:
4915:
4912:
4900:
4897:
4872:
4869:
4856:
4853:
4850:
4847:
4827:
4824:
4821:
4803:
4802:
4788:
4782:
4779:
4777:
4774:
4773:
4770:
4767:
4765:
4762:
4761:
4759:
4754:
4749:
4743:
4740:
4738:
4735:
4734:
4731:
4728:
4726:
4723:
4722:
4720:
4695:
4691:
4687:
4684:
4679:
4675:
4661:
4658:
4656:
4653:
4637:
4633:
4629:
4624:
4620:
4597:
4593:
4570:
4566:
4543:
4539:
4535:
4532:
4529:
4524:
4520:
4497:
4493:
4489:
4486:
4483:
4478:
4474:
4459:locally finite
4446:
4443:
4440:
4437:
4434:
4414:
4394:
4391:
4374:
4371:
4368:
4365:
4362:
4342:
4318:
4302:
4299:
4297:
4294:
4277:
4274:
4271:
4268:
4265:
4245:
4225:
4209:
4206:
4205:
4204:
4201:
4198:
4186:
4166:
4163:
4160:
4157:
4154:
4123:
4120:
4117:
4114:
4111:
4108:
4088:
4085:
4082:
4079:
4076:
4055:
4052:
4049:
4046:
4015:
4012:
4009:
4006:
4003:
3983:
3980:
3977:
3974:
3969:
3965:
3944:
3941:
3938:
3935:
3932:
3908:
3888:
3885:
3882:
3879:
3876:
3856:
3836:
3833:
3830:
3827:
3804:
3792:
3789:
3787:
3784:
3771:Main article:
3768:
3765:
3761:
3760:
3757:
3745:
3725:
3714:
3700:
3696:
3675:
3672:
3669:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3612:Main article:
3609:
3606:
3602:
3601:
3588:
3582:
3579:
3577:
3574:
3573:
3570:
3567:
3565:
3562:
3561:
3559:
3534:
3530:
3526:
3523:
3518:
3514:
3490:
3470:
3446:
3426:
3423:
3420:
3417:
3414:
3392:
3388:
3367:
3364:
3359:
3355:
3351:
3346:
3343:
3339:
3318:
3298:
3295:
3292:
3272:
3250:
3246:
3221:
3199:
3195:
3174:
3154:
3134:
3114:
3094:
3079:Main article:
3076:
3073:
3071:
3068:
3055:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
2991:
2980:
2979:
2978:
2977:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2894:
2890:
2886:
2883:
2860:
2857:
2854:
2834:
2812:
2809:
2791:
2777:
2776:
2763:
2760:
2757:
2754:
2749:
2745:
2741:
2738:
2735:
2715:
2712:
2709:
2706:
2701:
2697:
2671:
2668:
2665:
2662:
2657:
2653:
2632:
2629:
2624:
2620:
2616:
2613:
2610:
2605:
2601:
2597:
2594:
2591:
2568:
2565:
2562:
2559:
2556:
2536:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2492:
2489:
2466:
2463:
2460:
2457:
2454:
2449:
2445:
2424:
2419:
2415:
2411:
2408:
2405:
2400:
2396:
2392:
2389:
2386:
2383:
2361:
2357:
2336:
2316:
2311:
2307:
2303:
2300:
2278:
2274:
2253:
2233:
2228:
2224:
2220:
2217:
2214:
2209:
2205:
2182:
2178:
2157:
2137:
2113:
2101:
2098:
2082:Main article:
2079:
2076:
2061:
2057:
2036:
2016:
2013:
2010:
2006:
2002:
1998:
1994:
1991:
1988:
1985:
1982:
1978:
1974:
1970:
1966:
1955:
1954:
1943:
1937:
1933:
1929:
1925:
1922:
1919:
1915:
1911:
1908:
1905:
1902:
1897:
1893:
1867:
1863:
1841:
1820:
1817:
1814:
1810:
1806:
1802:
1798:
1795:
1792:
1789:
1784:
1780:
1763:
1760:
1755:
1754:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1719:
1715:
1710:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1683:
1679:
1675:
1672:
1669:
