Топологиялық кеңістіктердің автоморфизм тобының изотопия кластарының тобы

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.

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 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 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 homeomorphisms.

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 and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic groups 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 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 (which immediately implies that it injects in the mapping class group of the underlying topological surface).

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.

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.

If K ⊂ S3 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 (S3, K). The symmetry group of a hyperbolic knot is known to be dihedral 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 Z2.

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 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.