## Abstract

We show that smooth foliated manifolds are determined by their automorphism groups in the following sense. Theorem A Let 1 ≤ k ≤ ∞ and X_{1}, X_{2} be second countable C^{k} foliated manifolds. Denote by H^{k}(X_{i}) the groups of C^{k} auto-homeomorphisms of X_{i} which take every leaf of X_{i} to a leaf of X_{i}. Suppose that φ is an isomorphism between H^{k}(X_{1}) and H^{k}(X_{2}). Then there is a homeomorphism τ between X_{1} and X_{2} such that: (1) φ (g) = for every H^{k} (X) and (2) τ takes every leaf of X_{1} to a leaf of X_{2}. Theorem 1 combined with a theorem of Rybicki (Soochow J Math 22:525-542, 1996) yields the following corollary. Corollary B For i = 1, 2 let X_{1}, X_{2} be second countable C^{∞} foliated manifolds. Suppose that is an isomorphism between H^{∞}(X_{1}) and H^{∞}(X_{2}). Then there is a C^{∞} homeomorphism τ between X_{1} and X_{2} such that: (1) φ (g) = for every g H^{k} (X) and (2) τ takes every leaf of X_{1} to a leaf of X_{2}.

Original language | English |
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Pages (from-to) | 355-375 |

Number of pages | 21 |

Journal | Geometriae Dedicata |

Volume | 150 |

Issue number | 1 |

DOIs | |

State | Published - 1 Feb 2011 |

## Keywords

- Diffeomorphism groups
- Foliations
- Homeomorphism groups
- Reconstruction

## ASJC Scopus subject areas

- Geometry and Topology