1666:
1663:
1660:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1614:
1610:
1589:
1586:
1583:
1580:
1575:
1571:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1527:
1523:
1502:
1499:
1476:
1464:
1461:
1448:
1444:
1440:
1437:
1432:
1428:
1405:
1400:
1378:
1374:
1370:
1367:
1362:
1358:
1337:
1332:
1327:
1322:
1319:
1316:
1296:
1293:
1290:
1270:
1250:
1230:
1208:
1203:
1190:homology group
1175:
1170:
1165:
1162:
1159:
1156:
1151:
1146:
1141:
1138:
1116:
1111:
1089:
1085:
1081:
1078:
1073:
1069:
1065:
1062:
1042:
1037:
1032:
1027:
1024:
1021:
1018:
1015:
1011:
1007:
1004:
999:
995:
991:
988:
968:
964:
960:
957:
952:
948:
922:
917:
911:
905:
900:
895:
890:
885:
856:
852:
848:
843:
822:
819:
816:
796:
776:
754:
749:
727:
715:
712:
699:
696:
693:
690:
685:
681:
677:
674:
671:
668:
665:
660:
656:
644:diffeomorphism
628:
625:
622:
619:
614:
610:
588:
587:
575:
572:
569:
566:
561:
557:
552:
548:
545:
542:
539:
534:
530:
526:
523:
520:
517:
514:
511:
488:
468:
448:
445:
442:
439:
434:
430:
405:
402:
399:
396:
391:
387:
375:
374:
362:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
321:
315:
312:
309:
305:
301:
298:
295:
292:
289:
286:
283:
260:
240:
220:
200:
197:
194:
191:
186:
182:
149:
137:
134:
132:
129:
101:
98:
52:homeomorphisms
26:
9:
6:
4:
3:
2:
7165:
7154:
7151:
7149:
7146:
7145:
7143:
7130:
7125:
7121:
7117:
7113:
7108:
7104:
7100:
7095:
7090:
7086:
7082:
7078:
7073:
7069:
7065:
7061:
7057:
7053:
7049:
7044:
7039:
7035:
7031:
7027:
7022:
7018:
7014:
7010:
7006:
7001:
6996:
6992:
6988:
6983:
6979:
6975:
6971:
6967:
6963:
6959:
6955:
6951:
6946:
6941:
6937:
6933:
6928:
6924:
6920:
6916:
6912:
6908:
6904:
6899:
6894:
6890:
6886:
6881:
6877:
6871:
6867:
6863:
6859:
6854:
6849:
6845:
6841:
6837:
6833:
6829:
6825:
6821:
6815:
6811:
6806:
6802:
6796:
6792:
6788:
6787:Margalit, Dan
6784:
6780:
6776:
6772:
6768:
6764:
6760:
6756:
6751:
6747:
6743:
6739:
6735:
6730:
6725:
6721:
6717:
6712:
6708:
6702:
6698:
6693:
6690:
6684:
6679:
6675:
6672:(in German).
6671:
6667:
6663:
6659:
6655:
6651:
6647:
6642:
6638:
6634:
6630:
6626:
6621:
6616:
6612:
6608:
6604:
6600:
6596:
6595:
6590:
6586:
6582:
6578:
6574:
6570:
6566:
6562:
6557:
6556:
6544:
6539:
6532:
6527:
6520:
6515:
6508:
6503:
6496:
6491:
6484:
6479:
6472:
6467:
6460:
6455:
6448:
6443:
6436:
6434:
6428:
6421:
6416:
6409:
6404:
6397:
6392:
6385:
6380:
6373:
6368:
6361:
6356:
6349:
6344:
6337:
6332:
6325:
6320:
6313:
6308:
6301:
6296:
6289:
6284:
6277:
6272:
6265:
6260:
6253:
6248:
6244:
6231:
6225:
6221:
6213:
6197:
6194:
6191:
6186:
6177:
6160:
6154:
6151:
6143:
6138:
6137:open question
6128:
6125:
6123:
6119:
6109:
6092:
6089:
6086:
6080:
6072:
6056:
6048:
6043:
6026:
6022:
6010:
6005:
6002:
5998:
5994:
5986:
5975:
5972:
5968:
5961:
5955:
5952:
5929:
5923:
5920:
5895:
5884:
5881:
5867:
5853:
5850:
