Editing Mapping class group

{{short description|Group of isotopy classes of a topological automorphism group}}
In [[mathematics]], in the subfield of [[geometric topology]], the '''mapping class group''' is an important algebraic invariant of a [[topological space]].  Briefly, the mapping class group is a certain [[discrete group]] corresponding to symmetries of the space.

== Motivation ==
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_function#Continuous_functions_between_topological_spaces|continuous]] maps with continuous [[Inverse function|inverses]]: functions which stretch and deform the space continuously without breaking or gluing the space. This set of homeomorphisms can be thought of as a space itself. It forms a group under functional composition. We can also define a topology on this new space of homeomorphisms. The [[open set]]s of this new function space will be made up of sets of functions that map [[Compact space|compact]] subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite [[Intersection (set theory)|intersections]] (which must be open by definition of topology) and arbitrary [[Union (set theory)|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 homeomorphisms.

== Definition ==
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 <math>\operatorname{Aut}(X)/\operatorname{Aut}_0(X)</math>, where <math>\operatorname{Aut}_0(X)</math> is the [[connected space|path-component]] of the identity in <math>\operatorname{Aut}(X)</math>. (Notice that in the compact-open topology, path components and isotopy classes coincide, i.e., two maps ''f'' and ''g'' are in the same path-component [[iff]] they are isotopic{{Citation needed|date=October 2021}}).  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 <math>\pi_0(\operatorname{Aut}(X))</math>, where one substitutes for Aut the appropriate group for the [[category theory|category]] to which ''X'' belongs. Here <math>\pi_0</math> denotes the 0-th [[homotopy group]] of a space.

So in general, there is a [[Exact sequence#Short exact sequence|short]] [[exact sequence]] of groups:

:<math>1 \rightarrow \operatorname{Aut}_0(X) \rightarrow \operatorname{Aut}(X) \rightarrow \operatorname{MCG}(X) \rightarrow 1.</math>

Frequently this sequence is not [[split exact sequence|split]].<ref>
{{cite journal | last=Morita | first=Shigeyuki | title=Characteristic classes of surface bundles | journal=[[Inventiones Mathematicae]] | volume=90 | issue=3 | year=1987 | doi=10.1007/bf01389178 | pages=551–577 | bibcode=1987InMat..90..551M | mr=0914849| url=http://projecteuclid.org/euclid.bams/1183552184 }}
</ref>

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''.

There are many [[subgroup]]s of mapping class groups that are frequently studied.  If ''M'' is an oriented manifold, <math>\operatorname{Aut}(M)</math> 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 as the identity on all the [[Homology (mathematics)|homology groups]] of ''M'' is called the '''Torelli group''' of ''M''.

==Examples==

=== Sphere ===
In any category (smooth, PL, topological, homotopy)<ref>{{citation| mr=0212840|last1=Earle|first1= Clifford J.|author1-link=Clifford John Earle Jr.| last2= Eells|first2= James|author2-link=James Eells| 
title=The diffeomorphism group of a compact Riemann surface|
journal=[[Bulletin of the American Mathematical Society]] | volume=73|year=1967|issue=4 |pages=557–559|doi=10.1090/S0002-9904-1967-11746-4|doi-access=free}}</ref>
:<math>\operatorname{MCG}(S^2) \simeq \Z/2\Z,</math>
corresponding to maps of [[Degree of a continuous mapping|degree]]&nbsp;±1.

=== Torus ===
In the [[homotopy category]]
:<math> \operatorname{MCG}(\mathbf{T}^n) \simeq \operatorname{GL}(n,\Z). </math>
This is because the [[Torus#n-dimensional torus|n-dimensional torus]] <math>\mathbf{T}^n = (S^1)^n</math> is an [[Eilenberg–MacLane space]].

