3766:
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
4280:
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
4872:
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
5761:
4639:
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
4873:
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
4653:
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
1746:
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).
1740:
93:
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
2804:
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
2795:
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.
4788:
2963:
6029:
1040:
573:
102:: 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).
3770:
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.
4864:
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
5468:
5303:
697:
5648:
361:
922:
6128:
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
5360:
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
3588:
1175:
4055:
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
4130:
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:
1087:
1635:
1548:
2630:
5637:
5539:
5403:
5350:
3033:
1335:
4693:
3532:
1446:
1376:
966:
2514:
4497:
2761:
4543:
1643:
2713:
626:
3981:
2669:
1587:
446:
403:
198:
3643:
3365:
1941:
855:
2566:
1818:
3752:
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.
6163:
5932:
5827:
5571:
5139:
5046:
4922:
4444:
4372:
4275:
4164:
4086:
4013:
3424:
4637:
2422:
4701:
5494:. 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
2866:
2014:
3490:
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.
2231:
1405:
1208:
1116:
754:
5900:
6199:
4189:
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;
5937:
3296:
1840:
971:
4597:
4570:
1500:
1294:
494:
5791:
3392:
3250:
2061:
1867:
5175:
5087:
3942:
2314:
6095:
5853:
4825:
2534:
2464:
2334:
2251:
4121:
3673:
2858:
3886:
3700:
3199:
2361:
2278:
2182:
1248:
4854:
4053:
3834:
1268:
1228:
3607:
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
820:
6116:
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.
5198:
6056:
5492:
5238:
5218:
5010:
4990:
4970:
4950:
4412:
4340:
4316:
4243:
4223:
4184:
3906:
3854:
3802:
3743:
3723:
3488:
3468:
3444:
3316:
3270:
3219:
3172:
3152:
3132:
3112:
3092:
3053:
2989:
2832:
2789:
2155:
2135:
2111:
2034:
1474:
794:
774:
725:
486:
466:
258:
238:
218:
147:
6133:. The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of
6801:. Mathematical Notes. Vol. 48. translated from the 1979 French original by Djun M. Kim and Dan Margalit. Princeton University Press. pp. xvi+254.
4277:
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).
5411:
5246:
638:
822:. 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
5756:{\displaystyle \Phi _{n}:\operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} /n\mathbb {Z} )}
2791:
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
2768:
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.
5639:, and then, for any nontrivial element of the Torelli group, constructing by geometric means subgroups of finite index which does not contain it.
266:
220:. 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
866:
3540:
468:
which are isotopic to the identity. It is a normal subgroup of the group of positive homeomorphisms, and the mapping class group of
1121:
6168:
In the other direction there is a lower bound for the dimension of a (putative) faithful representation, which has to be at least
109:
who gave the subject a more geometric flavour and used this work to great effect in his program for the study of three-manifolds.
6581:
5863:
The mapping class group has only finitely many classes of finite groups, as follows from the fact that the finite-index subgroup
5855:(this follows easily from a classical result of Minkowski on linear groups and the fact that the Torelli group is torsion-free).
4134:
Finite-order elements fix a point inside TeichmĂĽller space (more concretely this means that any mapping class of finite order in
1045:
4319:
1592:
1505:
6862:
6806:
2574:
5598:
5500:
5364:
5311:
2994:
1299:
631:
This definition can also be made in the differentiable category: if we replace all instances of "homeomorphism" above with "
4657:
3496:
1410:
1340:
930:
5092:
The first homology of the mapping class group is finite and it follows that the first cohomology group is finite as well.
6787:
6693:
3602:
2469:
1735:{\displaystyle \operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S,\partial S)/\operatorname {Homeo} _{0}(S,\partial S)}
5573:
boil down to a statement about its
Torelli subgroup) and applications to 3-dimensional topology and algebraic geometry.
4456:
4295:
2718:
4502:
2077:
Braid groups can be defined as the mapping class groups of a disc with punctures. More precisely, the braid group on
2679:
592:
6130:
3947:
2635:
1553:
412:
369:
164:
3610:
3321:
7136:
5902:
is torsion-free, as discussed in the previous paragraph. Moreover, this also implies that any finite subgroup of
1875:
4892:
There are other interesting systems of generators for the mapping class group besides Dehn twists. For example,
825:
2539:
1762:
1477:
6136:
6059:
5905:
5800:
5544:
5112:
5019:
4895:
4783:{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}},{\begin{pmatrix}1&0\\1&1\end{pmatrix}}}
4417:
4345:
4248:
4137:
4059:
4019:
3986:
3397:
99:
6594:; Farb, Benson (2004). "Every mapping class group is generated by 3 torsion elements and by 6 involutions".
4602:
4374:
extends to an action on this complex. This complex is quasi-isometric to TeichmĂĽller space endowed with the
2958:{\displaystyle 1\to \pi _{1}(S,x)\to \operatorname {Mod} (S\setminus \{x\})\to \operatorname {Mod} (S)\to 1}
6775:
6034:
A bound on the order of finite subgroups can also be obtained through geometric means. The solution to the
2366:
6974:
Masur, Howard A.; Minsky, Yair N. (2000). "Geometry of the complex of curves II: Hierarchical structure".
4801:
The smallest number of Dehn twists that can generate the mapping class group of a closed surface of genus
4386:
The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. The
2671:. The Dehn–Nielsen–Baer theorem states that it is in addition surjective. In particular, it implies that:
7141:
6035:
1949:
7014:
2187:
1381:
1184:
1092:
730:
154:
6742:
Farb, Benson; Lubotzky, Alexander; Minsky, Yair (2001). "Rank-1 phenomena for mapping class groups".
6038:
implies that any such group is realised as the group of isometries of an hyperbolic surface of genus
6024:{\displaystyle \operatorname {Mod} (S)/\ker(\Phi _{3})\cong \operatorname {Sp} _{2g}(\mathbb {Z} /3)}
5866:
4375:
1035:{\displaystyle \Phi :\operatorname {SL} _{2}(\mathbb {Z} )\to \operatorname {Mod} (\mathbb {T} ^{2})}
6872:
Masbaum, Gregor; Reid, Alan W. (2012). "All finite groups are involved in the mapping class group".
6171:
4866:
4795:
2569:
568:{\displaystyle \operatorname {Mod} (S)=\operatorname {Homeo} ^{+}(S)/\operatorname {Homeo} _{0}(S)}
82:
3275:
1823:
6919:
Masur, Howard A.; Minsky, Yair N. (1999). "Geometry of the complex of curves. I. Hyperbolicity".
5587:
5013:
4575:
3944:
a homeomorphism, modulo a suitable equivalence relation. There is an obvious action of the group
4548:
1482:
1273:
70:(indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in
5769:
3370:
3228:
2039:
1845:
113:
5144:
5051:
3915:
3836:
is the space of marked complex (equivalently, conformal or complete hyperbolic) structures on
2283:
6065:
5832:
5241:
4804:
2519:
2427:
2319:
2236:
1756:
44:
4091:
3652:
2837:
7091:
7048:
6958:
6938:
6903:
6763:
6642:
6569:
4447:
3859:
3678:
3177:
2339:
2256:
2160:
1233:
4830:
4029:
3810:
3805:
1253:
1213:
8:
6703:
Eskin, Alex; Masur, Howard; Rafi, Kasra (2017). "Large-scale rank of TeichmĂĽller space".
5642:
An interesting class of finite-index subgroups is given by the kernels of the morphisms:
5357:
4599:, and this "minimally" (this is a technical condition which can be stated as follows: if
799:
63:
6942:
5180:
7052:
7026:
7001:
6983:
6962:
6928:
6907:
6881:
6730:
6712:
6621:
6603:
6041:
5477:
5223:
5203:
4995:
4975:
4955:
4935:
4397:
4325:
4301:
4228:
4208:
4169:
3891:
3839:
3787:
3728:
3708:
3473:
3453:
3429:
3301:
3255:
3204:
3157:
3137:
3117:
3097:
3077:
3038:
2974:
2817:
2774:
2140:
2120:
2096:
2019:
1637:
is the connected component of the identity. The mapping class group is then defined as
1459:
779:
759:
710:
471:
451:
406:
243:
223:
203:
132:
77:
The mapping class group of surfaces are related to various other groups, in particular
52:
28:
699:
induces an isomorphism between the quotients by their respective identity components.
7117:
7100:
6858:
6850:
6841:
6824:
6802:
6783:
6689:
6465:
5106:
4342:(isotopy classes of maximal systems of disjoint simple closed curves). The action of
4127:
4026:). It is compatible with various geometric structures (metric or complex) with which
3761:
7082:
7065:
7015:"A note on the abelianizations of finite-index subgroups of the mapping class group"
7005:
6966:
6911:
6734:
6625:
6548:
Birman, Joan (1969). "Mapping class groups and their relationship to braid groups".
4414:, which are acted upon by, and have trivial stabilisers in, the mapping class group
7112:
7077:
7056:
7036:
6993:
6946:
6891:
6836:
6820:
6751:
6722:
6666:
6617:
6613:
6557:
6106:
5495:
5353:
4881:
4880:
An example of a relation between Dehn twists occurring in this presentation is the
4281:
class group is not a hyperbolic group it has some properties reminiscent of those.
3222:
1502:
then the definition of the mapping class group needs to be more precise. The group
106:
7040:
6755:
3767:
on smaller surfaces which may themselves be either finite order or pseudo-Anosov.
7087:
7044:
6954:
6899:
6759:
6638:
6591:
6565:
5463:{\displaystyle \operatorname {Mod} (S)\to \operatorname {Sp} _{2g}(\mathbb {Z} )}
5298:{\displaystyle \operatorname {Mod} (S)\to \operatorname {GL} _{2g}(\mathbb {Z} )}
3909:
2114:
692:{\displaystyle \operatorname {Diff} ^{+}(S)\subset \operatorname {Homeo} ^{+}(S)}
150:
6414:
1178:
632:
6726:
1451:
7130:
6816:
4202:
925:
580:
158:
116:, where it provides a testing ground for various conjectures and techniques.
40:
6895:
112:
More recently the mapping class group has been by itself a central topic in
6825:"A presentation for the mapping class group of a closed orientable surface"
6561:
6125:
5581:
An example of application of the
Torelli subgroup is the following result:
4225:
is a complex whose vertices are isotopy classes of simple closed curves on
2072:
71:
56:
6950:
6857:. Translations of Mathematical Monographs. American Mathematical Society.
6633:
Brock, Jeff (2002). "Pants decompositions and the Weil–Petersson metric".
6586:. Annals of Mathematics Studies. Vol. 82. Princeton University Press.
