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| {{mergefrom|Teichmüller modular group|discuss=Talk:Mapping class group#Merger proposal|date=July 2011}}
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| In [[mathematics]], in the sub-field of [[geometric topology]], the '''mapping class group'''
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| is an important algebraic invariant of a [[topological space]]. Briefly, the mapping class group is a [[discrete group]] of 'symmetries' of the space.
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| Consider a topological space, that is, a space with some notion of closeness between points in the space. We can consider the set of [[homeomorphism]]s from the space into itself, that is, continuous functions with continuous inverses: functions which stretch and deform the space continuously without puncturing or breaking the space. This set of homeomorphisms can be thought of as a space itself. It can be seen fairly easily that this space forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The open sets of this new function space will be made up of sets of functions that map compact subsets K into open subsets U as K and U range throughout our original topological space, completed with their finite intersections (which must be open by definition of topology) and arbitrary unions (again which must be open). This gives a notion of continuity on the space of functions, so that we can consider continuous deformation of the homeomorphisms themselves: called [[Homotopy|homotopies]]. We define the mapping class group by taking homotopy classes of homeomorphisms, and inducing the group structure from the functional composition group structure already present on the space of automorphisms.
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| == Definition ==
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| The term '''mapping class group''' has a flexible usage. Most often it is used in the context of a [[manifold]] ''M''. The mapping class group of ''M'' is interpreted as the group of [[ambient isotopy|isotopy-classes]] of [[automorphism]]s of ''M''. So if ''M'' is a [[topological manifold]], the mapping class group is the group of isotopy-classes of [[Homeomorphism group|homeomorphisms]] of ''M''. If ''M'' is a [[smooth manifold]], the mapping class group is the group of isotopy-classes of [[diffeomorphism]]s of ''M''. Whenever the group of automorphisms of an object ''X'' has a natural [[topological space|topology]], the mapping class group of ''X'' is defined as Aut(''X'')/Aut<sub>0</sub>(''X'') where Aut<sub>0</sub>(''X'') is the [[connected space|path-component]] of the identity in Aut(''X''). For topological spaces, this is usually the [[compact-open topology]]. In the [[low-dimensional topology]] literature, the mapping class group of ''X'' is usually denoted MCG(''X''), although it is also frequently denoted π<sub>0</sub>(Aut(''X'')) where one substitutes for ''Aut'' the appropriate group for the [[category theory|category]] ''X'' is an object of. π<sub>0</sub> denotes the ''0''-th [[homotopy group]] of a space.
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| So in general, there is a short-exact sequence of groups:
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| :<math>1 \rightarrow {\rm Aut}_0(X) \rightarrow {\rm Aut}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.</math>
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| Frequently this sequence is not [[split exact sequence|split]].<ref>S.Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987)</ref>
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| If working in the [[homotopy category]], the mapping-class group of ''X'' is the group of [[homotopy|homotopy-classes]] of [[homotopy|homotopy-equivalences]] of ''X''.
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| There are many subgroups of mapping class groups that are frequently studied. If ''M'' is an oriented manifold, Aut(''M'') would be the orientation-preserving automorphisms of ''M'' and so the mapping class group of ''M'' (as an oriented manifold) would be index two in the mapping class group of ''M'' (as an unoriented manifold) provided ''M'' admits an orientation-reversing automorphism. Similarly, the subgroup that acts trivially on the [[Homology (mathematics)|homology]] of ''M'' is called the '''Torelli group''' of ''M'', one could think of this as the mapping class group of a homologically-marked surface.
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| ==Examples==
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| === Sphere ===
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| In any category (smooth, PL, topological, homotopy) <ref>MR0212840 (35 #3705)
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| Earle, C. J.; Eells, J.
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| The diffeomorphism group of a compact Riemann surface.
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| Bull. Amer. Math. Soc. 73 1967 557--559.</ref>
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| :<math>\scriptstyle {\rm MCG}(\mathbf{S}^2) \simeq {\mathbf Z}/2{\mathbf Z}, </math> | |
| corresponding to maps of [[degree of a continuous mapping|degree]] ±1.
