Application of A_∞-methods on topological invariants

Lusternik和Schnirelmann定义了L-S范畴，给出了拓扑空间的同伦不变量，对于流形M，它给出了M上C^∞-函数临界点个数的下界。在Ganea和许多其他人的研究之后，以下两个问题被认为是开放问题，并被列在货车Mill和M.里德[问题642]（Ganea猜想）空间X × S^n的L-S范畴等于X的L-S范畴加1吗？[问题643]闭流形的L-S范畴是否大于它的一次穿孔子流形的L-S范畴？在空间的环空间的射影空间上重新定义了高阶Hopf不变量，给出了判定L-S范畴的一个准则，这意味着问题642和问题643的否定答案。在本研究中，作者广泛应用上述思想，完全确定了球面上球丛的全空间流形的L-S范畴。因此，我们可以将流形构造为S^2-丛的全空间，这是Ganea猜想的反例。他与冈山大学的Mamoru Mimura和近畿福祉大学的Tetsu Nishimoto继续研究单连通悬挂空间或非单连通空间上的纤维丛的全空间，并将这一思想与高阶Hopf不变量相结合，与京都大学的Akira Kono一起得到了L-S范畴的一个新的上界。利用A_∞-方法定义了L-S范畴的一个新的下界。这些结果给出了除Sp（4），F_4，PSp（3）和PSp（4）以外的所有秩为4的紧致单李群的L-S范畴的确定. Kamata，Saeki，Sumi，Oda和Nishimoto等人利用A_∞观点得到了不同的结果。

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.