6318:
5595:
The proof proceeds first by using residual finiteness of the linear group
3534:
the Dehn twists correspond to unipotent matrices. For example, the matrix
2081:
strands is naturally isomorphic to the mapping class group of a disc with
6771:
6577:
6441:
4023:
2806:
78:
6246:
5308:
This map is in fact a surjection with image equal to the integer points
3035:
must be replaced by the finite-index subgroup of mapping classes fixing
1250:(proving in particular that it is injective) and it can be checked that
586:
If we modify the definition to include all homeomorphisms we obtain the
6997:
6671:
6654:
6110:
3069:
702:
48:
6655:"Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flächen"
6988:
6933:
6608:
3593:
corresponds to the Dehn twist about a horizontal curve in the torus.
6477:
4643:
4284:
356:{\displaystyle \delta (f,g)=\sup _{x\in S}\left(d(f(x),g(x))\right)}
200:
the group of orientation-preserving, or positive, homeomorphisms of
6650:
6453:
6390:
628:, which contains the mapping class group as a subgroup of index 2.
95:
67:
20:
7031:
6886:
6717:
6342:
6306:
6282:
6258:
2536:, but only up to conjugation. Thus we get a well-defined map from
2466:
to be the element of the fundamental group associated to the loop
917:{\displaystyle \mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}
105:
The Dehn–Nielsen theory was reinterpreted in the mid-seventies by
6513:
5576:
2792:
1589:
which restrict to the identity on the boundary, and the subgroup
6688:. translated and introduced by John Stillwell. Springer-Verlag.
6525:
6354:
2632:. This map is a morphism and its kernel is exactly the subgroup
47:) deformation. It is of fundamental importance for the study of
7101:"Mapping class group of a surface is generated by two elements"
6797:
Fathi, Albert; Laudenbach, François; Poénaru, Valentin (2012).
6217:
We describe here only "clean, complete" (in the terminology of
4018:
This action has many interesting properties; for example it is
3583:{\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}
1550:
of homeomorphisms relative to the boundary is the subgroup of
7066:"On the geometry and dynamics of diffeomorphisms of surfaces"
6782:. Princeton Mathematical Series. Princeton University press.
4450:
complex which is quasi-isometric to the mapping class group.
4166:
can be realised as an isometry for some hyperbolic metric on
1170:{\displaystyle x+\mathbb {Z} ^{2}\mapsto Ax+\mathbb {Z} ^{2}}
861:
124:
6234:
6109:: that is, any subgroup of it either contains a non-abelian
5541:
is comparatively very well understood, a lot of facts about
776:
is isotopic to either the identity or to the restriction to
6501:
6489:
6402:
3705:
reducible: there exists a set of disjoint closed curves on
1452:
Mapping class group of surfaces with boundary and punctures
1082:{\displaystyle A\in \operatorname {SL} _{2}(\mathbb {Z} )}
6330:
4794:
In particular, the mapping class group of a surface is a
6294:
6113:
subgroup or it is virtually solvable (in fact abelian).
5095:
4927:
4446:. It is (in opposition to the curve or pants complex) a
3493:
In the mapping class group of the torus identified with
2066:
1630:{\displaystyle \operatorname {Homeo} _{0}(S,\partial S)}
1543:{\displaystyle \operatorname {Homeo} ^{+}(S,\partial S)}
6378:
4192:
Reducible classes do not act minimally on the boundary.
2625:{\displaystyle \operatorname {Out} (\pi _{1}(S,x_{0}))}
1378:. In the same way, the extended mapping class group of
7063:
7012:
6796:
6447:
6429:
6348:
6324:
6252:
5632:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5534:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5398:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
5345:{\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}
4749:
4710:
3596:
3549:
3058:
3028:{\displaystyle \operatorname {Mod} (S\setminus \{x\})}
1330:{\displaystyle \operatorname {Mod} (\mathbb {T} ^{2})}
6174:
6139:
6068:
6044:
5940:
5908:
5869:
5835:
5803:
5772:
5651:
5601:
5547:
5503:
5480:
5414:
5367:
5314:
5249:
5226:
5206:
5183:
5147:
5115:
5054:
5022:
4998:
4978:
4958:
4938:
4898:
4833:
4807:
4704:
4688:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
4660:
4605:
4578:
4551:
4505:
4459:
4420:
4400:
4348:
4328:
4304:
4251:
4231:
4211:
4172:
4140:
4094:
4062:
4032:
3989:
3950:
3918:
3894:
3862:
3842:
3813:
3790:
3774:
3731:
3711:
3681:
3655:
3613:
3543:
3527:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
3499:
3476:
3456:
3432:
3400:
3373:
3324:
3304:
3278:
3258:
3231:
3207:
3180:
3160:
3140:
3120:
3100:
3080:
3041:
2997:
2977:
2869:
2840:
2820:
2777:
2721:
2682:
2638:
2577:
2542:
2522:
2472:
2430:
2369:
2342:
2322:
2286:
2259:
2239:
2190:
2163:
2143:
2123:
2099:
2042:
2022:
1952:
1878:
1848:
1826:
1765:
1646:
1595:
1556:
1508:
1485:
1462:
1441:{\displaystyle \operatorname {GL} _{2}(\mathbb {Z} )}
1413:
1384:
1371:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
1343:
1302:
1276:
1256:
1236:
1216:
1187:
1124:
1095:
1048:
974:
961:{\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}
933:
869:
828:
802:
782:
762:
733:
713:
641:
595:
497:
474:
454:
448:. By definition it is equal to the homeomorphisms of
415:
372:
269:
246:
226:
206:
167:
135:
4924:
can be generated by two elements or by involutions.
3446:
made above, and the resulting element is called the
3225:
orientation. This is used to define a homeomorphism
1750:
703:
The mapping class groups of the sphere and the torus
366:
is a distance inducing the compact-open topology on
2509:{\displaystyle {\bar {\gamma }}*f(\alpha )*\gamma }
260:inducing its topology then the function defined by
51:via their embedded surfaces and is also studied in
6193:
6157:
6089:
6050:
6023:
5926:
5894:
5847:
5821:
5785:
5755:
5631:
5565:
5533:
5486:
5462:
5397:
5344:
5297:
5232:
5212:
5192:
5169:
5141:acts by automorphisms on the first homology group
5133:
5081:
5040:
5004:
4984:
4964:
4944:
4916:
4848:
4819:
4782:
4687:
4631:
4591:
4564:
4537:
4492:{\displaystyle \alpha _{1},\ldots ,\alpha _{\xi }}
4491:
4438:
4406:
4366:
4334:
4310:
4269:
4237:
4217:
4178:
4158:
4115:
4080:
4047:
4007:
3975:
3936:
3900:
3880:
3848:
3828:
3796:
3737:
3717:
3694:
3667:
3637:
3582:
3526:
3482:
3462:
3438:
3418:
3386:
3359:
3310:
3290:
3264:
3244:
3213:
3193:
3166:
3146:
3126:
3106:
3086:
3047:
3027:
2983:
2957:
2852:
2826:
2783:
2755:
2707:
2663:
2624:
2560:
2528:
2508:
2458:
2416:
2355:
2328:
2308:
2272:
2245:
2225:
2176:
2149:
2129:
2105:
2055:
2028:
2008:
1946:which is the identity on both boundary components
1935:
1861:
1834:
1812:
1734:
1629:
1581:
1542:
1494:
1468:
1440:
1399:
1370:
1329:
1288:
1262:
1242:
1222:
1202:
1169:
1110:
1081:
1034:
960:
916:
849:
814:
788:
768:
748:
719:
691:
635:" we obtain the same group, that is the inclusion
620:
567:
480:
460:
440:
397:
355:
252:
232:
212:
192:
141:
6366:
6270:
4644:Generators and relations for mapping class groups
4453:A marking is determined by a pants decomposition
4285:Other complexes with a mapping class group action
2771:The conclusion of the theorem does not hold when
2756:{\displaystyle \operatorname {Out} (\pi _{1}(S))}
7128:
7019:Proceedings of the American Mathematical Society
6062:then implies that the maximal order is equal to
4538:{\displaystyle \beta _{1},\ldots ,\beta _{\xi }}
3755:
2088:
292:
6815:
6408:
3114:and one chooses a closed tubular neighbourhood
43:of the surface viewed up to continuous (in the
6702:
6550:Communications on Pure and Applied Mathematics
6336:
5577:Residual finiteness and finite-index subgroups
4887:
4196:
3983:on such pairs, which descends to an action of
3779:
2715:is isomorphic to the outer automorphism group
2708:{\displaystyle \operatorname {Mod} ^{\pm }(S)}
621:{\displaystyle \operatorname {Mod} ^{\pm }(S)}
7070:Bulletin of the American Mathematical Society
6918:
6741:
6531:
6360:
6100:
4648:
3976:{\displaystyle \operatorname {Homeo} ^{+}(S)}
2991:itself has punctures the mapping class group
2664:{\displaystyle \operatorname {Homeo} _{0}(S)}
2516:. This automorphism depends on the choice of
1582:{\displaystyle \operatorname {Homeo} ^{+}(S)}
1177:. The action of diffeomorphisms on the first
441:{\displaystyle \operatorname {Homeo} _{0}(S)}
398:{\displaystyle \operatorname {Homeo} ^{+}(S)}
193:{\displaystyle \operatorname {Homeo} ^{+}(S)}
6871:
6770:
6519:
6483:
6471:
6459:
6396:
6312:
6288:
6264:
3638:{\displaystyle g\in \operatorname {Mod} (S)}
3360:{\displaystyle f^{-1}\circ \tau _{0}\circ f}
3019:
3013:
2925:
2919:
2809:in 1969. The exact statement is as follows.
