Application of A_∞-methods on topological invariants

点击翻译按钮获取中文摘要

L-S category is defined by Lusternik and Schnirelmann to give a homotopy invariant of a topological space, which gives, for a manifold M, a lower bound of the number of the critical points of a C^∞-function on M. After the investigation of Ganea and many others, the following two problems were realised as open problems and listed in the book "Open Problems in Topology" by van Mill and M. Reed.[Problem 642] (Ganea conjecture) Is the L-S category of a space X × S^n equal to the L-S category of X plus 1?[Problem 643] Is the L-S category of a closed manifold greater than that of its once punctured submanifold?The higher Hopf invariant redefined on projective spaces of the loop space of a space, gave a criterion to determine L-S category, which implies negative answers to Problem 642 and Problem 643.In this research, extensively using the above idea, the present investigator completely determined the L-S category of a manifold which is the total space of a sphere-bundle over a sphere. As a result, we can construct manifolds as total spaces of S^2-bundles, which give counter examples to the Ganea conjecture. With Mamoru Mimura at Okayama University and Tetsu Nishimoto at Kinki Welfare University, he continued to investigate on the total space of a fibre bundle over a simply-connected suspension space or a non-simply-connected space.Combining this idea with the higher Hopf invariants, he together with Akira Kono at Kyoto University obtained a new upper bound for L-S category. They also defined a new lower bound for L-S category using the A_∞-method. These results yield the determination of L-S category of all compact simple Lie groups up to rank 4,except for Sp(4),F_4,PSp(3) and PSp(4). Using the A_∞-point of view, Kamata, Saeki, Sumi, Oda and Nishimoto have obtained various results.