Homeomorphism group

The group $ \mathfrak M ( X) $ of homeomorphic mappings of a topological space $ X $ onto itself (cf. also Homeomorphism). If $ X $ is a compact manifold, then the algebraic properties of $ \mathfrak M ( X) $, especially the structure of its normal subgroups, determine $ X $ up to a homeomorphism [1]. In particular, for $ n \neq 4 $ it is known that $ \mathfrak M ( S ^ {n} ) $ 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 [2]. For a manifold $ M $ the minimal normal subgroup in $ \mathfrak M ( M) $ is the subgroup generated by the homeomorphisms that are the identity outside domains in $ M $.

The group $ \mathfrak M ( X) $ may be topologized in different manners (cf. Space of mappings, topological). Of fundamental importance are the compact-open topology and (if $ X $ is metrizable) the fine $ C ^ {0} $-topology, in which neighbourhoods of the identity $ O _ {f} $ are defined by strictly-positive functions $ f: X \rightarrow ( 0, \infty ) $, and $ h \in \mathfrak M ( X) $ forms part of $ O _ {f} $ if $ \rho ( hx, x) < f( x) $ for all $ x $, where $ \rho $ is the metric in $ X $. However, $ \mathfrak M ( X) $ need not necessarily be a topological group in these topologies, since the mapping $ h \rightarrow h ^ {-} 1 $ is not always continuous, and even if it is, $ \mathfrak M ( X) $ need not be a topological group of transformations (cf. also Transformation group), i.e. the mapping $ ( h, x) \rightarrow hx $ can be discontinuous [3]. However, if $ X $ is a manifold, then $ \mathfrak M ( X) $ is a topological group of transformations in both these topologies. The study of the topological properties of $ \mathfrak M ( X) $ is of interest, in the first place, for a homogeneous space $ X $, i.e. such that the action of $ \mathfrak M ( X) $ on $ X $ is transitive. However, the available studies are far from complete even for simple manifolds. Thus, it is not known (1977) if $ \mathfrak M ( X) $ is an infinite-dimensional manifold, even though it is (for a metrizable manifold) locally contractible in the fine $ C ^ {0} $ topology [4]. 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 $ \Gamma ( X) $ of $ \mathfrak M ( X) $ by the component of the identity $ \mathfrak M _ {0} ( X) $ is called the homeotopy group of $ X $. Generally speaking, $ \mathfrak M _ {0} ( X) $ 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 $ S ^ {3} $, $ S ^ {2} \times S ^ {1} $, etc.). The homotopy properties of $ \mathfrak M _ {0} $ 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 $ \mathfrak M ( \mathbf R ^ {n} ) $, 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 ( $ ss $-groups) $ \mathrm{Top} _ {n} $, in which the $ k $-dimensional simplexes are the fibred homeomorphisms of $ \Delta ^ {k} \times \mathbf R ^ {n} $ that are stationary on the zero section (here $ \Delta ^ {k} $ is the standard $ k $-simplex). The boundary homeomorphisms and degenerations are defined with the aid of the standard mappings $ \Delta ^ {l} \times \mathbf R ^ {n} \rightarrow \Delta ^ {l} \times \mathbf R ^ {n} $. The $ ss $-monoids $ G _ {n} $ of the $ ss $-groups $ \mathrm{PL} _ {n} $, $ \mathrm{Diff} _ {n} $, $ O _ {n} $ (homotopy equivalences of $ S ^ {n- 1} $ of piecewise-linear, smooth and orthogonal mappings of $ \mathbf R ^ {n} $) are defined in the same manner, and

$$ G _ {n} \supset \mathrm{Top} _ {n} \supset \ \mathrm{PL} _ {n} \supset \mathrm{Diff} _ {n} \supset O _ {n} , $$

and the quotients $ G _ {n} / \mathrm{Top} _ {n} $, etc., have the natural structures of $ ss $-complexes, which makes it possible to study homotopy properties of these imbeddings.

The study of the various subgroups of $ \mathfrak M ( M) $ for manifolds $ M $ 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.

References

[1]	J.V. Whittaker, "On isomorphic groups and homogeneous spaces" Ann. of Math. (2) , 78 : 1 (1963) pp. 74–91
[2]	R.D. Anderson, "The algebraic simplicity of certain groups of homeomorphisms" Amer. J. Math. , 80 (1958) pp. 955–963
[3]	R.F. Arens, "Topologies for homeomorphism groups" Amer. J. Math. , 68 : 4 (1946) pp. 593–610
[4]	A.V. Chernavskii, "Local contractibility of the group of homeomorphisms of a manifold" Math. USSR-Sb. , 8 : 3 (1969) pp. 287–333 Mat. Sb. , 79 (121) : 3 (1969) pp. 307–356