5847:
5824:
5818:
5815:
5807:
5789:
5753:
5749:
5737:
5732:
5729:
5725:
5707:
5702:
5699:
5695:
5685:
5679:
5676:
5673:
5668:
5656:
5655:
5654:
5651:
5626:
5621:
5618:
5614:
5602:
5600:
5595:
5594:
5593:
5585:
5568:
5562:
5559:
5528:
5523:
5520:
5516:
5508:
5492:
5484:
5483:Torelli group
5457:
5452:
5449:
5445:
5435:
5429:
5426:
5417:
5392:
5387:
5384:
5380:
5370:
5366:
5339:
5334:
5331:
5327:
5317:
5292:
5287:
5284:
5280:
5270:
5264:
5261:
5254:
5238:
5218:
5198:
5195:
5172:
5164:
5160:
5136:
5130:
5127:
5119:
5104:
5101:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5043:
5037:
5034:
5026:
5010:
4990:
4970:
4950:
4936:
4919:
4913:
4910:
4896:
4894:
4889:
4887:
4881:
4879:
4868:
4854:
4851:
4848:
4845:
4825:
4822:
4819:
4810:
4808:
4786:
4780:
4775:
4768:
4763:
4757:
4752:
4747:
4741:
4736:
4729:
4724:
4718:
4709:
4708:
4707:
4682:
4677:
4673:
4652:
4635:
4631:
4627:
4622:
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4369:
4363:
4360:
4340:
4332:
4316:
4308:
4307:pants complex
4301:Pants complex
4293:
4289:
4272:
4266:
4263:
4243:
4223:
4216:of a surface
4215:
4214:curve complex
4202:
4199:
4184:
4161:
4155:
4152:
4144:
4143:
4142:
4140:
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3009:
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2135:
2127:
2111:
2097:
2095:
2091:
2085:
2075:
2059:
2055:
2034:
2011:
2008:
2000:
1989:
1983:
1980:
1972:
1941:
1931:
1923:
1920:
1917:
1913:
1909:
1903:
1895:
1891:
1883:
1882:
1881:
1865:
1861:
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1812:
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1769:
1759:
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1063:
1060:
1035:
1022:
1019:
1002:
997:
993:
989:
955:
950:
946:
938:
937:modular group
920:
909:
903:
893:
888:
874:
869:
850:
846:
820:
817:
814:
794:
774:
752:
725:
718:Suppose that
711:
694:
688:
683:
679:
675:
669:
663:
658:
654:
645:
640:
623:
617:
612:
608:
600:
595:
593:
570:
564:
559:
555:
550:
543:
537:
532:
528:
524:
518:
512:
509:
502:
501:
500:
499:is the group
486:
466:
443:
437:
432:
428:
419:
400:
394:
389:
385:
360:
350:
344:
341:
335:
329:
323:
319:
313:
310:
307:
299:
293:
290:
287:
281:
274:
273:
272:
258:
238:
218:
195:
189:
184:
180:
171:
167:
163:
147:
128:
126:
121:
119:
114:
112:
111:Jakob Nielsen
108:
97:
95:
91:
86:
84:
80:
76:
71:
69:
65:
61:
57:
53:
49:
45:
44:modular group
41:
37:
33:
19:
7119:
7115:
7084:
7080:
7033:
7029:
7000:math/9807150
6990:
6986:
6945:math/9804098
6935:
6931:
6888:
6884:
6865:
6843:
6839:
6809:
6790:
6783:Farb, Benson
6758:
6754:
6719:
6715:
6696:
6673:
6669:
6645:
6620:math/0307039
6610:
6606:
6593:
6564:
6560:
6538:
6526:
6521:, Theorem 1.
6514:
6509:, Theorem 4.
6502:
6490:
6478:
6466:
6454:
6442:
6432:
6427:
6415:
6403:
6391:
6379:
6367:
6355:
6343:
6338:, Chapter 9.