2799:
2003:
1981:
1975:
1953:
1807:
1779:
6973:
6384:
6218:
1936:{\displaystyle \tau _{0}(z)=e^{2i\pi |z|}z}
6649:
6590:
6435:
6240:
850:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
125:Mapping class group of orientable surfaces
119:
7116:
7081:
7030:
6987:
6932:
6885:
6840:
6716:
6670:
6635:Complex Manifolds and Hyperbolic Geometry
6607:
6119:
6006:
5746:
5733:
5703:
5622:
5524:
5453:
5388:
5335:
5288:
4678:
4245:. The action of the mapping class groups
3517:
2561:{\displaystyle \operatorname {Homeo} (S)}
1828:
1813:{\displaystyle A_{0}=\{1\leq |z|\leq 2\}}
1431:
1387:
1361:
1314:
1190:
1157:
1133:
1098:
1072:
1019:
998:
951:
904:
887:
872:
843:
830:
736:
7098:
6420:
4859:
6855:Subgroups of TeichmĂĽller Modular Groups
6583:Braids, links, and mapping class groups
6158:{\displaystyle \operatorname {Mod} (S)}
5927:{\displaystyle \operatorname {Mod} (S)}
5822:{\displaystyle \operatorname {Mod} (S)}
5566:{\displaystyle \operatorname {Mod} (S)}
5177:. This is a free abelian group of rank
5134:{\displaystyle \operatorname {Mod} (S)}
5109:is functorial, the mapping class group
5100:
5041:{\displaystyle \operatorname {Mod} (S)}
4917:{\displaystyle \operatorname {Mod} (S)}
4439:{\displaystyle \operatorname {Mod} (S)}
4367:{\displaystyle \operatorname {Mod} (S)}
4270:{\displaystyle \operatorname {Mod} (S)}
4159:{\displaystyle \operatorname {Mod} (S)}
4081:{\displaystyle \operatorname {Mod} (S)}
4008:{\displaystyle \operatorname {Mod} (S)}
3419:{\displaystyle \operatorname {Mod} (S)}
7129:
6849:
6576:
6547:
6507:
6495:
6300:
6276:
4856:; this was proven later by Humphries.
4632:{\displaystyle \alpha _{i},\beta _{i}}
4499:and a collection of transverse curves
3094:is an oriented simple closed curve on
1476:is a compact surface with a non-empty
19:In mathematics, and more precisely in
6632:
6372:
6105:The mapping class groups satisfy the
5829:. It is a torsion-free group for all
5096:Subgroups of the mapping class groups
4928:Cohomology of the mapping class group
2417:{\displaystyle \in \pi _{1}(S,x_{0})}
2067:Braid groups and mapping class groups
968:. It is easy to construct a morphism
6683:
6677:
6349:Fathi, Laudenbach & Poénaru 2012
6325:Fathi, Laudenbach & Poénaru 2012
5356:. This comes from the fact that the
4318:is a complex whose vertices are the
3725:which is preserved by the action of
6976:Geometric & Functional Analysis
6686:Papers on group theory and topology
5858:
4381:
3649:of finite order (i.e. there exists
3597:The Nielsen–Thurston classification
3059:Elements of the mapping class group
2157:then we can define an automorphism
2009:{\displaystyle \{|z|=1\},\{|z|=2\}}
407:connected component of the identity
13:
5971:
5934:is a subgroup of the finite group
5880:
5774:
5653:
3775:Actions of the mapping class group
2036:is then generated by the class of
1842:. One can define a diffeomorphism
1723:
1687:
1618:
1531:
1486:
1283:
1277:
1257:
1237:
1217:
975:
14:
7153:
6637:. American Mathematical Society.
3856:. These are represented by pairs
3426:does not depend on the choice of
3282:
3010:
2916:
2676:The extended mapping class group
2226:{\displaystyle \pi _{1}(S,x_{0})}
1751:Mapping class group of an annulus
1296:are inverse isomorphisms between
924:is naturally identified with the
6780:A primer on mapping class groups
6474:, Theorem 6.15 and Theorem 6.12.
6131:topological quantum field theory
4390:is a complex whose vertices are
4289:
1400:{\displaystyle \mathbb {T} ^{2}}
1203:{\displaystyle \mathbb {T} ^{2}}
1111:{\displaystyle \mathbb {T} ^{2}}
749:{\displaystyle \mathbb {R} ^{3}}
7083:10.1090/s0273-0979-1988-15685-6
6211:
5895:{\displaystyle \ker(\Phi _{3})}
3804:(usually without boundary) the
3603:Nielsen–Thurston classification
860:The mapping class group of the
857:, the cyclic group of order 2.
6618:10.1016/j.jalgebra.2004.02.019
6194:{\displaystyle 2{\sqrt {g-1}}}
6152:
6146:
6084:
6072:
6018:
6002:
5980:
5967:
5953:
5947:
5921:
5915:
5889:
5876:
5816:
5810:
5750:
5729:
5710:
5707:
5699:
5680:
5677:
5671:
5626:
5618:
5560:
5554:
5528:
5520:
5457:
5449:
5430:
5427:
5421:
5392:
5384:
5339:
5331:
5292:
5284:
5265:
5262:
5256:
5164:
5158:
5128:
5122:
5035:
5029:
4911:
4905:
4682:
4674:
4640:higher-dimensional simplices.
4572:intersects at most one of the
4433:
4427:
4361:
4355:
4264:
4258:
4153:
4147:
4075:
4069:
4042:
4036:
4002:
3996:
3970:
3964:
3928:
3875:
3863:
3823:
3817:
3632:
3626:
3521:
3513:
3413:
3407:
3134:then there is a homeomorphism
3063:
3022:
3004:
2949:
2946:
2940:
2931:
2928:
2910:
2901:
2898:
2886:
2873:
2750:
2747:
2741:
2728:
2702:
2696:
2658:
2652:
2619:
2616:
2597:
2584:
2555:
2549:
2497:
2491:
2479:
2453:
2450:
2444:
2441:
2411:
2392:
2376:
2370:
2303:
2290:
2220:
2201:
1993:
1985:
1965:
1957:
1924:
1916:
1895:
1889:
1797:
1789:
1759:is homeomorphic to the subset
1729:
1714:
1693:
1678:
1659:
1653:
1624:
1609:
1576:
1570:
1537:
1522:
1435:
1427:
1365:
1357:
1324:
1309:
1143:
1076:
1068:
1029:
1014:
1005:
1002:
994:
955:
947:
686:
680:
661:
655:
615:
609:
562:
556:
535:
529:
510:
504:
435:
429:
392:
386:
345:
342:
336:
327:
321:
315:
285:
273:
187:
181:
1:
7064:Thurston, William P. (1988).
7041:10.1090/s0002-9939-09-10124-7
6756:10.1215/s0012-7094-01-10636-4
4695:is generated by the matrices
3756:Pseudo-Anosov diffeomorphisms
2860:. There is an exact sequence
2089:The Dehn–Nielsen–Baer theorem
2016:. The mapping class group of
796:of the symmetry in the plane
409:for this topology is denoted
66:can be defined for arbitrary
7118:10.1016/0040-9383(95)00037-2
6842:10.1016/0040-9383(80)90009-9
6337:Eskin, Masur & Rafi 2017
6228:
3291:{\displaystyle S\setminus A}
1835:{\displaystyle \mathbb {C} }
1089:induces a diffeomorphism of
756:. Then any homeomorphism of
588:extended mapping class group
7:
6799:Thurston's work on surfaces
6448:Proc. Amer. Math. Soc. 2010
6409:Hatcher & Thurston 1980
6253:Bull. Amer. Math. Soc. 1988
6036:Nielsen realisation problem
5586:The mapping class group is
5408:The kernel of the morphism
5240:. This action thus gives a
5012:punctures then the virtual
4888:Other systems of generators
4592:{\displaystyle \alpha _{i}}
4545:such that every one of the
4197:Action on the curve complex
3780:Action on TeichmĂĽller space
3394:in the mapping class group
3298:it is the identity, and on
10:
7158:
6540:
6101:General facts on subgroups
4649:The Dehn–Lickorish theorem
4565:{\displaystyle \beta _{i}}
4126:The action extends to the
3784:Given a punctured surface
3759:
3600:
3067:
2070:
1869:by the following formula:
1495:{\displaystyle \partial S}
1289:{\displaystyle \Pi ,\Phi }
88:
6744:Duke Mathematical Journal
6727:10.1215/00127094-0000006X
6705:Duke Mathematical Journal
6219:Masur & Minsky (2000)
5786:{\displaystyle \Phi _{n}}
3387:{\displaystyle \tau _{c}}
3245:{\displaystyle \tau _{c}}
3174:to the canonical annulus
2834:be a compact surface and
2800:The Birman exact sequence
2184:of the fundamental group
2056:{\displaystyle \tau _{0}}
1862:{\displaystyle \tau _{0}}
83:outer automorphism groups
37:TeichmĂĽller modular group
6921:Inventiones Mathematicae
6484:Farb & Margalit 2012
6472:Farb & Margalit 2012
6460:Farb & Margalit 2012
6397:Farb & Margalit 2012
6313:Farb & Margalit 2012
6289:Farb & Margalit 2012
6265:Farb & Margalit 2012
6204:
5170:{\displaystyle H_{1}(S)}
5082:{\displaystyle 4g-4+b+k}
4992:boundary components and
4867:finitely presented group
4796:finitely generated group
3937:{\displaystyle f:S\to X}
2570:outer automorphism group
2363:representing an element
2309:{\displaystyle f(x_{0})}
6896:10.2140/gt.2012.16.1393
6874:Geometry & Topology
6385:Masur & Minsky 2000
6090:{\displaystyle 84(g-1)}
5848:{\displaystyle n\geq 3}
5014:cohomological dimension
4820:{\displaystyle g\geq 2}
3201:defined above, sending
2529:{\displaystyle \gamma }
2459:{\displaystyle f_{*}()}
2329:{\displaystyle \alpha }
2246:{\displaystyle \gamma }
2233:as follows: fix a path
120:Definition and examples
31:, sometimes called the
7137:Geometric group theory
6562:10.1002/cpa.3160220206
6195:
6159:
6120:Linear representations
6091:
6052:
6025:
5928:
5896:
5849:
5823:
5787:
5757:
5633:
5567:
5535:
5488:
5464:
5399:
5346:
5299:
5234:
5214:
5194:
5171:
5135:
5083:
5042:
5006:
4986:
4966:
4952:is a surface of genus
4946:
4918:
4850:
4821:
4784:
4689:
4633:
4593:
4566:
4539:
4493:
4440:
4408:
4368:
4336:
4312:
4271:
4239:
4219:
4180:
4160:
4117:
4116:{\displaystyle 3g-3+k}
4082:
4049:
4020:properly discontinuous
4015:on TeichmĂĽller space.
4009:
3977:
3938:
3902:
3882:
3850:
3830:
3798:
3739:
3719:
3696:
3669:
3668:{\displaystyle n>0}
3639:
3584:
3528:
3484:
3464:
3440:
3420:
3388:
3361:
3312:
3292:
3266:
3246:
3215:
3195:
3168:
3148:
3128:
3108:
3088:
3049:
3029:
2985:
2959:
2854:
2853:{\displaystyle x\in S}
2828:
2785:
2757:
2709:
2665:
2626:
2562:
2530:
2510:
2460:
2418:
2357:
2330:
2310:
2274:
2247:
2227:
2178:
2151:
2137:is a homeomorphism of
2131:
2107:
2057:
2030:
2010:
1937:
1863:
1836:
1814:
1736:
1631:
1583:
1544:
1496:
1470:
1442:
1401:
1372:
1331:
1290:
1270:is injective, so that
1264:
1244:
1224:
1204:
1171:
1112:
1083:
1036:
962:
918:
851:
816:
790:
770:
750:
727:is the unit sphere in
721:
693:
622:
569:
482:
462:
442:
399:
357:
254:
234:
214:
194:
143:
114:geometric group theory
16:Concept in mathematics
7013:Putman, Andy (2010).
