caa a22 4500 445884517 CHVBK 20180317145555.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10711-010-9509-4 doi (NATIONALLICENCE)springer-10.1007/s10711-010-9509-4 Rubin Matatyahu Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel aut A reconstruction theorem for smooth foliated manifolds [Elektronische Daten] [Matatyahu Rubin] 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 Ck foliated manifolds. Denote by H k(X i) the groups of Ck 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. Springer Science+Business Media B.V., 2010 Homeomorphism groups nationallicence Diffeomorphism groups nationallicence Foliations nationallicence Reconstruction nationallicence Geometriae Dedicata Springer Netherlands 150/1(2011-02-01), 355-375 0046-5755 150:1<355 2011 150 10711 https://doi.org/10.1007/s10711-010-9509-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10711-010-9509-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Rubin Matatyahu Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel aut NATIONALLICENCE 773 0- Geometriae Dedicata Springer Netherlands 150/1(2011-02-01), 355-375 0046-5755 150:1<355 2011 150 10711 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer