On the group of all homeomorphisms of a manifold

Abstract

Introduction.It is shown in this paper that the identity component of the group G(M) of all homeomorphisms of a closed manifold of dimension ^3 is(1) simple in the algebraic sense;(2) equal to the group of deformations of M (i.e., the group of homeomorphisms of M isotopic to the identity homeomorphism) ;(3) open in G(M).The proofs for dimensions 2 and 3 are given separately; the proof for dimension 1, the circle, is not given here, but can be modeled after the proof for dimension 2.Theorem 9 of this paper (the main tool in obtaining the above results in the case ra = 3) is very similar toa recent independent result of Kister appearing elsewhere in this journal (see also [l]).It is possible to deduce from the above results that the space of homeomorphisms of M is locally arcwise connected.However, since this is a quite special case of more general theorems of Hamstrom and Dyer [2 ] and Hamstrom [3], we omit the proof.It is also shown in this paper that if ra^3, then the group of deformations of the sphere 5" is of index 2 in G(Sn).Furthermore, we find a characterization of homeomorphisms of degree 1 on a closed orientable manifold M of dimension ^3 which admits a homeomorphism of degree -1.Finally, we conclude as a corollary that if ra^3, then two homeomorphisms of 5B are isotopic if and only if they are homotopic.These two sets of results are related by the following fact.In showing the results of the first paragraph, we show that if M is a closed manifold of dimension ^3, then a homeomorphism A of M is a deformation if and only if A = Ai ■ ■ ■ hk where A,-is a homeomorphism of M which is the identity outside some internal closed «-cell £,• in M,n = dim M (a closed cell is internal if it lies in an open cell ; actually, we do not use all of the internal closed ra-cells in M.) In showing the results of the second paragraph, we show that if M is a closed orientable manifold of dimension ^ 3 which admits a homeomorphism of degree -1, then a homeomorphism A of M is of degree 1 if and only

Locations

• Transactions of the American Mathematical Society - View - PDF