6951:10.1007/s002220050343
6522:, pp. 1393–1411.
6196:
6160:
6092:
6053:
6026:
5929:
5897:
5850:
5824:
5788:
5758:
5634:
5568:
5536:
5489:
5465:
5400:
5347:
5300:
5242:linear representation
5235:
5215:
5195:
5172:
5136:
5084:
5043:
5007:
4987:
4967:
4947:
4919:
4860:Finite presentability
4851:
4822:
4785:
4690:
4634:
4594:
4567:
4540:
4494:
4441:
4409:
4376:Weil–Petersson metric
4369:
4337:
4313:
4298:of a compact surface
4272:
4240:
4220:
4181:
4161:
4118:
4083:
4050:
4010:
3978:
3939:
3903:
3883:
3881:{\displaystyle (X,f)}
3851:
3831:
3799:
3740:
3720:
3697:
3695:{\displaystyle g^{n}}
3670:
3640:
3585:
3529:
3485:
3465:
3441:
3421:
3389:
3362:
3313:
3293:
3267:
3247:
3221:to a circle with the
3216:
3196:
3194:{\displaystyle A_{0}}
3169:
3149:
3129:
3109:
3089:
3050:
3030:
2986:
2960:
2855:
2829:
2786:
2758:
2710:
2666:
2627:
2563:
2531:
2511:
2461:
2419:
2358:
2356:{\displaystyle x_{0}}
2331:
2311:
2275:
2273:{\displaystyle x_{0}}
2248:
2228:
2179:
2177:{\displaystyle f_{*}}
2152:
2132:
2108:
2058:
2031:
2011:
1938:
1864:
1837:
1815:
1737:
1632:
1584:
1545:
1497:
1471:
1443:
1402:
1373:
1332:
1291:
1265:
1245:
1243:{\displaystyle \Phi }
1225:
1210:gives a left-inverse
1205:
1172:
1113:
1084:
1037:
963:
919:
852:
817:
791:
771:
751:
722:
694:
623:
570:
483:
463:
443:
400:
358:
255:
235:
215:
195:
144:
59:problems for curves.
45:compact-open topology
7099:Wajnryb, B. (1996).
6172:
6137:
6066:
6042:
5938:
5906:
5867:
5833:
5801:
5793:is usually called a
5770:
5649:
5599:
5545:
5501:
5478:
5412:
5365:
5312:
5247:
5224:
5204:
5181:
5145:
5113:
5101:The Torelli subgroup
5052:
5020:
4996:
4976:
4956:
4936:
4896:
4849:{\displaystyle 2g+1}
4831:
4805:
4702:
4658:
4603:
4576:
4549:
4503:
4457:
4418:
4398:
4346:
4326:
4320:pants decompositions
4302:
4249:
4229:
4209:
4170:
4138:
4092:
4060:
4048:{\displaystyle T(S)}
4030:
3987:
3948:
3916:
3892:
3860:
3840:
3829:{\displaystyle T(S)}
3811:
3788:
3729:
3709:
3679:
3653:
3611:
3541:
3497:
3474:
3454:
3430:
3398:
3371:
3322:
3302:
3276:
3256:
3229:
3205:
3178:
3158:
3138:
3118:
3098:
3078:
3039:
2995:
2975:
2867:
2838:
2818:
2775:
2719:
2680:
2636:
2575:
2540:
2520:
2470:
2428:
2367:
2340:
2320:
2284:
2257:
2237:
2188:
2161:
2141:
2121:
2097:
2040:
2020:
1950:
1876:
1846:
1824:
1763:
1644:
1593:
1554:
1506:
1483:
1460:
1411:
1382:
1341:
1300:
1274:
1263:{\displaystyle \Pi }
1254:
1234:
1223:{\displaystyle \Pi }
1214:
1185:
1122:
1093:
1046:
972:
931:
867:
826:
800:
780:
760:
731:
711:
639:
593:
495:
472:
452:
413:
370:
267:
244:
224:
204:
165:
133:
6943:1999InMat.138..103M
6534:, pp. 581–597.
6450:, pp. 753–758.
6426:, pp. 377–383.
6363:, pp. 103–149.
6303:, pp. 213–238.
6255:, pp. 417–431.
6243:, pp. 135–206.
5795:congruence subgroup
5358:intersection number
5220:is closed of genus
815:{\displaystyle z=0}
64:mapping class group
25:mapping class group
7142:Geometric topology
6998:10.1007/pl00001643
6684:Dehn, Max (1987).
6672:10.1007/bf02547712
6532:Duke Math. J. 2001
6361:Invent. Math. 1999
6191:
6155:
6087:
6048:
6021:
5924:
5892:
5845:
5819:
5783:
5753:
5629:
5563:
5531:
5484:
5460:
5395:
5342:
5295:
5230:
5210:
5193:{\displaystyle 2g}
5190:
5167:
5131:
5079:
5038:
5002:
4982:
4962:
4942:
4914:
4875:cut system complex
4846:
4817:
4780:
4774:
4735:
4685:
4629:
4589:
4562:
4535:
4489:
4436:
4404:
4364:
4332:
4308:
4267:
4235:
4215:
4176:
4156:
4113:
4078:
4045:
4005:
3973:
3934:
3898:
3878:
3846:
3826:
3794:
3735:
3715:
3692:
3665:
3635:
3580:
3574:
3524:
3480:
3460:
3436:
3416:
3384:
3357:
3308:
3288:
3262:
3242:
3211:
3191:
3164:
3144:
3124:
3104:
3084:
3045:
3025:
2981:
2971:In the case where
2955:
2850:
2824:
2781:
2753:
2705:
2661:
2622:
2558:
2526:
2506:
2456:
2414:
2353:
2326:
2306:
2270:
2243:
2223:
2174:
2147:
2127:
2103:
2053:
2026:
2006:
1933:
1859:
1832:
1810:
1732:
1627:
1579:
1540:
1492:
1466:
1456:In the case where
1438:
1397:
1368:
1327:
1286:
1260:
1240:
1220:
1200:
1167:
1108:
1079:
1032:
958:
914:
847:
812:
786:
766:
746:
717:
689:
618:
565:
478:
458:
438:
395:
353:
306:
250:
230:
210:
190:
139:
53:algebraic geometry
39:, is the group of
6864:978-1-4704-4526-3
6821:Thurston, William
6808:978-0-691-14735-2
6520:Geom. Topol. 2012
6189:
6051:{\displaystyle g}
5588:residually finite
5487:{\displaystyle S}
5233:{\displaystyle g}
5213:{\displaystyle S}
5107:singular homology
5005:{\displaystyle k}
4985:{\displaystyle b}
4965:{\displaystyle g}
4945:{\displaystyle S}
4407:{\displaystyle S}
4335:{\displaystyle S}
4311:{\displaystyle S}
4238:{\displaystyle S}
4218:{\displaystyle S}
4179:{\displaystyle S}
4128:Thurston boundary
4088:are of dimension
3901:{\displaystyle X}
3849:{\displaystyle S}
3806:TeichmĂĽller space
3797:{\displaystyle S}
3762:Pseudo-Anosov map
3748:or pseudo-Anosov.
3738:{\displaystyle g}
3718:{\displaystyle S}
3702:is the identity),
3483:{\displaystyle c}
3463:{\displaystyle c}
3439:{\displaystyle f}
3311:{\displaystyle A}
3265:{\displaystyle S}
3214:{\displaystyle c}
3167:{\displaystyle A}
3147:{\displaystyle f}
3127:{\displaystyle A}
3107:{\displaystyle S}
3087:{\displaystyle c}
3048:{\displaystyle x}
2984:{\displaystyle S}
2827:{\displaystyle S}
2784:{\displaystyle S}
2482:
2150:{\displaystyle S}
2130:{\displaystyle f}
2106:{\displaystyle S}
2029:{\displaystyle A}
1469:{\displaystyle S}
789:{\displaystyle S}
769:{\displaystyle S}
720:{\displaystyle S}
481:{\displaystyle S}
461:{\displaystyle S}
291:
253:{\displaystyle S}
233:{\displaystyle d}
213:{\displaystyle S}
142:{\displaystyle S}
7149:
7122:
7120:
7095:
7085:
7060:
7034:
7009:
6991:
6970:
6936:
6915:
6889:
6880:(3): 1393–1411.
6868:
6846:
6844:
6812:
6793:
6767:
6738:
6720:
6699:
6676:, translated in
6675:
6674:
6659:Acta Mathematica
6646:
6629:
6611:
6592:Brendle, Tara E.