For other categories if <math>n\ge  5</math>,<ref>{{cite book |first=A.E. |last=Hatcher |chapter=Concordance spaces, higher simple-homotopy theory, and applications |chapter-url={{GBurl|6hsDCAAAQBAJ|p=3}} |title=Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1 |series=Proceedings of Symposia in Pure Mathematics |publisher= |location= |date=1978 |volume=32 |issue=1 |isbn=978-0-8218-9320-3 |pages=3–21 |doi=10.1090/pspum/032.1/520490 |mr=0520490}}</ref> one has the following split-exact sequences:

In the [[category of topological spaces]]
:<math>0\to \Z_2^\infty\to \operatorname{MCG}(\mathbf{T}^n) \to \operatorname{GL}(n,\Z)\to 0</math>
In the [[Piecewise linear manifold|PL-category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>
(⊕ representing [[direct sum]]).
In the [[Smooth manifold|smooth category]]
:<math>0\to \Z_2^\infty\oplus\binom n2\Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1}\to \operatorname{MCG}(\mathbf{T}^n)\to \operatorname{GL}(n,\Z)\to 0</math>

where <math>\Gamma_i</math> are the Kervaire–Milnor finite abelian groups of [[homotopy sphere]]s and <math>\Z_2</math> is the group of order 2.

=== Surfaces ===
{{Main article | Mapping class group of a surface}}

The mapping class groups of [[Surface (topology)|surface]]s have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of <math>\operatorname{MCG}(\mathbf{T}^2)</math> above), since they act on [[Teichmüller space]] and the quotient is the [[moduli space]] of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to [[hyperbolic group]]s and to higher rank linear groups{{citation needed|date=July 2016}}. They have many applications in [[William Thurston|Thurston]]'s theory of geometric [[three-manifold]]s (for example, to [[surface bundle]]s).  The elements of this group have also been studied by themselves: an important result is the [[Nielsen–Thurston classification]] theorem, and a generating family for the group is given by [[Dehn twist]]s which are in a sense the "simplest" mapping classes. Every finite group is a subgroup of the mapping class group of a closed, orientable surface;<ref>{{cite book |first=Leon |last=Greenberg |chapter=Maximal groups and signatures |chapter-url={{GBurl|EFbQCwAAQBAJ|p=207}} |title=Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland |publisher=Princeton University Press |series=Annals of Mathematics Studies |volume=79 |date=1974 |isbn=978-1-4008-8164-2 |pages=207–226 |mr=0379835}}</ref> in fact one can realize any finite group as the group of isometries of some compact [[Riemann surface]] (which immediately implies that it injects in the mapping class group of the underlying topological surface).

==== Non-orientable surfaces ====
Some [[orientability|non-orientable]] surfaces have mapping class groups with simple presentations.  For example, every homeomorphism of the [[real projective plane]] <math>\mathbf{P}^2(\R)</math> is isotopic to the identity:

:<math> \operatorname{MCG}(\mathbf{P}^2(\R)) = 1. </math>

The mapping class group of the [[Klein bottle]] ''K'' is:

:<math> \operatorname{MCG}(K)= \Z_2 \oplus  \Z_2.</math>

The four elements are the identity, a [[Dehn twist]] on a 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.

We also remark that the closed [[genus (mathematics)|genus]] three non-orientable surface ''N''<sub>3</sub> (the connected sum of three projective planes) has:

:<math> \operatorname{MCG}(N_3) = \operatorname{GL}(2,\Z). </math>

This is because the surface ''N'' has a unique class of one-sided curves such that, when ''N'' is cut open along such a curve ''C'', the resulting surface <math>N\setminus C</math> is ''a torus with a disk removed''. As an unoriented surface, its mapping class group is <math>\operatorname{GL}(2,\Z)</math>. (Lemma 2.1<ref>{{cite journal |first=Martin |last=Scharlemann |title=The complex of curves on nonorientable surfaces |journal=Journal of the London Mathematical Society |volume=s2-25 |issue=1 |pages=171–184 |date=February 1982 |doi=10.1112/jlms/s2-25.1.171 |citeseerx=10.1.1.591.2588}}</ref>).

=== 3-Manifolds ===
Mapping class groups of [[3-manifold]]s 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>{{cite journal |first=S. |last=Kojima |title=Isometry transformations of hyperbolic 3-manifolds |journal=Topology and Its Applications |volume=29 |issue=3 |pages=297–307 |date=August 1988 |doi=10.1016/0166-8641(88)90027-2 |url=|doi-access= }}</ref>

== Mapping class groups of pairs ==
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''.

=== Symmetry group of knot and links ===
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|dihedral]] 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>.

== Torelli group ==
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'', named after the [[Torelli theorem]].

