A reconstruction theorem for smooth foliated manifolds

作者：Rubin Matatyahu^*

来源：Geometriae Dedicata, 2011, 150(1): 355-375.

DOI：10.1007/s10711-010-9509-4

• Homeomorphism groups
• Diffeomorphism groups
• Foliations
• Reconstruction

We show that smooth foliated manifolds are determined by their automorphism groups in the following sense. Theorem A Let 1 <= k <= infinity 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 phi is an isomorphism between H(k) (X(1)) and H(k) (X(2)). Then there is a homeomorphism tau between X(1) and X(2) such that: (1) phi(g) = tau circle g circle tau(-1) for every g is an element of H(k) (X) and (2) tau 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(infinity) foliated manifolds. Suppose that phi is an isomorphism between H(infinity)(X(1)) and H(infinity)(X(2)). Then there is a C(infinity) homeomorphism tau between X(1) and X(2) such that: (1) phi(g) = tau circle g circle tau(-1) for every g is an element of H(infinity)(X) and (2) tau takes every leaf of X(1) to a leaf of X(2).

• 出版日期2011-2