Research on higher dimensional polyhedra and operads

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The summary of reserch results is as follows.1. We study conditions for mod p finite H-spaces to be quasi p-regular, and we give a generalization to the theorem of Kumpel on p-regulraity. Our result includes results by Harper, McClearly and Wilkerson.2. We define unstable p-th order mod p cohomology operations for odd primes p. Then, by using the operations we show that there are no H-spaces with certain cohomology.3. Williams gave a definition of higher order homotopy commutativity of the multiplication on associative H-spaces. We generalize his definition to higher homotopy assocative H-spaces. Then, by using this definition we give the mod p torus theorem for H-spaces with finitely generated cohomology.4. We show that there is no even degree generators except for degree eight and twenty in the mod 3 cohomology of finite H-spaces. We also determine the cohomology strucre of them as algebras.5. We consider the inbedding of an An-space X to its loop space of the projective n-1-space ΩP_<n-1>X. Then we study relations between an A_<n-1>-cohomology class x∈H^*(X;F) and its lift x^^^∈H^*(ΩP_<n-1>X;F), where F is a field.6. It is well known that if a map f: X → Y between H-spaces is an H-map then the map induces a map between their projective planes. Moreover, it is also known that if Y is homotopy associative then the converse holds. We show that the assumption of homotopy associativity is necessary in showing the converse. We also generalize the result to the case of A_n-spaces.7. It is proved that the moduli space of a punctured Rieman sphere has cellular decomposition. We consider a decomposition of the associahedron by using a homotopy theoretical property, and then by using the decomposition we construct a handle decomposition of the moduli space which is a dual decomposition of the original decomposition.