6331:
6319:
6307:
6295:
6283:
6271:
6259:
6247:
6224:
6178:
6134:
6126:
6115:
6044:
5873:
5805:
5776:
5652:
5605:
5596:
5591:
5482:
5418:
5318:
5115:
5102:
5059:is equal to
4942:
4902:
4890:
4885:
4882:
4874:
4811:
4804:
4663:
4463:
4402:
4398:
4396:
4304:
4290:
4211:
4136:
4033:(though not
4028:
3794:
3780:
3776:
3762:
3617:
3603:
3503:
3458:
3084:
2981:
2824:
2814:
2781:
2778:
2686:
2103:
2093:
2089:
2087:
2084:Braid groups
1956:
1765:
1756:
1466:
870:
717:
641:
598:
596:
589:
376:
172:surface and
139:
122:
115:
103:
90:braid groups
87:
83:group theory
72:
47:
43:
35:
29:
6676:: 135–206.
6519:Ivanov 1992
6507:Ivanov 1992
6312:Birman 1969
6288:Birman 1974
6232:) markings.
3656:is either:
3075:Dehn twists
2818:Joan Birman
2096:punctures.
60:3-manifolds
7142:Categories
6607:J. Algebra
6384:Brock 2002
3686:such that
3459:Dehn twist
3081:Dehn twist
590:This is a
170:orientable
7043:0812.0017
6898:1106.4261
6729:1307.3733
6689:Dehn 1987
6662:Dehn, Max
6240:Citations
6195:−
6155:
6135:It is an
6090:−
6011:
5995:≅
5983:Φ
5976:
5956:
5924:
5892:Φ
5885:
5851:≥
5819:
5786:Φ
5738:
5722:→
5708:
5692:→
5680:
5665:Φ
5627:
5563:
5529:
5458:
5442:→
5430:
5393:
5340:
5293:
5277:→
5265:
5131:
5073:−
5038:
4914:
4823:≥
4683:
4632:β
4619:α
4592:α
4565:β
4542:ξ
4538:β
4531:…
4519:β
4496:ξ
4492:α
4485:…
4473:α
4436:
4364:
4267:
4156:
4113:−
4078:
4005:
3973:
3940:→
3635:
3629:∈
3522:
3416:
3387:τ
3363:∘
3354:τ
3350:∘
3342:−
3294:∖
3245:τ
3022:∖
3013:
2961:→
2949:
2943:→
2928:∖
2919:
2913:→
2889:π
2885:→
2856:∈
2744:π
2737:
2705:
2700:±
2661:
2600:π
2593:
2558:
2535:γ
2515:γ
2512:∗
2506:α
2497:∗
2491:¯
2488:γ
2459:α
2448:∗
2395:π
2391:∈
2385:α
2347:based at
2335:α
2252:γ
2204:π
2181:∗
2056:τ
1924:π
1892:τ
1862:τ
1813:≤
1797:≤
1735:∂
1723:
1699:∂
1687:
1662:
1630:∂
1618:
1579:
1543:∂
1531:
1498:∂
1436:
1366:
1318:
1295:Φ
1289:Π
1269:Π
1249:Φ
1229:Π
1155:↦
1077:
1064:∈
1023:
1017:→
1003:
987:Φ
956:
689:
676:⊂
664:
618:
613:±
592:countable
565:
538:
513:
438:
395:
311:∈
282:δ
190:
162:connected
79:manifolds
7116:Topology
7017:14834205
6978:16199015
6923:17330187
6864:(1992).
6840:Topology
6834:(1980).
6789:(2012).
6746:15393033
6664:(1938).
6637:14784932
6591:(1974).
6433:Topology
4403:markings
2264:between
1489:boundary
1053:: every
118:Thurston
107:Max Dehn
32:topology
7103:0956596
7068:2047111
7060:2557192
6970:1714338
6950:Bibcode
6915:2967055
6775:1813237
6654:1940162
6581:0243519
6552:Sources
5363:of the
2804:Out(Fn)
2579:to the
2435:define
1768:annulus
594:group.
100:History
40:surface
7101:
7066:
7058:
7015:
6976:
6968:
6921:
6913:
6872:
6816:
6797:
6773:
6744:
6703:
6652:
6635:
6579:
3899:where
3461:about
2126:closed
416:. The
166:closed
68:moduli
34:, the
7064:S2CID
7038:arXiv
7013:S2CID
6995:arXiv
6974:S2CID
6940:arXiv
6919:S2CID
6893:arXiv
6742:S2CID
6724:arXiv
6722:(8).
