A reconstruction theorem for smooth foliated manifolds

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 ${\varphi}$ 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) ${\varphi(g) = \tau {\raise1pt\hbox{\scriptsize\kern1.5pt$\circ$\kern1.5pt}} g {\raise1pt\hbox{\scriptsize\kern1.5pt$\circ$\kern1.5pt}} \tau^{-1}}$ for every ${g \in H^k(X)}$ 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 ${\varphi}$ 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) ${\varphi(g) = \tau {\raise1pt\hbox{\scriptsize\kern1.5pt$\circ$\kern1.5pt}} g {\raise1pt\hbox{\scriptsize\kern1.5pt$\circ$\kern1.5pt}} \tau^{-1}}$ for every ${g \in H^{\infty}(X)}$ and (2) τ takes every leaf of X ₁ to a leaf of X ₂.