Hopf spaces and higher homotopy

研究结果总结如下：1.给出了An-空间和An-空间之间的An-映射的另一种定义。为了给出另一种定义，我们使用了一个比Stasheff复形K_n具有更高双折射率的复形。我们使用的复形是点（1，2，.）的轨道的凸船体，n）在n维欧几里得空间中在n阶对称群的作用下。利用这种替代定义，可以给出An-空间上高阶同伦交换性的组合定义。利用Harper-Zabrodsky上同调运算的不稳定形式研究了模3有限Hopf空间的上同调.我们的结果是：设X是H_*（X ; Z/3）上的单连通模3有限Hopf空间，其结合Pontryagin积为H_*（X ; Z/3）.然后，上同调H*（X ; Z/3）作为代数同构于哈珀的Hopf空间的乘积空间、例外李群E_8s和奇球面的乘积空间.这项研究包括圣地亚哥加州大学的林教授。作为将上述结果推广到大于3的奇素数的第一步，我们得到了Steenrd代数在5挠自由模5有限圈空间的上同调上的作用.研究了对奇素数p具有平凡Steenrod作用的mod p有限Ap-space，得到如下结果：设X是单连通的mod p有限Ap-space.则X是ρ正则的当且仅当Steenrod代数在H*（#ZX ; Z/p）上的作用是平凡的。

The summary of research results is as follows.1. We gave an alternative definition of An-spaces and An-maps between An-spaces. To give the alternative definition we used a complex with higher symmetricity than the Stasheff's complex K_n. The complex we used is the convex hull of the orbit of the point (1,2 , ..., n) in the n dimensional Euclid space under the action of the n-th symmetric group. Using this alternative definition makes it possible to give a combinatorial definition of higher homotopy commutativity on An-spaces.2. We studied the cohomology of mod 3 finite Hopf spaces by using the unstable version of the Harper-Zabrodsky cohomology operation. Our result is the following : Let X be a simply connected mod 3 finite Hopf space with associative Pontryagin product on H_*(X ; Z/3). Then, the cohomology H*(X ; Z/3) is isomorphic as an algebra to the one of the product space of Harper's Hopf spaces, the exceptional Lie group E_8s and odd spheres. This research includes Professor Lin of University of California at San Diego.3. For the first step to extend the above result to odd primes greater than three, we got a result on the action of the Steenrd algebra on the cohomology of 5 torsion free mod 5 finite loop spaces.4. We studied mod p finite Ap-spaces with trivial Steenrod action for odd prime p. Our result is as follows : Let X be a simply connected mod p finite Ap-space. Then, X is rho-regular if and only if the action of the Steenrod algebra on H*(#ZX ; Z/p) is trivial.