## TSUBOI, Takashi

 Title Professor Field Topology, Dynamical Systems Research interests Dynamical study of the behavior of leaves of foliations, quantitative study of foliations such as their characteristic classes, the relationship between dynamics and characteristic classes, dynamical study of infinite group actions, classifying spaces for diffeomorphism groups and the invariants of transformation groups. Current research   I am studying the structure of foliations on manifold, groups of diffeomorphisms, dynamical systems and related subjects. A foliations is given by a completely integrable plane field on a manifold and it is the family of maximal integral manifolds of it. One of the quantitative theories of foliations is the theory of foliated cobordisms. Two foliations are said cobordant if their disjoint union is the restriction to the boundary of a foliation transverse to the boundary. Let FΩ(3,1) denote the group of cobordism classes of oriented codimension-1 foliations on oriented 3-dimensional manifolds. The Godbillon-Vey invariant is a foliated cobordism invariant for FΩ(3,1). Thurston showed that GV:FΩ(3,1)→R is surjective. The injectivity of this homomorphism is one of the central problems in this theory. I contributed to determine the natural class of foliations for which the Godbillon-Vey invariant is defined. This was a question posed by Hurder-Katok and Ghys, and I gave a characterization of the Godbillon-Vey class by the foliated cobordism group in this class of foliations Namely, for a codimension-1 foliation F on an oriented 3-dimensional manifold M, its Godbillon-Vey invariant is 0 if and only if (M, F) is cobordant to the limit of foliations which are null cobordant. Foliations are naturally a Γ_n-structure for the topological groupoid Γ_n consisting of germs of local diffeomorphisms of R^n and it is classified by the classifying space BΓ_n for Γ_n-structures. The space BΓ_n played an essential role in the theory of existence of foliations and classification of foliations. The cohomology classes of BΓ_n is called the characteristic classes for codimension-n foliations. The existence of the Godbillon-Vey class implies that BΓ_n^r for C^r class Γ_n-structures is not (2n+1)-connected. It is an important problem to study the connectivity of BΓ_n^r. Mather, Herman and Thurston showed that BΓ_n^r is (n+1)-connected if r≠n+1. I showed that for BΓ_n^r for the Γ_n^r -structures with trivial normal bundles, it is approximately (n+n/r)-connected. Moreover, when r=1 it is contractible. The difference of the regularity reflects on the dynamical property. On the torus T², Denjoy showed that there exist nonsigular flow of class C¹ without closed orbits nor dense orbits, but there exist no such flows of class C². The contractibility of BΓ_n1 was shown by using this dynamics. It implies that any tangent plane field on a manifold is homotopic to the tangent bundle of a foliation of class C¹.   Another important problem was to find the relationship between the value of the Godbillon-Vey invariant and qualitative behavior of foliations. In the work in collaboration with Mizutani and Morita, we showed the Godbillon-Vey invariant vanishes for the foliations without holonomy or almost without holonomy. It was the first step leading to Duminy's theorem which says that if GV≠ 0 then there exists a leaf which attracts itself by its holonomy. I also looked at the rationality or irrationality of the Godbillon-Vey class of codimension-1 foliations of higher dimensional manifolds and I gave an interesting family of foliations which exhibits the rationality as well as examples of foliations which exhibit irrationality.   Interesting foliations appear in relation with dynamical systems. In the work in collaboration with Noda, we studied regular projectively Anosov flows and we classified regular projectively Anosov flows on T² bundles over the circle, Seifert fibered spaces over 2-dimensional hyperbolic orbifolds. Related with this work, I studied with Matsumoto on the uniqueness of the transverse intersections of two foliations in a 3-dimensional manifold. It is also important as a research on typical foliations.   I studied the perfectness of the identity component of the group of diffeomorphisms, and showed the perfectness for the groups of foliation preserving diffeomorphisms, contactmorphisms with low regularity, etc. A group is said to be perfect if its abelianization is trivial, that is, any element can be written as a product of commutators. It is called uniformly perfect if the number of commutators is bounded. Brago-Iwanov-Polterovich showed that the identity component of the group of diffeomorphisms of an n-dimensional sphere or a 3-dimensional manifold is uniformly perfect. I generalized their result and showed that the identity component of the group of diffeomorphisms of a compact odd dimensional manifold or a compact even dimensional manifold which has a handle-decomposition without handles of the middle index is uniformly perfect. In these cases, the bound of number of commutators is at most 4. I showed also the uniform perfectness of the group for a compact even dimensional manifold of dimension not equal to 2 or 4. The number of commutators in this case may have the information on manifolds. For a space such as spheres I showed that any (orientation preserving) homeomorphism can be written as one commutator.   In 1974, Herman showed that the identity component of the group real analytic diffeomorphisms of the n-dimensional torus is simple. I showed the perfectness of the identity component of the group real analytic diffeomorphisms for many real analytic manifolds with circle action. This is one of few results known for the group of real analytic diffeomorphisms. Selected publications The Godbillon-Vey classes of codimension one foliations which are almost without holonomy (with T. Mizutani and S. Morita), Annals of Mathematics 113 (1981), 515-527. On 2-cycles of B Diff (S¹) which are represented by foliated S¹-bundles over T², Annales de l'Institut Fourier, 31 (2) (1981), 1-59. On the homology of classifying spaces for foliated products, Advanced Studies in Pure Mathematics 5, Foliations, (1985), 7-120. On the foliated products of class C¹, Annals of Mathematics, 130, (1989), 227-271. CR-structures on Seifert manifolds (with Y. Kamishima), Inventiones mathematicae 104 (1991), 149-163. The Godbillon-Vey invariant and the foliated cobordism group, Proceedings of the Japan Academy, 68, Ser A (1992) 85-90. Area functionals and Godbillon-Vey cocycles, Annale de l'Institut Fourier 42 (1992) 421-447. Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle, J. Math. Soc. Japan. 47 (1995) 1-30. Small commutators in piecewise linear homeomorphisms of the real line, Topology 34 (1995) 815-857. Acyclicity of the groups of homeomorphisms of the Menger compact spaces (with V. Sergiescu) American Journal of Mathematics 118 (1996) 1299-1312. The Calabi invariant and the Euler class, Transactions Amer. Math. Soc. 352 (2000), no. 2, 515--524. Transverse intersections of foliations in three-manifolds, (with S. Matsumoto) Monographie de L'Enseignement Mathematique 38 (2001), 503-525. Regular projectively Anosov flows on the Seifert fibered 3-manifolds, J. Math. Soc. Japan. 56 (2004), 1233-1253. On the group of real analytic diffeomorphisms, Annales Scientifiques de l'Ecole Normale Superieure, 49, (2009) 601--651. On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Commentarii Mathematici Helvetici, 87, (2012) 141-185. Homeomorphism groups of commutator width one, Proceedings Amer. Math. Soc. 141, (2013) 1839-1847. Memberships, awards and activities Member of Mathematical Society of Japan, American Mathematical Society Awarded the Geometry Prize of the Mathematical Society of Japan in 1991 for the original research achievements on the foliations of class C¹. 2000-2004, 2009- Member of Board of Trustees of the Mathematical Society of Japan. 2009-2011 President of the Mathematical Society of Japan.