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]]:
:<math>1 \to \operatorname{Tor}(\Sigma) \to \operatorname{MCG}(\Sigma) \to \operatorname{Sp}(H^1(\Sigma)) \cong \operatorname{Sp}_{2g}(\mathbf{Z}) \to 1</math>

One can extend this to
:<math>1 \to \operatorname{Tor}(\Sigma) \to \operatorname{MCG}^*(\Sigma) \to \operatorname{Sp}^{\pm}(H^1(\Sigma)) \cong \operatorname{Sp}^{\pm}_{2g}(\mathbf{Z}) \to 1</math>

The [[symplectic group]] is well understood.  Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group.

Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes.

== Stable mapping class group ==
{{Expand section|date=December 2009}}
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 (disambiguation)|Mumford conjecture]]s). The integral (not just rational) cohomology ring was computed in 2002 by [[Ib Madsen]] and [[Michael Weiss (mathematician)|Michael Weiss]], proving Mumford's conjecture.

==See also==
*[[Braid group]]s, the mapping class groups of punctured discs
*[[Homotopy group]]s
*[[Homeotopy]] groups
*[[Lantern relation]]

==References==
{{reflist}}
{{refbegin}}
* {{cite book|first=Joan|last= Birman|authorlink=Joan Birman|title=Braids, links and mapping class groups|title-link= Braids, Links, and Mapping Class Groups |series=Annals of Mathematical Studies|volume=82|publisher=[[Princeton University Press]]|location=Princeton, N.J. | year=1974|isbn=978-0691081496|mr=0375281}}
*{{cite book |author1-link=Andrew Casson |author2-link=Steve Bleiler |first1=Andrew |last1=Casson |first2=Steve |last2=Bleiler |title=Automorphisms of surfaces after Nielsen and Thurston |publisher=Cambridge University Press |orig-year=1988 |date=2014 |isbn=978-1-299-70610-1 |pages= |url={{GBurl|uHlW4kRUjfEC|pg=PP1}}}}
*{{cite book |author1-link=Nikolai V. Ivanov |first=Nikolai V. |last=Ivanov |chapter=9. Mapping class groups and arithmetic groups |chapter-url={{GBurl|8OYxdADnhZoC|p=618}} |editor= |title=Handbook of Geometric Topology |publisher=Elsevier |date=2001 |isbn=978-0-08-053285-1 |pages=618–624 |url=}}
*{{cite book |author1-link=Benson Farb |first1=Benson |last1=Farb |first2=Dan |last2=Margalit |title=A Primer on Mapping Class Groups |publisher=Princeton University Press |location= |date=2012 |isbn=978-0-691-14794-9 |url={{GBurl|FmRMgAV8JwoC|pg=PR7}}}}
*{{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| url=https://hal.archives-ouvertes.fr/hal-00250897/file/athanase.pdf }}
*{{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| arxiv=math/0511271 | last1=Lawton | first1=Sean | last2=Peterson | first2=Elisha }}
*{{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}}
*{{Citation | editor1-last=Papadopoulos | editor1-first=Athanase | title=Handbook of Teichmüller theory. Vol. IV | publisher=European Mathematical Society (EMS), Zürich | series=IRMA Lectures in Mathematics and Theoretical Physics | isbn=978-3-03719-117-0 | doi= 10.4171/117  | year=2014 | volume=19}}
{{refend}}

=== Stable mapping class group ===
*{{cite journal |author1-link=Ib Madsen |author2-link=Michael Weiss (mathematician) |last1=Madsen |first1=Ib |first2=Michael |last2=Weiss |title=The stable moduli space of Riemann surfaces: Mumford's conjecture |journal=Annals of Mathematics |volume= 165|issue= 3|pages=843–941 |date=2007 |doi=10.4007/annals.2007.165.843 |jstor=20160047 |arxiv=math/0212321 |citeseerx=10.1.1.236.2025|s2cid=119721243 }}

== External links ==
* [https://web.archive.org/web/20090623072924/http://math.ucsd.edu/~justin/madsenweissS06.html Madsen-Weiss MCG Seminar]; many references

{{DEFAULTSORT:Mapping Class Group}}
[[Category:Geometric topology]]
[[Category:Homeomorphisms]]