Foliation Theory

本研究的目的是从广义上研究叶理的结构和性质，首席研究员T.Nishimori一直在研究相似伪群的定性理论，以便将余维1叶理的定性理论推广到更高余维的叶理。为了给试验打下坚实的基础，他在C^<1+Lipschitz>类范畴的推广形式中详细证明了Hector的一致收敛定理。研究者T.Suwa将Baum和Bott的一个定理推广到开流形，将奇异全纯叶理的剩余与它们的拓扑不变量联系起来，并将他自己的一个定理推广到高维奇异簇。此外，他还通过引入Nash剩余部分地回答了Baum和Bott的合理性猜想。Investigator I.Nakai用假台球的形式给出了曲率形式的几何解释，并证明了弱结合n-网是结合的，如果其成员的三元组的Chern联络不是平坦的，然后由1-形式的束（dim 1的投影线性族）的成员定义叶理。这一结果完成了Poincare，Mayrhofer和Reidemeister对平面弱结合4-网的分类。研究者H.Minakawa构造了可定向闭曲面的基本群到圆的同态群的例外同态，该圆的Euler类满足等式边界Ghys不等式（源于Milnor-Wood不等式）。此外，他分类奇异圈分段线性同胚群的一个圈。

The purpose of this research was to investigate the structures and properties of foliations in a broad sense.Head Investigator T.Nishimori has been studying the qualitative theory of similarity pseudogroups in order to extend the qualitative theory of codimension one foliations for foliations of higher codimension. To prepare a firm base for the trial, he gave a detailed proof for Hector's Uniform Convergence Theorem in an extended form of class C^<1+Lipschitz> category.Investigator T.Suwa extended a theorem of Baum and Bott, connecting the residue of singular holomorphic foliations and their topological invariants, to open manifolds, and extended one of his own theorems to singular varieties of higher dimension. Furthermore, by introducing Nash residue, he gave a partial answer to the rationality Conjecture of Baum and Bott.Invesigator I.Nakai gave a geometric interpretation of the curvature form in terms of fake billiard and proved that a weakly associative n-web is associative if Chern connections of triples of the members are not flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This results completed the classification of weakly associative 4-webs initiated by Poincare, Mayrhofer and Reidemeister for the flat case.Investigator H.Minakawa constructed exceptional homomorphisms of the fundamental group of a closed orientable surface to the diffeomorphism group of a circle whose Euler class satisfies the equality bounding Ghys Inequality (arising from Milnor-Wood Inequality). Furthermore he classified exotic circles in the piecewise liniear homeomorphism group of a circle.