## Abstract

For a group G of homeomorphisms of a regular topological space X and a subset U⊆X, set G:={g∈G|g{up harpoon right}(X\U)=Id}. We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, ∪_{U∈U}GU generates G.Theorem ILet G, H be factorizable groups of homeomorphisms of X and Y respectively, and suppose that G, H do not have fixed points. Let φ be an isomorphism between G and H. Then there is a homeomorphism τ between X and Y such that φ(g)=τ○g○τ^{-1} for every g∈G.Theorem A strengthens known theorems in which the existence of τ is concluded from the assumption of factorizability and some additional assumptions.Theorem IIFor ℓ=1,2 let (X_{ℓ},Φ_{ℓ}) be a countably paracompact foliated (not necessarily smooth) manifold and G_{ℓ} be any group of foliation-preserving homeomorphisms of (X_{ℓ},Φ_{ℓ}) which contains the group H0(Xℓ,Φℓ) of all foliation-preserving homeomorphisms which take every leaf to itself. Let φ be an isomorphism between G_{1} and G_{2}. Then there is a foliation-preserving homeomorphism τ between X_{1} and X_{2} such that φ(g)=τ○g○τ^{-1} for every g∈G_{1}.In both Theorems I and II, τ is unique.

Original language | English |
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Pages (from-to) | 1664-1679 |

Number of pages | 16 |

Journal | Topology and its Applications |

Volume | 157 |

Issue number | 9 |

DOIs | |

State | Published - 1 Jun 2010 |

## Keywords

- Factorizable homeomorphism group
- Foliated manifold
- Foliation-preserving homeomorphism group
- Homeomorphism group
- Locally moving group
- Reconstruction

## ASJC Scopus subject areas

- Geometry and Topology