Geometry of Groups and Moduli Spaces

点击翻译按钮获取中文摘要

We investigated mapping class group of surfaces and moduli space of Riemann surfaces as well as related groups and moduli spaces mainly from the viewpoint of topology. In a joint work with Penner, the representative constructed a canonical 1-cocycle on the Teichmuller space. Kawazumi, together with Penner and Bene, realized higher Johnson homomorphisms combinatorially on the Teichmuller space. Furuta developed theory of Floer homotopy type associated to the Seiberg-Witten theory and Tsuboi obtained a remarkable result concerning the structure of real analytic diffeomorphism group of manifolds. Fujiwara, Kohno, and Matsumoto obtained deep results in combinatorial structure of hyperbolic as well as the mapping class group, study of configuration spaces and arithmetic mapping class group, respectively. Mitsumatsu, Kitano, Akita with-Kawazumi, Hirose, and Murakami obtained interesting results, respectively, in the studies of 3-dimensional contact geometry, twisted Alexander polynomials, integral Morita-Mumford classes, mapping class group of surfaces in 4-manifolds, and volumes in hyperbolic geometry. Also the representative studied the structure of the Lie algebra consisting of symplectic derivations of the tensor algebra, without unit, generated by the homology of surfaces and, as an application, constructed unstable homology classes of genus 1 moduli spaces. Based on these results, further study of our theme has come into view.