Hopf invariants and their application

1. 我们在低维共h空间上得到了“Ganea猜想”的肯定答案，在证明中，我们找到了一个在高维共h空间上的一个基本定理，而在高维上找到了一个反例，用上述基本定理证明了这个例子是一个共h空间。研究了Lusternik和Schnirelmann定义的L-S范畴——一个数值不变量，即“高Hopf不变量”和“环空间上的A_∞-结构”。结果，我们找到了Gamea关于L-S范畴猜想的反例，即单连通闭流形。当空间X处于“元稳定”范围时，End （ΣX）不是一个环，而是一个“方形”环。如果X是一个球映射的映射锥（在元稳定范围内），我们完全用Hopf不变量来确定End （ΣX）的结构。

1. We obtained an affirmative answer to "the Ganea conjecture" on co-H-space of low dimensions, In the proof, we found an fundamental theorem on a co-Hopf space, However a counter-example was found in higher dimensions, The above fundamental theorem is used to show the example is a co-H-space.2. L-S category - a numrical invariant defined by Lusternik and Schnirelmann is studied in terms of "higher Hopf inuariuat" and "A_∞-structures on loop spaces". As a result, we found a counter exampled to Gamea's conjecture on L-S category as a simply connected closed manifold.3. when a space X is in "meta stable" range, End (ΣX) is not a ring but a "square" ring. We determine the structure of End (ΣX) completely in terms of a Hopf invariant, if X is a mapping cone of a sphere-mapping (in meta stable range).