6587:
6573:
6535:
6529:
6523:
6517:
6511:
6505:
6499:
6493:
6487:
6481:
6475:
6469:
6463:
6457:
6451:
6445:
6439:
6433:
6427:
6418:
6412:
6406:
6400:
6394:
6388:
6382:
6376:
6370:
6364:
6358:
6352:
6346:
6340:
6334:
6328:
6322:
6316:
6310:
6304:
6298:
6292:
6286:
6280:
6274:
6268:
6262:
6256:
6250:
6244:
6238:
6222:
6215:
6200:
6198:
6197:
6192:
6190:
6179:
6164:
6162:
6161:
6156:
6107:Tits alternative
6096:
6094:
6093:
6088:
6057:
6055:
6054:
6049:
6030:
6028:
6027:
6022:
6014:
6009:
5998:
5997:
5979:
5978:
5960:
5933:
5931:
5930:
5925:
5901:
5899:
5898:
5893:
5888:
5887:
5859:Finite subgroups
5854:
5852:
5851:
5846:
5828:
5826:
5825:
5820:
5792:
5790:
5789:
5784:
5782:
5781:
5762:
5760:
5759:
5754:
5749:
5741:
5736:
5725:
5724:
5706:
5695:
5694:
5661:
5660:
5638:
5636:
5635:
5630:
5625:
5614:
5613:
5572:
5570:
5569:
5564:
5540:
5538:
5537:
5532:
5527:
5516:
5515:
5496:arithmetic group
5493:
5491:
5490:
5485:
5469:
5467:
5466:
5461:
5456:
5445:
5444:
5404:
5402:
5401:
5396:
5391:
5380:
5379:
5354:symplectic group
5351:
5349:
5348:
5343:
5338:
5327:
5326:
5304:
5302:
5301:
5296:
5291:
5280:
5279:
5239:
5237:
5236:
5231:
5219:
5217:
5216:
5211:
5199:
5197:
5196:
5191:
5176:
5174:
5173:
5168:
5157:
5156:
5140:
5138:
5137:
5132:
5088:
5086:
5085:
5080:
5047:
5045:
5044:
5039:
5011:
5009:
5008:
5003:
4991:
4989:
4988:
4983:
4971:
4969:
4968:
4963:
4951:
4949:
4948:
4943:
4923:
4921:
4920:
4915:
4882:lantern relation
4855:
4853:
4852:
4847:
4826:
4824:
4823:
4818:
4789:
4787:
4786:
4781:
4779:
4778:
4740:
4739:
4694:
4692:
4691:
4686:
4681:
4670:
4669:
4638:
4636:
4635:
4630:
4628:
4627:
4615:
4614:
4598:
4596:
4595:
4590:
4588:
4587:
4571:
4569:
4568:
4563:
4561:
4560:
4544:
4542:
4541:
4536:
4534:
4533:
4515:
4514:
4498:
4496:
4495:
4490:
4488:
4487:
4469:
4468:
4445:
4443:
4442:
4437:
4413:
4411:
4410:
4405:
4388:markings complex
4382:Markings complex
4373:
4371:
4370:
4365:
4341:
4339:
4338:
4333:
4317:
4315:
4314:
4309:
4276:
4274:
4273:
4268:
4244:
4242:
4241:
4236:
4224:
4222:
4221:
4216:
4185:
4183:
4182:
4177:
4165:
4163:
4162:
4157:
4122:
4120:
4119:
4114:
4087:
4085:
4084:
4079:
4054:
4052:
4051:
4046:
4014:
4012:
4011:
4006:
3982:
3980:
3979:
3974:
3960:
3959:
3943:
3941:
3940:
3935:
3907:
3905:
3904:
3899:
3887:
3885:
3884:
3879:
3855:
3853:
3852:
3847:
3835:
3833:
3832:
3827:
3803:
3801:
3800:
3795:
3744:
3742:
3741:
3736:
3724:
3722:
3721:
3716:
3701:
3699:
3698:
3693:
3691:
3690:
3674:
3672:
3671:
3666:
3644:
3642:
3641:
3636:
3589:
3587:
3586:
3581:
3579:
3578:
3533:
3531:
3530:
3525:
3520:
3509:
3508:
3489:
3487:
3486:
3481:
3469:
3467:
3466:
3461:
3445:
3443:
3442:
3437:
3425:
3423:
3422:
3417:
3393:
3391:
3390:
3385:
3383:
3382:
3366:
3364:
3363:
3358:
3350:
3349:
3337:
3336:
3317:
3315:
3314:
3309:
3297:
3295:
3294:
3289:
3271:
3269:
3268:
3263:
3251:
3249:
3248:
3243:
3241:
3240:
3223:counterclockwise
3220:
3218:
3217:
3212:
3200:
3198:
3197:
3192:
3190:
3189:
3173:
3171:
3170:
3165:
3153:
3151:
3150:
3145:
3133:
3131:
3130:
3125:
3113:
3111:
3110:
3105:
3093:
3091:
3090:
3085:
3054:
3052:
3051:
3046:
3034:
3032:
3031:
3026:
2990:
2988:
2987:
2982:
2964:
2962:
2961:
2956:
2885:
2884:
2859:
2857:
2856:
2851:
2833:
2831:
2830:
2825:
2790:
2788:
2787:
2782:
2762:
2760:
2759:
2754:
2740:
2739:
2714:
2712:
2711:
2706:
2692:
2691:
2670:
2668:
2667:
2662:
2648:
2647:
2631:
2629:
2628:
2623:
2615:
2614:
2596:
2595:
2567:
2565:
2564:
2559:
2535:
2533:
2532:
2527:
2515:
2513:
2512:
2507:
2484:
2483:
2475:
2465:
2463:
2462:
2457:
2440:
2439:
2423:
2421:
2420:
2415:
2410:
2409:
2391:
2390:
2362:
2360:
2359:
2354:
2352:
2351:
2335:
2333:
2332:
2327:
2315:
2313:
2312:
2307:
2302:
2301:
2279:
2277:
2276:
2271:
2269:
2268:
2252:
2250:
2249:
2244:
2232:
2230:
2229:
2224:
2219:
2218:
2200:
2199:
2183:
2181:
2180:
2175:
2173:
2172:
2156:
2154:
2153:
2148:
2136:
2134:
2133:
2128:
2112:
2110:
2109:
2104:
2062:
2060:
2059:
2054:
2052:
2051:
2035:
2033:
2032:
2027:
2015:
2013:
2012:
2007:
1996:
1988:
1968:
1960:
1942:
1940:
1939:
1934:
1929:
1928:
1927:
1919:
1888:
1887:
1868:
1866:
1865:
1860:
1858:
1857:
1841:
1839:
1838:
1833:
1831:
1819:
1817:
1816:
1811:
1800:
1792:
1775:
1774:
1741:
1739:
1738:
1733:
1710:
1709:
1700:
1674:
1673:
1636:
1634:
1633:
1628:
1605:
1604:
1588:
1586:
1585:
1580:
1566:
1565:
1549:
1547:
1546:
1541:
1518:
1517:
1501:
1499:
1498:
1493:
1475:
1473:
1472:
1467:
1447:
1445:
1444:
1439:
1434:
1423:
1422:
1406:
1404:
1403:
1398:
1396:
1395:
1390:
1377:
1375:
1374:
1369:
1364:
1353:
1352:
1336:
1334:
1333:
1328:
1323:
1322:
1317:
1295:
1293:
1292:
1287:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1230:to the morphism
1229:
1227:
1226:
1221:
1209:
1207:
1206:
1201:
1199:
1198:
1193:
1176:
1174:
1173:
1168:
1166:
1165:
1160:
1142:
1141:
1136:
1117:
1115:
1114:
1109:
1107:
1106:
1101:
1088:
1086:
1085:
1080:
1075:
1064:
1063:
1041:
1039:
1038:
1033:
1028:
1027:
1022:
1001:
990:
989:
967:
965:
964:
959:
954:
943:
942:
923:
921:
920:
915:
913:
912:
907:
901:
896:
895:
890:
881:
880:
875:
856:
854:
853:
848:
846:
838:
833:
821:
819:
818:
813:
795:
793:
792:
787:
775:
773:
772:
767:
755:
753:
752:
747:
745:
744:
739:
726:
724:
723:
718:
698:
696:
695:
690:
676:
675:
651:
650:
627:
625:
624:
619:
605:
604:
574:
572:
571:
566:
552:
551:
542:
525:
524:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
425:
424:
404:
402:
401:
396:
382:
381:
362:
360:
359:
354:
352:
348:
305:
259:
257:
256:
251:
239:
237:
236:
231:
219:
217:
216:
211:
199:
197:
196:
191:
177:
176:
148:
146:
145:
140:
7157:
7156:
7152:
7151:
7150:
7148:
7147:
7146:
7127:
7126:
7125:
6865:
6851:Ivanov, Nikolai
6809:
6790:
6789:978-069114794-9
6696:
6695:978-038796416-4
6578:Birman, Joan S.
6543:
6538:
6530:
6526:
6518:
6514:
6506:
6502:
6494:
6490:
6486:, Theorem 6.11.
6482:
6478:
6470:
6466:
6458:
6454:
6446:
6442:
6436:J. Algebra 2004
6434:
6430:
6419:
6415:
6407:
6403:
6395:
6391:
6383:
6379:
6371:
6367:
6359:
6355:
6347:
6343:
6335:
6331:
6323:
6319:
6311:
6307:
6299:
6295:
6287:
6283:
6275:
6271:
6263:
6259:
6251:
6247:
6241:Acta Math. 1938
6239:
6235:
6231:
6226:
6225:
6216:
6212:
6207:
6178:
6173:
6170:
6169:
6138:
6135:
6134:
6122:
6103:
6067:
6064:
6063:
6060:Hurwitz's bound
6043:
6040:
6039:
6010:
6005:
5990:
5986:
5974:
5970:
5956:
5939:
5936:
5935:
5907:
5904:
5903:
5883:
5879:
5868:
5865:
5864:
5861:
5834:
5831:
5830:
5802:
5799:
5798:
5777:
5773:
5771:
5768:
5767:
5745:
5737:
5732:
5717:
5713:
5702:
5687:
5683:
5656:
5652:
5650:
5647:
5646:
5621:
5606:
5602:
5600:
5597:
5596:
5579:
5546:
5543:
5542:
5523:
5508:
5504:
5502:
5499:
5498:
5479:
5476:
5475:
5452:
5437:
5433:
5413:
5410:
5409:
5387:
5372:
5368:
5366:
5363:
5362:
5334:
5319:
5315:
5313:
5310:
5309:
5287:
5272:
5268:
5248:
5245:
5244:
5225:
5222:
5221:
5205:
5202:
5201:
5182:
5179:
5178:
5152:
5148:
5146:
5143:
5142:
5114:
5111:
5110:
5103:
5098:
5053:
5050:
5049:
5021:
5018:
5017:
4997:
4994:
4993:
4977:
4974:
4973:
4957:
4954:
4953:
4937:
4934:
4933:
4930:
4897:
4894:
4893:
4890:
4862:
4832:
4829:
4828:
4806:
4803:
4802:
4773:
4772:
4767:
4761:
4760:
4755:
4745:
4744:
4734:
4733:
4728:
4722:
4721:
4716:
4706:
4705:
4703:
4700:
4699:
4677:
4665:
4661:
4659:
4656:
4655:
4651:
4646:
4623:
4619:
4610:
4606:
4604:
4601:
4600:
4583:
4579:
4577:
4574:
4573:
4556:
4552:
4550:
4547:
4546:
4529:
4525:
4510:
4506:
4504:
4501:
4500:
4483:
4479:
4464:
4460:
4458:
4455:
4454:
4419:
4416:
4415:
4399:
4396:
4395:
4384:
4347:
4344:
4343:
4327:
4324:
4323:
4303:
4300:
4299:
4292:
4287:
4250:
4247:
4246:
4230:
4227:
4226:
4210:
4207:
4206:
4199:
4171:
4168:
4167:
4139:
4136:
4135:
4093:
4090:
4089:
4061:
4058:
4057:
4031:
4028:
4027:
3988:
3985:
3984:
3955:
3951:
3949:
3946:
3945:
3917:
3914:
3913:
3910:Riemann surface
3893:
3890:
3889:
3861:
3858:
3857:
3841:
3838:
3837:
3812:
3809:
3808:
3789:
3786:
3785:
3782:
3777:
3764:
3758:
3730:
3727:
3726:
3710:
3707:
3706:
3686:
3682:
3680:
3677:
3676:
3654:
3651:
3650:
3612:
3609:
3608:
3605:
3599:
3573:
3572:
3567:
3561:
3560:
3555:
3545:
3544:
3542:
3539:
3538:
3516:
3504:
3500:
3498:
3495:
3494:
3475:
3472:
3471:
3455:
3452:
3451:
3431:
3428:
3427:
3399:
3396:
3395:
3378:
3374:
3372:
3369:
3368:
3367:. The class of
3345:
3341:
3329:
3325:
3323:
3320:
3319:
3318:it is equal to
3303:
3300:
3299:
3277:
3274:
3273:
3272:as follows: on
3257:
3254:
3253:
3236:
3232:
3230:
3227:
3226:
3206:
3203:
3202:
3185:
3181:
3179:
3176:
3175:
3159:
3156:
3155:
3139:
3136:
3135:
3119:
3116:
3115:
3099:
3096:
3095:
3079:
3076:
3075:
3072:
3066:
3061:
3040:
3037:
3036:
2996:
2993:
2992:
2976:
2973:
2972:
2880:
2876:
2868:
2865:
2864:
2839:
2836:
2835:
2819:
2816:
2815:
2802:
2776:
2773:
2772:
2735:
2731:
2720:
2717:
2716:
2687:
2683:
2681:
2678:
2677:
2643:
2639:
2637:
2634:
2633:
2610:
2606:
2591:
2587:
2576:
2573:
2572:
2541:
2538:
2537:
2521:
2518:
2517:
2474:
2473:
2471:
2468:
2467:
2435:
2431:
2429:
2426:
2425:
2405:
2401:
2386:
2382:
2368:
2365:
2364:
2347:
2343:
2341:
2338:
2337:
2321:
2318:
2317:
2316:and for a loop
2297:
2293:
2285:
2282:
2281:
2264:
2260:
2258:
2255:
2254:
2238:
2235:
2234:
2214:
2210:
2195:
2191:
2189:
2186:
2185:
2168:
2164:
2162:
2159:
2158:
2142:
2139:
2138:
2122:
2119:
2118:
2098:
2095:
2094:
2091:
2075:
2069:
2047:
2043:
2041:
2038:
2037:
2021:
2018:
2017:
1992:
1984:
1964:
1956:
1951:
1948:
1947:
1923:
1915:
1905:
1901:
1883:
1879:
1877:
1874:
1873:
1853:
1849:
1847:
1844:
1843:
1827:
1825:
1822:
1821:
1796:
1788:
1770:
1766:
1764:
1761:
1760:
1753:
1705:
1701:
1696:
1669:
1665:
1645:
1642:
1641:
1600:
1596:
1594:
1591:
1590:
1561:
1557:
1555:
1552:
1551:
1513:
1509:
1507:
1504:
1503:
1484:
1481:
1480:
1461:
1458:
1457:
1454:
1430:
1418:
1414:
1412:
1409:
1408:
1391:
1386:
1385:
1383:
1380:
1379:
1360:
1348:
1344:
1342:
1339:
1338:
1318:
1313:
1312:
1301:
1298:
1297:
1275:
1272:
1271:
1255:
1252:
1251:
1235:
1232:
1231:
1215:
1212:
1211:
1194:
1189:
1188:
1186:
1183:
1182:
1161:
1156:
1155:
1137:
1132:
1131:
1123:
1120:
1119:
1102:
1097:
1096:
1094:
1091:
1090:
1071:
1059:
1055:
1047:
1044:
1043:
1023:
1018:
1017:
997:
985:
981:
973:
970:
969:
950:
938:
934:
932:
929:
928:
908:
903:
902:
897:
891:
886:
885:
876:
871:
870:
868:
865:
864:
842:
834:
829:
827:
824:
823:
801:
798:
797:
781:
778:
777:
761:
758:
757:
740:
735:
734:
732:
729:
728:
712:
709:
708:
705:
671:
667:
646:
642:
640:
637:
636:
600:
596:
594:
591:
590:
547:
543:
538:
520:
516:
496:
493:
492:
473:
470:
469:
453:
450:
449:
420:
416:
414:
411:
410:
377:
373:
371:
368:
367:
311:
307:
295:
268:
265:
264:
245:
242:
241:
225:
222:
221:
205:
202:
201:
172:
168:
166:
163:
162:
134:
131:
130:
127:
122:
91:
55:in relation to
17:
12:
11:
5:
7155:
7145:
7144:
7139:
7124:
7123:
7111:(2): 377–383.
7096:
7076:(2): 417–431.
7061:
7025:(2): 753–758.
7010:
6982:(4): 902–974.
6971:
6927:(1): 103–149.
6916:
6869:
6863:
6847:
6835:(3): 221–237.
6817:Hatcher, Allen
6813:
6807:
6794:
6788:
6768:
6750:(3): 581–597.
6739:
6700:
6694:
6681:
6647:
6630:
6588:
6574:
6556:(2): 213–238.
6544:
6542:
6539:
6537:
6536:
6524:
6512:
6500:
6488:
6476:
6464:
6462:, Theorem 6.4.
6452:
6440:
6428:
6413:
6401:
6399:, Theorem 4.1.
6389:
6377:
6365:
6353:
6341:
6329:
6317:
6315:, Theorem 4.6.
6305:
6293:
6291:, Theorem 8.1.
6281:
6269:
6267:, Theorem 2.5.
6257:
6245:
6232:
6230:
6227:
6224:
6223:
6209:
6208:
6206:
6203:
6188:
6185:
6182:
6177:
6154:
6151:
6148:
6145:
6142:
6121:
6118:
6102:
6099:
6086:
6083:
6080:
6077:
6074:
6071:
6047:
6020:
6017:
6013:
6008:
6004:
6001:
5996:
5993:
5989:
5985:
5982:
5977:
5973:
5969:
5966:
5963:
5959:
5955:
5952:
5949:
5946:
5943:
5923:
5920:
5917:
5914:
5911:
5891:
5886:
5882:
5878:
5875:
5872:
5860:
5857:
5844:
5841:
5838:
5818:
5815:
5812:
5809:
5806:
5780:
5776:
5766:The kernel of
5764:
5763:
5752:
5748:
5744:
5740:
5735:
5731:
5728:
5723:
5720:
5716:
5712:
5709:
5705:
5701:
5698:
5693:
5690:
5686:
5682:
5679:
5676:
5673:
5670:
5667:
5664:
5659:
5655:
5628:
5624:
5620:
5617:
5612:
5609:
5605:
5593:
5592:
5578:
5575:
5562:
5559:
5556:
5553:
5550:
5530:
5526:
5522:
5519:
5514:
5511:
5507:
5483:
5470:is called the
5459:
5455:
5451:
5448:
5443:
5440:
5436:
5432:
5429:
5426:
5423:
5420:
5417:
5394:
5390:
5386:
5383:
5378:
5375:
5371:
5341:
5337:
5333:
5330:
5325:
5322:
5318:
5294:
5290:
5286:
5283:
5278:
5275:
5271:
5267:
5264:
5261:
5258:
5255:
5252:
5229:
5209:
5189:
5186:
5166:
5163:
5160:
5155:
5151:
5130:
5127:
5124:
5121:
5118:
5102:
5099:
5097:
5094:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5037:
5034:
5031:
5028:
5025:
5001:
4981:
4961:
4941:
4929:
4926:
4913:
4910:
4907:
4904:
4901:
4889:
4886:
4861:
4858:
4845:
4842:
4839:
4836:
4816:
4813:
4810:
4792:
4791:
4777:
4771:
4768:
4766:
4763:
4762:
4759:
4756:
4754:
4751:
4750:
4748:
4743:
4738:
4732:
4729:
4727:
4724:
4723:
4720:
4717:
4715:
4712:
4711:
4709:
4684:
4680:
4676:
4673:
4668:
4664:
4650:
4647:
4645:
4642:
4626:
4622:
4618:
4613:
4609:
4586:
4582:
4559:
4555:
4532:
4528:
4524:
4521:
4518:
4513:
4509:
4486:
4482:
4478:
4475:
4472:
4467:
4463:
4448:locally finite
4435:
4432:
4429:
4426:
4423:
4403:
4383:
4380:
4363:
4360:
4357:
4354:
4351:
4331:
4307:
4291:
4288:
4286:
4283:
4266:
4263:
4260:
4257:
4254:
4234:
4214:
4198:
4195:
4194:
4193:
4190:
4187:
4175:
4155:
4152:
4149:
4146:
4143:
4112:
4109:
4106:
4103:
4100:
4097:
4077:
4074:
4071:
4068:
4065:
4044:
4041:
4038:
4035:
4004:
4001:
3998:
3995:
3992:
3972:
3969:
3966:
3963:
3958:
3954:
3933:
3930:
3927:
3924:
3921:
3897:
3877:
3874:
3871:
3868:
3865:
3845:
3825:
3822:
3819:
3816:
3793:
3781:
3778:
3776:
3773:
3760:Main article:
3757:
3754:
3750:
3749:
3746:
3734:
3714:
3703:
3689:
3685:
3664:
3661:
3658:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3601:Main article:
3598:
3595:
3591:
3590:
3577:
3571:
3568:
3566:
3563:
3562:
3559:
3556:
3554:
3551:
3550:
3548:
3523:
3519:
3515:
3512:
3507:
3503:
3479:
3459:
3435:
3415:
3412:
3409:
3406:
3403:
3381:
3377:
3356:
3353:
3348:
3344:
3340:
3335:
3332:
3328:
3307:
3287:
3284:
3281:
3261:
3239:
3235:
3210:
3188:
3184:
3163:
3143:
3123:
3103:
3083:
3068:Main article:
3065:
3062:
3060:
3057:
3044:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2980:
2969:
2968:
2967:
2966:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2883:
2879:
2875:
2872:
2849:
2846:
2843:
2823:
2801:
2798:
2780:
2766:
2765:
2752:
2749:
2746:
2743:
2738:
2734:
2730:
2727:
2724:
2704:
2701:
2698:
2695:
2690:
2686:
2660:
2657:
2654:
2651:
2646:
2642:
2621:
2618:
2613:
2609:
2605:
2602:
2599:
2594:
2590:
2586:
2583:
2580:
2557:
2554:
2551:
2548:
2545:
2525:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2481:
2478:
2455:
2452:
2449:
2446:
2443:
2438:
2434:
2413:
2408:
2404:
2400:
2397:
2394:
2389:
2385:
2381:
2378:
2375:
2372:
2350:
2346:
2325:
2305:
2300:
2296:
2292:
2289:
2267:
2263:
2242:
2222:
2217:
2213:
2209:
2206:
2203:
2198:
2194:
2171:
2167:
2146:
2126:
2102:
2090:
2087:
2071:Main article:
2068:
2065:
2050:
2046:
2025:
2005:
2002:
1999:
1995:
1991:
1987:
1983:
1980:
1977:
1974:
1971:
1967:
1963:
1959:
1955:
1944:
1943:
1932:
1926:
1922:
1918:
1914:
1911:
1908:
1904:
1900:
1897:
1894:
1891:
1886:
1882:
1856:
1852:
1830:
1809:
1806:
1803:
1799:
1795:
1791:
1787:
1784:
1781:
1778:
1773:
1769:
1752:
1749:
1744:
1743:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1708:
1704:
1699:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1672:
1668:
1664:
1661:
1658:
1655:
1652:
1649:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1603:
1599:
1578:
1575:
1572:
1569:
1564:
1560:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1516:
1512:
1491:
1488:
1465:
1453:
1450:
1437:
1433:
1429:
1426:
1421:
1417:
1394:
1389:
1367:
1363:
1359:
1356:
1351:
1347:
1326:
1321:
1316:
1311:
1308:
1305:
1285:
1282:
1279:
1259:
1239:
1219:
1197:
1192:
1179:homology group
1164:
1159:
1154:
1151:
1148:
1145:
1140:
1135:
1130:
1127:
1105:
1100:
1078:
1074:
1070:
1067:
1062:
1058:
1054:
1051:
1031:
1026:
1021:
1016:
1013:
1010:
1007:
1004:
1000:
996:
993:
988:
984:
980:
977:
957:
953:
949:
946:
941:
937:
911:
906:
900:
894:
889:
884:
879:
874:
845:
841:
837:
832:
811:
808:
805:
785:
765:
743:
738:
716:
704:
701:
688:
685:
682:
679:
674:
670:
666:
663:
660:
657:
654:
649:
645:
633:diffeomorphism
617:
614:
611:
608:
603:
599:
577:
576:
564:
561:
558:
555:
550:
546:
541:
537:
534:
531:
528:
523:
519:
515:
512:
509:
506:
503:
500:
477:
457:
437:
434:
431:
428:
423:
419:
394:
391:
388:
385:
380:
376:
364:
363:
351:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
310:
304:
301:
298:
294:
290:
287:
284:
281:
278:
275:
272:
249:
229:
209:
189:
186:
183:
180:
175:
171:
138:
126:
123:
121:
118:
90:
87:
41:homeomorphisms
15:
9:
6:
4:
3:
2:
7154:
7143:
7140:
7138:
7135:
7134:
7132:
7119:
7114:
7110:
7106:
7102:
7097:
7093:
7089:
7084:
7079:
7075:
7071:
7067:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7033:
7028:
7024:
7020:
7016:
7011:
7007:
7003:
6999:
6995:
6990:
6985:
6981:
6977:
6972:
6968:
6964:
6960:
6956:
6952:
6948:
6944:
6940:
6935:
6930:
6926:
6922:
6917:
6913:
6909:
6905:
6901:
6897:
6893:
6888:
6883:
6879:
6875:
6870:
6866:
6860:
6856:
6852:
6848:
6843:
6838:
6834:
6830:
6826:
6822:
6818:
6814:
6810:
6804:
6800:
6795:
6791:
6785:
6781:
6777:
6776:Margalit, Dan
6773:
6769:
6765:
6761:
6757:
6753:
6749:
6745:
6740:
6736:
6732:
6728:
6724:
6719:
6714:
6710:
6706:
6701:
6697:
6691:
6687:
6682:
6679:
6673:
6668:
6664:
6661:(in German).
6660:
6656:
6652:
6648:
6644:
6640:
6636:
6631:
6627:
6623:
6619:
6615:
6610:
6605:
6601:
6597:
6593:
6589:
6585:
6584:
6579:
6575:
6571:
6567:
6563:
6559:
6555:
6551:
6546:
6545:
6533:
6528:
6521:
6516:
6509:
6504:
6497:
6492:
6485:
6480:
6473:
6468:
6461:
6456:
6449:
6444:
6437:
6432:
6425:
6423:
6417:
6410:
6405:
6398:
6393:
6386:
6381:
6374:
6369:
6362:
6357:
6350:
6345:
6338:
6333:
6326:
6321:
6314:
6309:
6302:
6297:
6290:
6285:
6278:
6273:
6266:
6261:
6254:
6249:
6242:
6237:
6233:
6220:
6214:
6210:
6202:
6186:
6183:
6180:
6175:
6166:
6149:
6143:
6140:
6132:
6127:
6126:open question
6117:
6114:
6112:
6108:
6098:
6081:
6078:
6075:
6069:
6061:
6045:
6037:
6032:
6015:
6011:
5999:
5994:
5991:
5987:
5983:
5975:
5964:
5961:
5957:
5950:
5944:
5941:
5918:
5912:
5909:
5884:
5873:
5870:
5856:
5842:
5839:
5836:
5813:
5807:
5804:
5796:
5778:
5742:
5738:
5726:
5721:
5718:
5714:
5696:
5691:
5688:
5684:
5674:
5668:
5665:
5662:
5657:
5645:
5644:
5643:
5640:
5615:
5610:
5607:
5603:
5591:
5589:
5584:
5583:
5582:
5574:
5557:
5551:
5548:
5517:
5512:
5509:
5505:
5497:
5481:
5473:
5472:Torelli group
5446:
5441:
5438:
5434:
5424:
5418:
5415:
5406:
5381:
5376:
5373:
5369:
5359:
5355:
5328:
5323:
5320:
5316:
5306:
5281:
5276:
5273:
5269:
5259:
5253:
5250:
5243:
5227:
5207:
5187:
5184:
5161:
5153:
5149:
5125:
5119:
5116:
5108:
5093:
5090:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5032:
5026:
5023:
5015:
4999:
4979:
4959:
4939:
4925:
4908:
4902:
4899:
4885:
4883:
4878:
4876:
4870:
4868:
4857:
4843:
4840:
4837:
4834:
4814:
4811:
4808:
4799:
4797:
4775:
4769:
4764:
4757:
4752:
4746:
4741:
4736:
4730:
4725:
4718:
4713:
4707:
4698:
4697:
4696:
4671:
4666:
4662:
4641:
4624:
4620:
4616:
4611:
4607:
4584:
4580:
4557:
4553:
4530:
4526:
4522:
4519:
4516:
4511:
4507:
4484:
4480:
4476:
4473:
4470:
4465:
4461:
4451:
4449:
4430:
4424:
4421:
4401:
4393:
4389:
4379:
4377:
4358:
4352:
4349:
4329:
4321:
4305:
4297:
4296:pants complex
4290:Pants complex
4282:
4278:
4261:
4255:
4252:
4232:
4212:
4205:of a surface
4204:
4203:curve complex
4191:
4188:
4173:
4150:
4144:
4141:
4133:
4132:
4131:
4129:
4124:
4110:
4107:
4104:
4101:
4098:
4095:
4072:
4066:
4063:
4039:
4033:
4025:
4021:
4016:
3999:
3993:
3990:
3967:
3961:
3956:
3952:
3931:
3925:
3922:
3919:
3911:
3895:
3872:
3869:
3866:
3843:
3820:
3814:
3807:
3791:
3772:
3768:
3763:
3753:
3747:
3732:
3712:
3704:
3687:
3683:
3662:
3659:
3656:
3648:
3647:
3646:
3629:
3623:
3620:
3617:
3614:
3604:
3594:
3575:
3569:
3564:
3557:
3552:
3546:
3537:
3536:
3535:
3510:
3505:
3501:
3491:
3477:
3457:
3449:
3433:
3410:
3404:
3401:
3379:
3375:
3354:
3351:
3346:
3342:
3338:
3333:
3330:
3326:
3305:
3285:
3279:
3259:
3237:
3233:
3224:
3208:
3186:
3182:
3161:
3141:
3121:
3101:
3081:
3071:
3056:
3042:
3016:
3007:
3001:
2998:
2978:
2952:
2943:
2937:
2934:
2922:
2913:
2907:
2904:
2895:
2892:
2889:
2881:
2877:
2870:
2863:
2862:
2861:
2847:
2844:
2841:
2821:
2812:
2811:
2810:
2808:
2797:
2794:
2778:
2769:
2764:
2744:
2736:
2732:
2725:
2722:
2699:
2693:
2688:
2684:
2674:
2673:
2672:
2655:
2649:
2644:
2640:
2611:
2607:
2603:
2600:
2592:
2588:
2581:
2578:
2571:
2552:
2546:
2543:
2523:
2503:
2500:
2494:
2488:
2485:
2476:
2447:
2436:
2432:
2406:
2402:
2398:
2395:
2387:
2383:
2379:
2373:
2348:
2344:
2323:
2298:
2294:
2287:
2265:
2261:
2240:
2215:
2211:
2207:
2204:
2196:
2192:
2169:
2165:
2144:
2124:
2116:
2100:
2086:
2084:
2080:
2074:
2064:
2048:
2044:
2023:
2000:
1997:
1989:
1978:
1972:
1969:
1961:
1930:
1920:
1912:
1909:
1906:
1902:
1898:
1892:
1884:
1880:
1872:
1871:
1870:
1854:
1850:
1804:
1801:
1793:
1785:
1782:
1776:
1771:
1767:
1758:
1748:
1726:
1720:
1717:
1711:
1706:
1702:
1697:
1690:
1684:
1681:
1675:
1670:
1666:
1662:
1656:
1650:
1647:
1640:
1639:
1638:
1621:
1615:
1612:
1606:
1601:
1597:
1573:
1567:
1562:
1558:
1534:
1528:
1525:
1519:
1514:
1510:
1489:
1479:
1463:
1449:
1424:
1419:
1415:
1392:
1354:
1349:
1345:
1319:
1306:
1303:
1280:
1195:
1180:
1162:
1152:
1149:
1146:
1138:
1128:
1125:
1103:
1065:
1060:
1056:
1052:
1049:
1024:
1011:
1008:
991:
986:
982:
978:
944:
939:
935:
927:
926:modular group
909:
898:
892:
882:
877:
863:
858:
839:
835:
809:
806:
803:
783:
763:
741:
714:
707:Suppose that
700:
683:
677:
672:
668:
664:
658:
652:
647:
643:
634:
629:
612:
606:
601:
597:
589:
584:
582:
559:
553:
548:
544:
539:
532:
526:
521:
517:
513:
507:
501:
498:
491:
490:
489:
488:is the group
475:
455:
432:
426:
421:
417:
408:
389:
383:
378:
374:
349:
339:
333:
330:
324:
318:
312:
308:
302:
299:
296:
288:
282:
279:
276:
270:
263:
262:
261:
247:
227:
207:
184:
178:
173:
169:
160:
156:
152:
136:
117:
115:
110:
108:
103:
101:
100:Jakob Nielsen
97:
86:
84:
80:
75:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
33:modular group
30:
26:
22:
7108:
7104:
7073:
7069:
7022:
7018:
6989:math/9807150
6979:
6975:
6934:math/9804098
6924:
6920:
6877:
6873:
6854:
6832:
6828:
6798:
6779:
6772:Farb, Benson
6747:
6743:
6708:
6704:
6685:
6662:
6658:
6634:
6609:math/0307039
6599:
6595:
6582:
6553:
6549:
6527:
6515:
6510:, Theorem 1.
