# Homeomorphism group

The group of homeomorphic mappings of a topological space onto itself (cf. also Homeomorphism). If is a compact manifold, then the algebraic properties of , especially the structure of its normal subgroups, determine up to a homeomorphism . In particular, for it is known that is a simple group. This is also true of the Cantor set, the Menger curve, the Sierpiński curve, and the sets of rational or irrational points on the real line . For a manifold the minimal normal subgroup in is the subgroup generated by the homeomorphisms that are the identity outside domains in .

The group may be topologized in different manners (cf. Space of mappings, topological). Of fundamental importance are the compact-open topology and (if is metrizable) the fine -topology, in which neighbourhoods of the identity are defined by strictly-positive functions , and forms part of if for all , where is the metric in . However, need not necessarily be a topological group in these topologies, since the mapping is not always continuous, and even if it is, need not be a topological group of transformations (cf. also Transformation group), i.e. the mapping can be discontinuous . However, if is a manifold, then is a topological group of transformations in both these topologies. The study of the topological properties of is of interest, in the first place, for a homogeneous space , i.e. such that the action of on is transitive. However, the available studies are far from complete even for simple manifolds. Thus, it is not known (1977) if is an infinite-dimensional manifold, even though it is (for a metrizable manifold) locally contractible in the fine topology . In particular, two sufficiently-close homeomorphisms can be connected by an isotopy (cf. Isotopy (in topology)). For an open manifold which is the interior of a compact manifold this is also true in the compact-open topology.

The quotient group of by the component of the identity is called the homeotopy group of . Generally speaking, is not identical with the group of homeomorphisms that are homotopic to the identity, but they coincide for two-dimensional and some three-dimensional manifolds (e.g. for , , etc.). The homotopy properties of have been studied for two-dimensional manifolds; this proved useful in establishing homological properties of braid groups (cf. Braid theory).

Of special importance in the theory of manifolds is the study of certain subgroups of the group , e.g. of the subgroup of diffeomorphisms. This study is made more difficult by the fact that the subgroups are not closed, while the topology of quotient spaces is unsatisfactory. For this reason one considers the semi-simplicial groups ( -groups) , in which the -dimensional simplexes are the fibred homeomorphisms of that are stationary on the zero section (here is the standard -simplex). The boundary homeomorphisms and degenerations are defined with the aid of the standard mappings . The -monoids of the -groups , , (homotopy equivalences of of piecewise-linear, smooth and orthogonal mappings of ) are defined in the same manner, and and the quotients , etc., have the natural structures of -complexes, which makes it possible to study homotopy properties of these imbeddings.

The study of the various subgroups of for manifolds forms the subject of several disciplines. In particular, the study of homeomorphism groups which preserve certain structures belongs to the corresponding branches of mathematics. Of considerable interest are algebraic problems connected with automorphism groups of trees and other graphs.