 Mapping class 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 discrete group of 'symmetries' of the space.
Contents
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 homeomorphisms 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. 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 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.
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 isotopyclasses of automorphisms of M. So if M is a topological manifold, the mapping class group is the group of isotopyclasses of homeomorphisms of M. If M is a smooth manifold, the mapping class group is the group of isotopyclasses of diffeomorphisms of M. Whenever the group of automorphisms of an object X has a natural topology, the mapping class group of X is defined as Aut(X) / Aut_{0}(X) where Aut_{0}(X) is the pathcomponent of the identity in Aut(X). For topological spaces, this is usually the compactopen topology. In the lowdimensional topology literature, the mapping class group of X is usually denoted MCG(X), although it is also frequently denoted π_{0}Aut(X) where one substitutes Aut for the appropriate notion of automorphism for whichever category X is an object of. π_{0} denotes the 0th homotopy group of a space.
So in general, there is a shortexact sequence of groups:
Frequently this sequence is not split.^{[1]}
If working in the homotopy category, the mappingclass group of X is the group of homotopyclasses of homotopyequivalences of X.
There are many subgroups of mapping class groups that are frequently studied. If M is an oriented manifold, Aut(M) would be the orientationpreserving 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 orientationreversing automorphism. Similarly, the subgroup that acts trivially on the homology of M is called the Torelli group of M, one could think of this as the mapping class group of a homologicallymarked surface.
Examples
Sphere
In any category (smooth, PL, topological, homotopy) ^{[2]}
corresponding to maps of degree .
Torus
In the homotopy category
This is because T^{n} = (S^{1})^{n} is an EilenbergMacLane space.
Provided that ,^{[3]} there are splitexact sequences:
In the category of topological spaces
In the PLcategory
In the smooth category
where Γ_{i} are KervaireMilnor finite abelian groups of homotopy spheres, is the group of order 2.
Surfaces
The mapping class groups of surfaces have been heavily studied, and are called Teichmüller modular groups. (Note the special case of MCG(T^{2}) above.) This is perhaps due to their strange similarity to higher rank linear groups as well as many applications, via surface bundles, in Thurston's theory of geometric threemanifolds. For more information on this topic see the Nielsen–Thurston classification theorem and the article on Dehn twists. Every finite group is a subgroup of the mapping class group of a closed, orientable surface,^{[4]} moreover one can realize any finite group as the group of isometries of some compact Riemann surface.
Nonorientable surfaces
Some nonorientable surfaces have mapping class groups with simple presentations. For example, every homeomorphism of the real projective plane is isotopic to the identity:
The mapping class group of the Klein bottle K is:
The four elements are the identity, a Dehn twist on the twosided curve which does not bound a Möbius strip, the yhomeomorphism of Lickorish, and the product of the twist and the yhomeomorphism. 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 three nonorientable surface N_{3} has:
This is because the surface has a unique onesided curve that, when cut open, yields a onceholed torus. This is discussed in a paper of Martin Scharlemann.
3Manifolds
Mapping class groups of 3manifolds have received considerable study as well, and are closely related to mapping class groups of 2manifolds. For example, any finite group can be realized as the mapping class group (and also the isometry group) of a compact hyperbolic 3manifold.^{[5]}
Mappingclass groups of pairs
Given a pair of spaces (X,A) the mapping class group of the pair is the isotopyclasses of automorphisms of the pair, where an automorphism of (X,A) is defined as an automorphism of X that preserves A, i.e. is invertible and f(A) = A.
Symmetry group of knot and links
If is a knot or a link, the symmetry group of the knot (resp. link) is defined to be the mapping class group of the pair (S^{3},K). The symmetry group of a hyperbolic knot is known to be dihedra or 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 .
Torelli group
Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo_{0} 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.
In the case of orientable surfaces, this is the action on first cohomology . Orientationpreserving maps are precisely those that act trivially on top cohomology . has a 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:
One can extend this to
The symplectic group is wellunderstood. 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
One can embed the surface Σ_{g,1} of genus g and 1 boundary component into Σ_{g + 1,1} by attaching an additional hole on the end (i.e, gluing together Σ_{g,1} and Σ_{1,2}), 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 conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Madsen and Weiss, proving Mumford's conjecture.
See also
 Braid groups, the mapping class groups of punctured discs
 Homotopy groups
 Homeotopy groups
 Lantern relation
References
 ^ S.Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987)
 ^ MR0212840 (35 #3705) Earle, C. J.; Eells, J. The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc. 73 1967 557559.
 ^ MR0520490 (80f:57014) Hatcher, A. E. Concordance spaces, higher simplehomotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 321, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. (Reviewer: Gerald A. Anderson) 57R52
 ^ L.Greenberg, Maximal groups and signatures, Ann. Math. Studies 79 (1974) 207226
 ^ S.Kojima, Topology and its Applications Volume 29, Issue 3, August 1988, Pages 297307
 Braids, Links, and Mapping Class Groups by Joan Birman.
 Automorphisms of surfaces after Nielsen and Thurston by Andrew Casson and Steve Bleiler.
 "Mapping Class Groups" by Nikolai V. Ivanov in the Handbook of Geometric Topology.
 A Primer on Mapping Class Groups by Benson Farb and Dan Margalit
 Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 9783037190296, MR2284826
 Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, doi:10.4171/055, ISBN 9783037190555, MR2524085
Stable mapping class group
 The stable moduli space of Riemann surfaces: Mumford's conjecture, by Ib Madsen and Michael S. Weiss, 2002
 Published as: The stable moduli space of Riemann surfaces: Mumford's conjecture, by Ib Madsen and Michael S. Weiss, 2007, Annals of Mathematics
External links
 MadsenWeiss MCG Seminar; many references
Categories: Geometric topology
 Homeomorphisms
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