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| === Torus ===
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| In the [[homotopy category]]
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| :<math> {\rm MCG}(\mathbf{T}^n) \simeq {\rm SL}(n, {\mathbf Z}). </math>
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| This is because '''T'''<sup>''n''</sup> = ('''S'''<sup>1</sup>)<sup>''n''</sup> is an [[Eilenberg-MacLane space]].
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| For other categories if ''n'' ≥ 5,<ref>MR0520490 (80f:57014) Hatcher, A. E. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 3--21, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. (Reviewer: Gerald A. Anderson) 57R52</ref> one has the following split-exact sequences:
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| In the [[category of topological spaces]]
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| :<math>0\to \mathbf Z_2^\infty\to MCG(\mathbf{T}^n) \to GL(n,\mathbf Z)\to 0</math>
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| In the [[Piecewise linear manifold|PL-category]]
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| :<math>0\to \mathbf Z_2^\infty\oplus\binom n2\mathbf Z_2\to MCG (\mathbf{T}^n)\to GL(n,\mathbf Z)\to 0</math>
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| (⊕ representing [[direct sum]]).
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| In the [[Smooth manifold|smooth category]]
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| :<math>0\to \mathbf Z_2^\infty\oplus\binom n2\mathbf Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to MCG(\mathbf{T}^n)\to GL(n,\mathbf Z)\to 0</math>
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| where Γ<sub>''i''</sub> are Kervaire-Milnor finite abelian groups of [[homotopy sphere]]s and '''Z'''<sub>2</sub> is the group of order 2.
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| === Surfaces ===
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| The mapping class groups of [[surface]]s have been heavily studied, and are called [[Teichmüller modular group]]s. (Note the special case of MCG('''T'''<sup>2</sup>) above.) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via [[surface bundle]]s, in [[William Thurston|Thurston]]'s theory of geometric [[three-manifold]]s. For more information on this topic see the [[Nielsen–Thurston classification]] theorem and the article on [[Dehn twist]]s. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,<ref>L.Greenberg, Maximal groups and signatures, Ann. Math. Studies 79 (1974) 207--226</ref> moreover one can realize any finite group as the group of isometries of some compact [[Riemann surface]].
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| ==== Non-orientable surfaces ====
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| Some [[orientability|non-orientable]] surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the [[real projective plane]] '''P'''<sup>2</sup>('''R''') is isotopic to the identity:
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| :<math> {\rm MCG}(\mathbf{P}^2(\mathbf{R})) = 1. </math>
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| The mapping class group of the [[Klein bottle]] ''K'' is:
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| :<math> {\rm MCG}(K)= \mathbf{Z}_2 \oplus \mathbf{Z}_2.</math>
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| The four elements are the identity, a [[Dehn twist]] on the two-sided curve which does not bound a [[Möbius strip]], the [[y-homeomorphism]] of [[Lickorish]], and the product of the twist and the y-homeomorphism. It is a nice exercise to show that the square of the Dehn twist is isotopic to the identity.
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| We also remark that the closed [[genus]] three non-orientable surface ''N''<sub>3</sub> has:
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| :<math>{\rm MCG(N_3)} = {\rm GL}(2, {\mathbf Z}). </math>
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| This is because the surface has a unique one-sided curve that, when cut open, yields a once-holed torus. This is discussed in a paper of [[Martin Scharlemann]].
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| === 3-Manifolds ===
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| Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3-manifold.<ref>S.Kojima, Topology and its Applications Volume 29, Issue 3, August 1988, Pages 297-307</ref>
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| == Mapping-class groups of pairs ==
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| Given a [[pair of spaces]] ''(X,A)'' the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of ''(X,A)'' is defined as an automorphism of ''X'' that preserves ''A'', i.e. ''f'': ''X'' → ''X'' is invertible and ''f(A)'' = ''A''.
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| === Symmetry group of knot and links ===
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| If ''K'' ⊂ '''S'''<sup>3</sup> is a [[knot (mathematics)|knot]] or a [[link (knot theory)|link]], the '''symmetry group of the knot (resp. link)''' is defined to be the mapping class group of the pair ('''S'''<sup>3</sup>, ''K''). The symmetry group of a [[hyperbolic knot]] is known to be [[dihedral group|dihedra]] or [[cyclic group|cyclic]], moreover every dihedral and cyclic group can be realized as symmetry groups of knots. The symmetry group of a [[torus knot]] is known to be of order two '''Z'''<sub>2</sub>.
