It forms a group under functional composition.
The term mapping class group has a flexible usage.
(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]).
, where one substitutes for Aut the appropriate group for the category to which X belongs.
There are many subgroups of mapping class groups that are frequently studied.
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. In any category (smooth, PL, topological, homotopy)[2] corresponding to maps of degree ±1.
,[3] one has the following split-exact sequences: In the category of topological spaces In the PL-category (⊕ representing direct sum).
are the Kervaire–Milnor finite abelian groups of homotopy spheres and
The mapping class groups of surfaces have been heavily studied, and are sometimes called Teichmüller modular groups (note the special case of
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 theorem, and a generating family for the group is given by Dehn twists which are in a sense the "simplest" mapping classes.
Some non-orientable surfaces have mapping class groups with simple presentations.
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 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.
As an unoriented surface, its mapping class group is
Mapping class groups of 3-manifolds have received considerable study as well, and are closely related to mapping class groups of 2-manifolds.
link) is defined to be the mapping class group of the pair (S3, K).
The symmetry group of a torus knot is known to be of order two Z2.
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 Homeo0 acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy).
In the case of orientable surfaces, this is the action on first cohomology H1(Σ) ≅ Z2g.
Orientation-preserving maps are precisely those that act trivially on top cohomology H2(Σ) ≅ Z. H1(Σ) 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 well understood.
by attaching an additional hole on the end (i.e., gluing together
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 Ib Madsen and Michael Weiss, proving Mumford's conjecture.