In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the subfield of
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originate ...
, the mapping class group is an important algebraic invariant of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
. 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
s from the space into itself, that is,
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
maps with continuous
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
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s of this new function space will be made up of sets of functions that map
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
subsets ''K'' into open subsets ''U'' as ''K'' and ''U'' range throughout our original topological space, completed with their finite
intersections
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
(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 homeomorphisms.
Definition
The term mapping class group has a flexible usage. Most often it is used in the context of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M''. The mapping class group of ''M'' is interpreted as the group of
isotopy classes of
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of ''M''. So if ''M'' is a
topological manifold, the mapping class group is the group of isotopy classes of
homeomorphisms
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
of ''M''. If ''M'' is a
smooth manifold, the mapping class group is the group of isotopy classes of
diffeomorphisms of ''M''. Whenever the group of automorphisms of an object ''X'' has a natural
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the mapping class group of ''X'' is defined as
, where
is the
path-component of the identity in
. (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). 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
, where one substitutes for Aut the appropriate group for the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
to which ''X'' belongs. Here
denotes the 0-th
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of a space.
So in general, there is a
short exact sequence of groups:
:
Frequently this sequence is not
split.
If working in the
homotopy category, the mapping class group of ''X'' is the group of
homotopy classes
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
of
homotopy equivalences of ''X''.
There are many
subgroups of mapping class groups that are frequently studied. If ''M'' is an oriented manifold,
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 groups of ''M'' is called the Torelli group of ''M''.
Examples
Sphere
In any category (smooth, PL, topological, homotopy)
:
corresponding to maps of
degree ±1.
Torus
In the
homotopy category
:
This is because the
n-dimensional torus is an
Eilenberg–MacLane space.
For other categories if
, one has the following split-exact sequences:
In the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again cont ...
:
In the
PL-category
:
(⊕ representing
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
).
In the
smooth category
:
where
are the Kervaire–Milnor finite abelian groups of
homotopy spheres and
is the group of order 2.
Surfaces
The mapping class groups of
surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of
above), since they act on
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...
and the quotient is the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to
hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s and to higher rank linear groups. They have many applications in
Thurston's theory of geometric
three-manifolds (for example, to
surface bundles). The elements of this group have also been studied by themselves: an important result is the
Nielsen–Thurston classification
In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by .
Given a homeomorphism ''f'' : ''S'' → ''S'', ther ...
theorem, and a generating family for the group is given by
Dehn twists 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,; in fact one can realize any finite group as the group of isometries of some compact
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
(which immediately implies that it injects in the mapping class group of the underlying topological surface).
Non-orientable surfaces
Some
non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
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
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...
''K'' is:
:
The four elements are the identity, a
Dehn twist on a two-sided curve which does not bound a
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
, 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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
three non-orientable surface ''N''
3 (the connected sum of three projective planes) has:
:
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
is ''a torus with a disk removed''. As an unoriented surface, its mapping class group is
. (Lemma 2.1).
3-Manifolds
Mapping class groups of
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
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.
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
3 is a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
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
dihedral or
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
, 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
2.
Torelli group
Notice that there is an induced action of the mapping class group on the
homology (and
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
) 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'', named after the
Torelli theorem
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) ''C'' is determined by ...
.
In the case of orientable surfaces, this is the action on first cohomology ''H''
1(Σ) ≅ Z
2''g''. Orientation-preserving maps are precisely those that act trivially on top cohomology ''H''
2(Σ) ≅ Z. ''H''
1(Σ) has a
symplectic structure, coming from the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
; 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 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
One can embed the surface
of genus ''g'' and 1 boundary component into
by attaching an additional hole on the end (i.e., gluing together
and
), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
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
Ib Madsen and
Michael Weiss, proving Mumford's conjecture.
See also
*
Braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
s, the mapping class groups of punctured discs
*
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s
*
Homeotopy groups
*
Lantern relation
In geometric topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The most general version of the relation involves seven Dehn twists. The relati ...
References
*
*
*
*
*
*
*
*
Stable mapping class group
*
External links
Madsen-Weiss MCG Seminar many references
{{DEFAULTSORT:Mapping Class Group
Geometric topology
Homeomorphisms