6503:
6498:, Theorem 4.
6491:
6479:
6467:
6455:
6443:
6431:
6421:
6416:
6404:
6392:
6380:
6368:
6356:
6344:
6332:
6327:, Chapter 9.
6320:
6308:
6296:
6284:
6272:
6260:
6248:
6236:
6213:
6167:
6123:
6115:
6104:
6033:
5862:
5794:
5765:
5641:
5594:
5585:
5580:
5471:
5407:
5307:
5104:
5091:
5048:is equal to
4931:
4891:
4879:
4874:
4871:
4863:
4800:
4793:
4652:
4452:
4391:
4387:
4385:
4293:
4279:
4200:
4125:
4022:(though not
4017:
3783:
3769:
3765:
3751:
3606:
3592:
3492:
3447:
3073:
2970:
2813:
2803:
2770:
2767:
2675:
2092:
2082:
2078:
2076:
2073:Braid groups
1945:
1754:
1745:
1455:
859:
706:
630:
587:
585:
578:
365:
161:surface and
128:
111:
104:
92:
79:braid groups
76:
72:group theory
61:
36:
32:
24:
18:
6665:: 135–206.
6508:Ivanov 1992
6496:Ivanov 1992
6301:Birman 1969
6277:Birman 1974
6221:) markings.
3645:is either:
3064:Dehn twists
2807:Joan Birman
2085:punctures.
49:3-manifolds
7131:Categories
6596:J. Algebra
6373:Brock 2002
3675:such that
3448:Dehn twist
3070:Dehn twist
579:This is a
159:orientable
7032:0812.0017
6887:1106.4261
6718:1307.3733
6678:Dehn 1987
6651:Dehn, Max
6229:Citations
6184:−
6144:
6124:It is an
6079:−
6000:
5984:≅
5972:Φ
5965:
5945:
5913:
5881:Φ
5874:
5840:≥
5808:
5775:Φ
5727:
5711:→
5697:
5681:→
5669:
5654:Φ
5616:
5552:
5518:
5447:
5431:→
5419:
5382:
5329:
5282:
5266:→
5254:
5120:
5062:−
5027:
4903:
4812:≥
4672:
4621:β
4608:α
4581:α
4554:β
4531:ξ
4527:β
4520:…
4508:β
4485:ξ
4481:α
4474:…
4462:α
4425:
4353:
4256:
4145:
4102:−
4067:
3994:
3962:
3929:→
3624:
3618:∈
3511:
3405:
3376:τ
3352:∘
3343:τ
3339:∘
3331:−
3283:∖
3234:τ
3011:∖
3002:
2950:→
2938:
2932:→
2917:∖
2908:
2902:→
2878:π
2874:→
2845:∈
2733:π
2726:
2694:
2689:±
2650:
2589:π
2582:
2547:
2524:γ
2504:γ
2501:∗
2495:α
2486:∗
2480:¯
2477:γ
2448:α
2437:∗
2384:π
2380:∈
2374:α
2336:based at
2324:α
2241:γ
2193:π
2170:∗
2045:τ
1913:π
1881:τ
1851:τ
1802:≤
1786:≤
1724:∂
1712:
1688:∂
1676:
1651:
1619:∂
1607:
1568:
1532:∂
1520:
1487:∂
1425:
1355:
1307:
1284:Φ
1278:Π
1258:Π
1238:Φ
1218:Π
1144:↦
1066:
1053:∈
1012:
1006:→
992:
976:Φ
945:
678:
665:⊂
653:
607:
602:±
581:countable
554:
527:
502:
427:
384:
300:∈
271:δ
179:
151:connected
68:manifolds
7105:Topology
7006:14834205
6967:16199015
6912:17330187
6853:(1992).
6829:Topology
6823:(1980).
6778:(2012).
6735:15393033
6653:(1938).
6626:14784932
6580:(1974).
6422:Topology
4392:markings
2253:between
1478:boundary
1042:: every
107:Thurston
96:Max Dehn
21:topology
7092:0956596
7057:2047111
7049:2557192
6959:1714338
6939:Bibcode
6904:2967055
6764:1813237
6643:1940162
6570:0243519
6541:Sources
5352:of the
2793:Out(Fn)
2568:to the
2424:define
1757:annulus
583:group.
89:History
29:surface
7090:
7055:
7047:
7004:
6965:
6957:
6910:
6902:
6861:
6805:
6786:
6762:
6733:
6692:
6641:
6624:
6568:
3888:where
3450:about
2115:closed
405:. The
155:closed
57:moduli
23:, the
7053:S2CID
7027:arXiv
7002:S2CID
6984:arXiv
6963:S2CID
6929:arXiv
6908:S2CID
6882:arXiv
6731:S2CID
6713:arXiv
6711:(8).
6622:S2CID
6604:arXiv
6205:Notes
4972:with
3953:Homeo
3908:is a
3470:. If
3154:from
2641:Homeo
2544:Homeo
1703:Homeo
1667:Homeo
1598:Homeo
1559:Homeo
1511:Homeo
862:torus
669:Homeo
545:Homeo
518:Homeo
418:Homeo
375:Homeo
170:Homeo
149:be a
27:of a
6859:ISBN
6803:ISBN
6784:ISBN
6690:ISBN
6424:1996
6111:free
4294:The
4201:The
4024:free
3912:and
3660:>
2814:Let
2280:and
2117:and
1755:Any
1337:and
1118:via
644:Diff
129:Let
98:and
81:and
62:The
7113:doi
7078:doi
7037:doi
7023:138
6994:doi
6947:doi
6925:138
6892:doi
6837:doi
6752:doi
6748:106
6723:doi
6709:166
6667:doi
6614:doi
6600:278
6558:doi
6141:Mod
5962:ker
5942:Mod
5910:Mod
5871:ker
5805:Mod
5797:of
5666:Mod
5549:Mod
5474:of
5416:Mod
5251:Mod
5200:if
5117:Mod
5105:As
5024:Mod
5016:of
4932:If
4900:Mod
4827:is
4422:Mod
4394:of
4350:Mod
4322:of
4253:Mod
4142:Mod
4064:Mod
3991:Mod
3621:Mod
3402:Mod
3252:of
3074:If
2999:Mod
2935:Mod
2905:Mod
2723:Out
2685:Mod
2579:Out
2113:is
2093:If
1820:of
1648:Mod
1407:is
1304:Mod
1181:of
1009:Mod
598:Mod
499:Mod
293:sup
240:on
35:or
7133::
7109:35
7107:.
7103:.
7088:MR
7086:.
7074:19
7072:.
7068:.
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7045:MR
7043:.
7035:.
7021:.
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7000:.
6992:.
6980:10
6978:.
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6819:;
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6758:.
6746:.
6729:.
6721:.
6707:.
6663:69
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6566:MR
6564:.
6554:22
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6201:.
6165:.
6097:.
6070:84
6058:.
6031:.
5988:Sp
5715:Sp
5685:Sp
5604:Sp
5590:.
5506:Sp
5435:Sp
5405:.
5370:Sp
5317:Sp
5305:.
5270:GL
5089:.
4884:.
4877:.
4869:.
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4663:SL
4378:.
4186:);
4123:.
3502:SL
3055:.
2763:.
2063:.
1448:.
1416:GL
1346:SL
1057:SL
983:SL
936:SL
157:,
153:,
85:.
74:.
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6737:.
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6279:.
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2083:n
2079:n
2049:0
2024:A
2004:}
2001:2
1998:=
1994:|
1990:z
1986:|
1982:{
1979:,
1976:}
1973:1
1970:=
1966:|
1962:z
1958:|
1954:{
1931:z
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1907:2
1903:e
1899:=
1896:)
1893:z
1890:(
1885:0
1855:0
1829:C
1808:}
1805:2
1798:|
1794:z
1790:|
1783:1
1780:{
1777:=
1772:0
1768:A
1742:.
1730:)
1727:S
1721:,
1718:S
1715:(
1707:0
1698:/
1694:)
1691:S
1685:,
1682:S
1679:(
1671:+
1663:=
1660:)
1657:S
1654:(
1625:)
1622:S
1616:,
1613:S
1610:(
1602:0
1577:)
1574:S
1571:(
1563:+
1538:)
1535:S
1529:,
1526:S
1523:(
1515:+
1490:S
1464:S
1436:)
1432:Z
1428:(
1420:2
1393:2
1388:T
1366:)
1362:Z
1358:(
1350:2
1325:)
1320:2
1315:T
1310:(
1281:,
1196:2
1191:T
1163:2
1158:Z
1153:+
1150:x
1147:A
1139:2
1134:Z
1129:+
1126:x
1104:2
1099:T
1077:)
1073:Z
1069:(
1061:2
1050:A
1030:)
1025:2
1020:T
1015:(
1003:)
999:Z
995:(
987:2
979::
956:)
952:Z
948:(
940:2
910:2
905:Z
899:/
893:2
888:R
883:=
878:2
873:T
844:Z
840:2
836:/
831:Z
810:0
807:=
804:z
784:S
764:S
742:3
737:R
715:S
687:)
684:S
681:(
673:+
662:)
659:S
656:(
648:+
616:)
613:S
610:(
575:.
563:)
560:S
557:(
549:0
540:/
536:)
533:S
530:(
522:+
514:=
511:)
508:S
505:(
476:S
456:S
436:)
433:S
430:(
422:0
393:)
390:S
387:(
379:+
350:)
346:)
343:)
340:x
337:(
334:g
331:,
328:)
325:x
322:(
319:f
316:(
313:d
309:(
303:S
297:x
289:=
286:)
283:g
280:,
277:f
274:(
248:S
228:d
208:S
188:)
185:S
182:(
174:+
137:S
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