Hopf invariants and their application

1.我们在低维的co-H-空间上对Ganea猜想给出了肯定的回答，在证明中，我们发现了一个关于co-Hopf空间的基本定理，而在高维空间中发现了一个反例，利用上述基本定理说明了这个例子是一个co-H-空间.利用“高阶Hopf不变量”和“loop空间上的A_∞-结构”研究了Lusternik和Schnirelmann定义的数值不变量L-S范畴。结果，我们在L-S范畴上找到了一个反例，证明了Gamea的猜想是单连通闭流形.当空间X在“Meta稳定”范围内时，End（X）不是环而是“方”环.如果X是球映射的映射锥（在Meta稳定域），则完全用一个Hopf不变量确定End（End X）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).