Hopf spaces and higher homotopy

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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.