6633:S2CID
6615:arXiv
6216:Notes
4983:with
3964:Homeo
3919:is a
3481:. If
3165:from
2652:Homeo
2555:Homeo
1714:Homeo
1678:Homeo
1609:Homeo
1570:Homeo
1522:Homeo
873:torus
680:Homeo
556:Homeo
529:Homeo
429:Homeo
386:Homeo
181:Homeo
160:be a
38:of a
6870:ISBN
6814:ISBN
6795:ISBN
6701:ISBN
6435:1996
6122:free
4305:The
4212:The
4035:free
3923:and
3671:>
2825:Let
2291:and
2128:and
1766:Any
1348:and
1129:via
655:Diff
140:Let
109:and
92:and
73:The
7124:doi
7089:doi
7048:doi
7034:138
7005:doi
6958:doi
6936:138
6903:doi
6848:doi
6763:doi
6759:106
6734:doi
6720:166
6678:doi
6625:doi
6611:278
6569:doi
6152:Mod
5973:ker
5953:Mod
5921:Mod
5882:ker
5816:Mod
5808:of
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5560:Mod
5485:of
5427:Mod
5262:Mod
5211:if
5128:Mod
5116:As
5035:Mod
5027:of
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4911:Mod
4838:is
4433:Mod
4405:of
4361:Mod
4333:of
4264:Mod
4153:Mod
4075:Mod
4002:Mod
3632:Mod
3413:Mod
3263:of
3085:If
3010:Mod
2946:Mod
2916:Mod
2734:Out
2696:Mod
2590:Out
2124:is
2104:If
1831:of
1659:Mod
1418:is
1315:Mod
1192:of
1020:Mod
609:Mod
510:Mod
304:sup
251:on
46:or
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7120:35
7118:.
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7099:MR
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5446:Sp
5416:.
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5281:GL
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3066:.
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1357:SL
1068:SL
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2015:}
2012:2
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2005:|
2001:z
1997:|
1993:{
1990:,
1987:}
1984:1
1981:=
1977:|
1973:z
1969:|
1965:{
1942:z
1936:|
1932:z
1928:|
1921:i
1918:2
1914:e
1910:=
1907:)
1904:z
1901:(
1896:0
1866:0
1840:C
1819:}
1816:2
1809:|
1805:z
1801:|
1794:1
1791:{
1788:=
1783:0
1779:A
1753:.
1741:)
1738:S
1732:,
1729:S
1726:(
1718:0
1709:/
1705:)
1702:S
1696:,
1693:S
1690:(
1682:+
1674:=
1671:)
1668:S
1665:(
1636:)
1633:S
1627:,
1624:S
1621:(
1613:0
1588:)
1585:S
1582:(
1574:+
1549:)
1546:S
1540:,
1537:S
1534:(
1526:+
1501:S
1475:S
1447:)
1443:Z
1439:(
1431:2
1404:2
1399:T
1377:)
1373:Z
1369:(
1361:2
1336:)
1331:2
1326:T
1321:(
1292:,
1207:2
1202:T
1174:2
1169:Z
1164:+
1161:x
1158:A
1150:2
1145:Z
1140:+
1137:x
1115:2
1110:T
1088:)
1084:Z
1080:(
1072:2
1061:A
1041:)
1036:2
1031:T
1026:(
1014:)
1010:Z
1006:(
998:2
990::
967:)
963:Z
959:(
951:2
921:2
916:Z
910:/
904:2
899:R
894:=
889:2
884:T
855:Z
851:2
847:/
842:Z
821:0
818:=
815:z
795:S
775:S
753:3
748:R
726:S
698:)
695:S
692:(
684:+
673:)
670:S
667:(
659:+
627:)
624:S
621:(
586:.
574:)
571:S
568:(
560:0
551:/
547:)
544:S
541:(
533:+
525:=
522:)
519:S
516:(
487:S
467:S
447:)
444:S
441:(
433:0
404:)
401:S
398:(
390:+
361:)
357:)
354:)
351:x
348:(
345:g
342:,
339:)
336:x
333:(
330:f
327:(
324:d
320:(
314:S
308:x
300:=
297:)
294:g
291:,
288:f
285:(
259:S
239:d
219:S
199:)
196:S
193:(
185:+
148:S
20:)
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