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| == Torelli group ==
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| Notice that there is an induced action of the mapping class group on the [[homology (mathematics)|homology]] (and [[cohomology]]) of the space ''X''. This is because (co)homology is functorial and Homeo<sub>0</sub> acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy). The kernel of this action is the ''Torelli group''.
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| In the case of orientable surfaces, this is the action on first cohomology ''H''<sup>1</sup>(Σ) ≅ '''Z'''<sup>2''g''</sup>. Orientation-preserving maps are precisely those that act trivially on top cohomology ''H''<sup>2</sup>(Σ) ≅ '''Z'''. ''H''<sup>1</sup>(Σ) has a [[Symplectic geometry|symplectic]] structure, coming from the [[cup product]]; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the [[short exact sequence]]:
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| :<math>1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}(\Sigma) \to \mbox{Sp}(H^1(\Sigma)) \cong \mbox{Sp}_{2g}(\mathbf{Z}) \to 1</math> | |
| One can extend this to
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| :<math>1 \to \mbox{Tor}(\Sigma) \to \mbox{MCG}^*(\Sigma) \to \mbox{Sp}^{\pm}(H^1(\Sigma)) \cong \mbox{Sp}^{\pm}_{2g}(\mathbf{Z}) \to 1</math>
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| The [[symplectic group]] is well-understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.
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| Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.
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| == Stable mapping class group ==
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| {{Expand section|date=December 2009}}
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| One can embed the surface <math>\Sigma_{g,1}</math> of genus ''g'' and 1 boundary component into <math>\Sigma_{g+1,1}</math> by attaching an additional hole on the end (i.e., gluing together <math>\Sigma_{g,1}</math> and <math>\Sigma_{1,2}</math>), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the [[direct limit]] of these groups and inclusions yields the '''stable mapping class group,''' whose rational cohomology ring was conjectured by [[David Mumford]] (one of conjectures called the [[Mumford conjecture]]s). The integral (not just rational) cohomology ring was computed in 2002 by [[Ib Madsen|Madsen]] and Weiss, proving Mumford's conjecture.
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| ==See also==
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| *[[Braid group]]s, the mapping class groups of punctured discs
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| *[[Homotopy group]]s
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| *[[Homeotopy]] groups
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| *[[Lantern relation]]
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| ==References==
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| {{reflist}}
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| *''Braids, Links, and Mapping Class Groups'' by [[Joan Birman]].
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| *''Automorphisms of surfaces after Nielsen and Thurston'' by [[Andrew Casson]] and [[Steve Bleiler]].
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| * "Mapping Class Groups" by [[Nikolai V. Ivanov]] in the ''Handbook of Geometric Topology''.
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| * [http://www.math.utah.edu/~margalit/primer/ ''A Primer on Mapping Class Groups''] by [[Benson Farb]] and Dan Margalit
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. I | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-029-6 | doi=10.4171/029 | mr=2284826 | year=2007 | volume=11}}
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. II | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-055-5 | doi=10.4171/055 | mr=2524085 | year=2009 | volume=13}}
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| *{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. III | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-103-3 | doi=10.4171/103 | mr=2961353 | year=2012 | volume=17}}
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| === Stable mapping class group ===
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| * [http://arxiv.org/abs/math.AT/0212321 The stable moduli space of Riemann surfaces: Mumford's conjecture], by [[Ib Madsen]] and Michael S. Weiss, 2002
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| *:Published as: [http://annals.princeton.edu/annals/2007/165-3/p04.xhtml The stable moduli space of Riemann surfaces: Mumford's conjecture], by Ib Madsen and Michael S. Weiss, 2007, Annals of Mathematics
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| == External links ==
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| * [http://math.ucsd.edu/~justin/madsenweissS06.html Madsen-Weiss MCG Seminar]; many references
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| {{DEFAULTSORT:Mapping Class Group}}
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| [[Category:Geometric topology]]
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| [[Category:Homeomorphisms]]
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