A reconstruction theorem for smooth foliated manifolds

We show that smooth foliated manifolds are determined by their automorphism groups in the following sense. Theorem A Let 1 ≤ k ≤ ∞ and X₁, X₂ 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(X₁) and H^k(X₂). Then there is a homeomorphism τ between X₁ and X₂ such that: (1) φ (g) = for every H^k (X) and (2) τ takes every leaf of X₁ to a leaf of X₂. 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 X₁, X₂ be second countable C^∞ foliated manifolds. Suppose that is an isomorphism between H^∞(X₁) and H^∞(X₂). Then there is a C^∞ homeomorphism τ between X₁ and X₂ such that: (1) φ (g) = for every g H^k (X) and (2) τ takes every leaf of X₁ to a leaf of X₂.